Whittaker History Aether Vol2

A History of the Theories of

AETHER'and

ELECTRICITY

SIR EDMUND WHITTAKER

VOLUME TWO

The Modern Theories

1900-1926iy

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Contents

I THE AGE OF RUTHERFORD

H. Becquerel discovers a New Property of Uranium ,Rutherford identifies Alpha and Beta Rays . 2

The Curies discover Polonium and Radium . 2

Villard observes Gamma Rays 3Rutherford and Soddy's Theory of Radio-activity 5Rutherford proves that Alpha Particles are Helium Atoms 7The Discovery of Isotopes. "Lead as the Final Product of Radio-activity. '3Barkla detects the Polarisation of X-rays '4He discovers the K- and L-Series . 16Bragg shows that the Velocity of the Electrons produced by X-rays reaches

the Velocity of the Primary Electrons of the X-ray Tube '7Laue suggests that Crystals might diffract X-rays ,8Rutherford discovers the Structure of the Atom 22

Moseley proves that the Number of Electrons which circulate round theNucleus of an Atom is equal to the Ordinal Number of the Element inthe Periodic Table 24

Rutherford disintegrates Atoms by bombarding them. 26

II THE RELATIVITY THEORY OF POINCARE AND LORENTZRelative Motion of Earth and Aether 27The Body Alpha 28Trouton's Experiment 29The Search for Double Refraction due to the Earth's Motion 29Poincare enunciates the Principle of Relativity 30The Lorentz Transformation 3'The Electric and Magnetic Forces constitute a Six-vector 35Paradoxical Consequences of the New Theory 36The Relativist View of the FitzGerald Contraction 37No Velocity can exceed the Velocity of Light 38Poincare discovers the Behaviour of Electric Charge and Current under

the Lorentz Transformation 39Einstein gives the Relativist Formula: for Aberration and the Doppler

E~ct ~The Transverse Doppler Effect 42The Vibrations of a Rotating Rod . 43The Rate of Disintegration of Mesons 44Planck discovers Relativist Dynamics 44The Relativist Kinetic Energy and Momentum of a Particle 46Lewis and Tolman's Proof 48The Connection of Mass with Energy 51Planck's New Definition of Momentum 54The Bent Lever of Lewis and Tolman 55Felix Klein's Definition of a Vector 56The Tensor-calculus of Ricci 58

CONTENTS

Metric of a SpaceMinkowski and Hargreaves introduce Space-timeMinkowski discovers the Energy-tensorHis treatment of Relativist DynamicsRelativist force on a Charged ParticleConnection of the Energy-tensor and the Energy-momentum VectorThe Electromagnetic Potential-vectorThe Energy-tensor in terms of the Lagrangean Function

III THE BEGINNINGS OF QUANTUM THEORY

Planck's discovery of the Formula for Pure-temperature RadiationHis Theoretical Justification for it .He determines various Fundamental ConstantsHis More Accurate Formulation of the Stefan-Boltzmann Law.A Later Proof of Planck's LawEinstein introduces PhotonsHis Theory of Photo-electricityPhoto-chemical DecompositionMomentum of a Photon .Corpuscular Derivation of the Doppler EffectThe Wave and Particle Theories of LightThomson's Concept of Needle RadiationEinstein's Application of Quantum Theory to Specific Heats of SolidsMolecular Heat of Gases .Einstein's Formula for Fluctuations in the Energy of RadiationPlanck's Second Theory of RadiationHis Third TheorySommerfeld connects Quantum Theory with Hamilton's Theory

Action

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788,84-858687899'9'929394-96

100101J03104-

of104-

IV SPECTROSCOPY IN THE OLDER QUANTUM THEORYConway discovers the correct Principle for the Interpretation of Spectra 106

Nicholson makes use of the Rutherford Atom and Quantum Rules J07Bjerrum's work on Molecular Spectra 108

Ehrenfest states the Law of Quantification of Angular Momentum 109

Bohr's Theory of the Balmer Series of Hydrogen 110Pickering's ~ Puppis Series J 13A. Fowler's work on the Spectra of Ionised Atoms 115The Stark Effect "7The Wilson-Sommerfeld Conditions for Quantification 118Sommerfeld's Elliptic Orbits 119Schwarzschild and Epstein's Theory of the Stark Effect 121Adiabatic Invariants J22Systems in which the Variables can be Separated 126Sommerfeld and Debye's Quantum Theory of the Zeeman Effect 128Space Quantification: the Stern-Gerlach Effect '30Selection-principles '31The Inner Quantum Number 133The Spin of the Electron. 134-The Paschen-Back Effect and the Anomalous Zeeman Effect 135The Structure of the Atom 137

CONTENTS

VII MAGNETISM AND ELECTROMAGNETISM, 19oo-I926Langevin's Application of Electron-theory to Magnetism 239The Bohr Magneton 242Tolman and Stewart's Experiment. 243Barnett's Experiment 243The Richardson Effectsin Magnetisation 244Problems related to Unipolar Induction 245Field due to a Moving Electron 246Leigh Page's Deduction of the Maxwellian Theory 247Cunningham's Theory of Electromagnetic Actions 248The EmissionTheory of Electromagnetism . 249Leathem's Phenomenon . 249Hargreaves' Integral-form of the Maxwell-LorentzTheory 250His Values for the Potentials of a Moving Electron 252

VIII THE DISCOVERY OF MATRIX-MECHANICSHeisenberg'sFoundation of Quantum-mechanics on the Virtual Orchestra 253His Introduction of Non-commutative Multiplication. 255Matrix-theory 256The Energy Matrix is Diagonal 259The Commutation Rule . 260

Born and Jordan's Solution for the Harmonic Oscillator 261The Paper of Born, Heisenberg and Jordan . 263Dirac's Approach to the Problem. . 265To Every Physical Quantity there correspondsan Operator 267

IX THE DISCOVERY OF WAVE-MECHANICSDe Broglie'sPrinciple embodied in Partial Differential Equations 268SchrOdinger'sEquation: Connection with Hamilton's Principal Function 270Quantification as a Problem of Proper Values 272The Hydrogen Atom treated by Wave-mechanics 273PhysicalSignificanceof the Wave-function . 275Normalisation and Orthogonality of Wave-functions 276The General Wave-equation 278Rigorous Equivalence of Schrodinger's and Hamilton's Equations 279The Wentzel-Kramers-BrillouinApproximation 282Superpositionof States 283Connection of Wave-mechanicswith Matrix-mechanics 285The Harmonic Oscillator treated by Wave-mechanics. 288Wave-packets 290

The Quanb.un Theory of Collisions 291The Treatment of Perturbations 296The Stark Effect treated by Wave-mechanics 297The Wave-mechanicalTheory of the Zeeman Effect 300

Schrodinger's Derivation of the Kramers-Heisenberg Formula forScattering 302

Wave-explanationof the Compton-effect 305Systemscontaining Several Identical Particles 306

NAME INDEX 309

SUBJECT INDEX 315

Memorandum on Notation

VECfORS are denoted by letters in black type, as E.

The three components of a vector E are denoted by Ez, Ey, E1 :

and the magnitude of the vector is denoted by E, so that

E' -E .•.• +E .•• +E.".

The vector prodUJ:t of two vectors E and H, which is denotedby [E • 11], is the vector whose component, are

(EyH.•.-E;H... E.H.-EzH., E.•.H.•-E .•H.•.).

Its direction is at right angles to the direction of E and H, andits magnitude is represented by twice the area of the triangle formedby them.

The scalar prodUJ:t of E and H is ExHz+EyHy+ E.H.. I t isdenoted by (E. H).

Th . oE... oEy oE.. d d b d' Ee quantIty ox +~ +8; 18 enote y IV •

(OE, _~.!

C!Y oz'is denoted by curl E.

The vector whose components areoE. oE.oz -ax' oE.,_oE.)

ox C!Y

If V denote a scalar quantity, the vector whose components are

( OV oV oV).- ox' - C!Y' - oz 18 denoted by grad V.

The symbol d is used to denote the vector operator whosea a a

components are a:? ~' 81:..

Chapter ITHE AGE OF RUTHERFORD

WHEN Rontgen announced his discovery of the X-rays 1 it wasnatural to suspect some connection between these rays and thefluorescence (or, as it was generally called at that time, phosphor-escence) of the part of the vacuum tube from which they wereemitted. Accordingly, a number of workers tried to find whetherphosphorescent bodies in general emitted radiations which couldpass through opaque bodies and then either affect photographicplates or excite phosphorescence in other bodies.

In particular, Henri Becquerel of Paris (1852-1908) resolved toexamine the radiations which are emitted, after exposure to thesun, by the double sulphate of uranium and potassium, a substancewhich had been shown by his father, Edmond Becquerel (1820-91),to have the property of phosphorescence. The result was com-municated to the French Academy on 24 February 1896.2 'Leta photographic plate,' he said, 'be wrapped in two sheets of verythick black paper, such that the plate is not affected by exposureto the sun for a day. Outside the paper place a quantity of thephosphorescent substance, and expose the whole to the sun forseveral hours. When the plate is developed, it displays a silhouetteof the phosphorescent substance. So the latter must emit radiationswhich are capable of passing through paper opaque to ordinarylight, and of affecting salts of silver.'

At this time Becquerel supposed the radiation to have beenexcited by the exposure of the phosphorescent substance to the sun ;but a week later he announced 3 that in one experiment the sunhad become obscured almost as soon as the exposure was begun,and yet that when the photographic plate was developed, the intensityof the silhouette was as strong as in the other cases: and moreover,he had found that the radiation persisted for an indefinite time afterthe substance had been removed from the sunlight, and after theluminosity which properly cowtitutes phosphorescence had diedaway; and he was thus led to conclude that the activity was spon-taneous and permanent. It was soon found that those salts ofuranium which do not :phosphoresce-that is, the uranous seriesof salts-and the metal Itself, all emit the rays; and it becameevident that what Becquerel had discovered was a radically newproperty, possessed by the element uranium in all its chemicalcompounds.

1 cf. Vol. I, p. 357 I Comptes Rendus, cxxii (1896), p. 420I CAmptes Rendus, cxxii (2 March 1896), p. 501

I

AETHER AND ELECTRICITY

Very soon he found 1 that the new rays, like the Rontgen andcathode rays, impart conductivity to gases. The conductivity dueto X-rays was at that time being investigated at the CavendishLaboratory, Cambridge, by J. J. Thomson, who had been joined inthe summer of 1895 by a young research student from New Zealandnamed Ernest Rutherford (1871-1937). They found that theconductivity is due to ions, or particles carrying electric charges,which are produced in the gas by the radiation, and which are setin motion when an electric field is applied. Rutherford went on toexamine the conductivity produced by the rays from uranium (which,as he showed, is likewise due to ionisation), and the absorption ofthese rays by matter: he found 2 that the rays are not all of thesame kind, but that at least two distinct types are present: one ofthese, to which he gave the name a-rays, is readily absorbed; while-another, which he named ,B-radiation, has a penetrating power ahundred times as great as the a-rays.

Early in 1898 two new workers entered the field. MaryaSklodowska, born in Warsaw in 1867 (d. 1934), had studied physicsin Paris, and in 1895 had married a young French physicist, PierreCurie (1859-1906). She now resolved to search for other substanceshaving the properties that Becquerel had found in uranium, andshowed in April 1898 that these properties were possessed by com-pounds of thorium,3 the element whIch, of the elements known atthat time, stood next to uranium in the order of atomic weights ;the same discovery was made simultaneously by G. C. Schmidt 4

in Germany. Madame Curie went on to show that, since theemission of rays by uranium and thorium is unaffected by chemicalchanges, it must be essentially an atomic property.5 Now the mineralpitchblende, from which the uranium was derived, was found tohave an activity much greater than could be accounted for by theuranium contained in it: and from this fact she inferred that thepitchblende must contain yet another 'radio-active' element.Making a systematic chemical analysis, she and her husband inJuly 1898 discovered a new element which, in honour of her nativecountry, she named polonium, 6 and then in December another,having an activity many million times as great as uranium: tothis the name radium was given.7 Its spectrum was examined byF. A. Demarc;:ay,8and a spectral line was found which was not other-wise identifiable. The next three and a half years were spent chieflyin determining its atomic weight, by a laborious series of successive

1 Comptes Rendus, cxxii (1896), p. 559I This paper was publIshed in Phil. Mag.(5) xlvii (1899), p. 109, after Rutherford

had left Cambridge for a chair in McGill University.I Comptes Rendus, cxxvi (12 April 1898), p. 1101• Ann. d. Phys. !xv (19 April 1898), p. 141• Some years later, the Curies described the ideas that had inspired their researches

in Comptes Rendus cxxxiv (1902), p. 85.• Comptes Rendus, cxxvii (1898), p. 175 ' Ibid. cxxvii (1898), p. 1215• Comptes Rendus, cxxvii (1898), p. 1218

2

THE AGE OF RUTHERFOROfractionations: the value found was 225.1 Meanwhile anotherFrench physicist, Andre Debierne (b. 1874), discovered in theuranium residues yet a further radio-active element,2 to which hegave the name actinium.

Attention was now directed to the a- and f3-raysof Rutherford.A few months after their discovery it was shown by Giesel, Becquereland others, that part of the radiation (the f3-rays)was deflected bya magnetic field,3 while part (the a-rays) was not appreciablydeflected.' After this Monsieur and Madame Curie 6 found that thedeviable rays carry negative electric charges, and Becquerel 8

succeeded in deviating them by an electrostatic field. The deviableor f3-rays were thus clearly of the same nature as cathode rays;and when measurements of the electric and magnetic deviations gavefor the ratio mfe a value 7 of the order 10-7, the identity of thef3-particles with the cathode-ray corpuscles was fully established.They differ only in velocity, the f3-rays being very much theswifter.

The a-rays were at this time supposed to be not deviated by amagnetic field: the deviation is in fact small, even when the fieldis powerful: but in February 1903 Rutherford 8 announced thathe had succeeded in deviating them by both magnetic and electro-static fields. The deviation was in the opposite sense to that of thecathode rays, so the a-radiations must consist of positively chargedparticles projected with great velocity,9and the smallness of the devia-tion suggested that the expelled particles were massive compared tothe electron. A method of observing them was discovered in 1903 bySir W. Crookes 10 and independently by J. Elster and W. Geitel,llwho found that when a radio-active substance was brought neara screen of Sidot's hexagonal blende (zinc sulphide), bright scintilla-tions were observed, due to the cleavage of the blende under thebombardment. Rutherford suggested that this property might beused for counting the number of a-particles in the rays.

Meanwhile it had been discovered by P. Villard 12 that in additionto the alpha and beta rays, radium emits a third type of radiation,much more penetrating than either of them, in fact 160 times aspenetrating as the beta rays. The thickness of aluminium traversedbefore the intensity is reduced to one-half is approximately 0·0005 cm.for the a-rays, 0·05 cm. for the f3-raysand 8 cm. for the y-rays, as

1 Later raised to 226 • Comptes Rendus, cxxx (2 April 1900), p. 906• F. O. Giesel, Ann d. Phys. !xix (1899), p. 834 (working with polonium); Becquerel,

Compus Rendus, cxxix (1899), p. 996 (working with radium); S. Meyer andE. v. Scbweidler,Phys. ZS. (1899), p. 113 (working with polonium and radium)

• Becquerel, Comptes Rendus, cxxix (1899), p. 1205; cxxx (1900), pp. 206, 372;Curie, ibid., cxxx (1900), p. 73

• Compus Rendus, cxxx (1900), p. 647 • CompUs Rendus, cx:xx(1900), p. 809• cr. W. Kaufmann, Verh. Deutsch Phys. Ges. ix (1907), p. 667• Phil. Mag.(6) v (Feb. 1903), p. 177• This had been conjectured by R. J. Strutt in Phil. Trans. cxcvi (1901), p. 507.

,. Proc. R.S. !xxi (30 April 1903), p. 405 11 Phys. ZS. iv (1 May 1903), p. 43911 CompUs Rendus, cxxx (30 April 1900), p. 1178

3

4


Villard's radiation was called. Villard found that the y-radiation is,like the X-rays, not deviable by magnetic forces.

In 1898 Rutherford was appointed to a chair in McGill Uni-versity, Montreal, and there with R. B. Owens, the professor ofelectrical engineering, began an investigation into the radio-activityof the thorium compounds. The conductivity produced by theoxide thoria in the air was found 1 to vary in an unexpected andperplexing manner: it could be altered considerably by slight draughtscaused by opening or shutting a door. Eventually Rutherford con-cluded that thoria eInitted 2 very small amounts of some materialsubstance which was itself radio-active, and which could be carriedaway in an air current: this, to which he gave the name thoriumemanation, was shown to be a gas belonging to the same cheInicalfamily as helium and argon, but of high molecular weight.3

Meanwhile in Cambridge C. T. R. Wilson had been developing 4

his cloud-chamber, which was to provide the most powerful of allmethods of investigation in atomic physics. In moist air, if a certaindegree of supersaturation is exceeded (this can be secured by a suddenexpansion of the air) condensation takes place on dust-nuclei, whenany are present: if by preliIninary operations condensation is madeto take place on the dust-nuclei, and the resulting droplets areallowed to settle, the air in the chamber is thereby freed from dust.If now X-rays or radiations from a radio-active substance are passedinto the chamber, and if the degree of supersaturation is sufficient,condensation again takes place: this is due to the production ofions by the radiation. Thus the tracks of ionising radIations can bemade visible by the sudden expansion of a moist gas, each ionbecoIning the centre of a visible globule of water. Wilson showedthat the ions produced by uranium radiation were identical withthose produced by X-rays. J. J. Thomson in July 1899 wrotepointing out the advantages of the Wilson chamber to Rutherford,who henceforth profited immensely by its use. In this way thetrack of a single atoInic projectile or electron could be renderedvisible.

An important property, discovered for the first time in connectionwith thorium emanation, was that the radio-activity connected withit rapidly decreased. This behaviour was found later to be character-istic of all radio-active substances: but in the earliest known cases,uranium and thorium, the half-period (i.e. the time required for theactivity to be reduced by one-half) is of the order of millions of years,so the property had not hitherto been noticed. Rutherford found 5

that the intensity of the 'induced radiation' of thorium falls off

1 Owens, Phil. Mag.(5) xlviii (Oct. 1899),p. 360I Phil. Mag.(5) xlix (Jan. 1900),p. 1I Soon after this, Friedrich Ernst Dam of Halle found that radium, like thorium,

produced an emanation: Halle Nat. Ges. Abh. xxiii (1900).• Phil. Trans. ClxxxiX(A)(1897), p. 265; Proc. Comb. P.S. ix (1898), p. 333• Phil. Mag. xlix (Feb. 1900),p. 161

(996)

THE AGE OP RUTHERFORD

exponentially with the time: so that if 11 is the intensity at any timeand 12 the intensity after the lapse of a time t then

Iz=11e-Mwhen ,\ is a constant.

In May 1900 Sir W. Crookes 1 showed that it was possible bychemical means to separate from uranium a small fraction, whichhe called uranium X, which possessed the whole of the photographicactivity of the original substance. He found, moreover, that theactivity of the uranium X gradually decayed, while the full activityof the residual uranium was gradually renewed, so that after asufficient lapse of time it was possible to separate from it a freshsupply of uranium X. These facts had an important share in theformation of the theory.

It was at first supposed that the pure uranium, immediately afterthe separation, is not radio-active: but F. Soddy (b. 1877) observedthat though photographically inactive, it is active when tested bythe electrical method. Now the a-rays are active electrically butnot photographically, whereas the {J-rays are active photographi-cally: and the conclusion was drawn 2 that pure uranium emitsonly a-rays and uranium X only {J-rays. Soddy had joined thestaff of McGill University in 1900 as Demonstrator in Chemistry,and at once began to assistRutherford in his work on radio-activity.Further experiments on the thorium emanation involved condensingit by extreme cold, and it was discovered 3 that the emanation wasproduced not directly by the thorium but by an intermediate sub-stance which, as it had many of the characters of Crookes' uraniumX, was named thorium X. This was the first indication that radio-activity involves a chain of transformations of chemical elements.

The work of Rutherford and Soddy on thorium and its radio-active derivatives led them to a general theory of radio-activity,which was published in September 1902-May 1903.' The greatestobstacle to a clear understanding of the subject had been, curiouslyenough, the intense belief of everybody in the principle of conserva-tion of energy: here was an enormous amount of energy beingoutpoured, and no-one could see where it came from. So long asit was attributed to the absorption of some unknown kind of externalradiation, the essenceof the matter could not be discovered. Ruther-ford and Soddy now swept this notion away, and asserted that:

(i) In the radio-active elements radium, thorium and uranium,there is a continuous production of new kinds of matter, which arethemselves radio-active.

(ii) When several changes occur together these are not simul-! Proc. R.S.(A), !xvi (1900), p. 4Q9• Soddy, ]oum. Chern. Soc. !xxxi and !xxxii (July 1902), p. 860; Rutherford and

A. G. Grier, Phil. Mag'(6) iv (Sept. 1902), p. 315• Phil. Mag.(6) iv (Sept. 1902), p. 370• Phil. Mag.(6) iv (Sc;:pt.1902), p. 370; ibid. (Nov. 1902), p. 569; ibid.(6) v (April

1903), pp. 441,445; IbId. (May 1903), pp. 561, 576(996) 5

AETHER AND ELEarRICITY

taneous, but successive; thus thorium produces thorium X, thethorium X produces the thorium emanation and the latter producesan excited activity.

(iii) The phenomenon of radio-activity consists in this, that acertain proportion of the atoms undergo spontaneous transformationinto atoms of a different nature: these changes are different incharacter from any changes that have been dealt with before inchemistry, for the energy comes from intra-atomic sources whichare not concerned in chemical reactions.

(iv) The number of atoms that disintegrate in unit time is adefinite proportion of the atoms that are present and have not yetdisintegrated. The proportion is characteristic of the radio-activebody, and is constant for that body. This leads at once to an expon-ential law of decay with the time: thus if no is the initial numberof atoms, and n is the number at time t afterwards, then

n = noe-)J

where A is the fraction of the total number which disintegrates inunit time, so the average life of an atom is l/A.

(v) The a-rays consist of positively charged particles, whose ratioof mass to charge is over 1,000 times as great as for the electrons incathode rays. If it is assumed that the value of the charge is thesame as for the electron, then the a-ray particles must have a massof the same order as that of the hydrogen atom.

(vi) The rays emitted are an accompaniment of the change ofthe atom into the one next produced, and there is every reason tosuppose, not merely that the expulsion of a charged particle accom-panies the change, but that this expulsion actually is the change.

The authors remarked (in the paper of November 1902) that innaturally occurring minerals containing radio-elements, the radio-active changes must have been taking place over a very long period,and it was therefore possible that the ultimate products might haveaccumulated in sufficient quantity to be detected. As helium isusually found in such minerals, it was suggested that helium mightbe such a product. Several years passed before this was finallyestablished, but its probability was continually increasing. Soddyleft Montreal in 1903 to work with Sir William Ramsay at UniversityCollege, London, and Rutherford, who was in England in thesummer of that year, called on Ramsay and Soddy, and with themdetected (by its spectrum) the presence of helium in the emanationof radium. It seemed certain, therefore, that helium occupied someplace in the sequence of linear descent which begins with radium,and at first the general expectation was that it would prove to bean end-product. Rutherford, however, entertained the idea thatit might be formed from the a-particles, 1 which, as we have seen,were known to be of the same order of mass as hydrogen or heliumatoms; that the a-p2rticles, in fact, might be positively charged

1 Nature, !xviii (20 Aug. 1903), p. 3666

THE AGE OF RUTHERFORDatoms of helium: and for some years this supposition was debatedwithout a definite conclusion being reached. In 1906 Rutherforddetermined 1 with greater accuracy the ratio elm of the a-particlesfrom radium C, and found it to be between 5·0 X WI and 5·2 X 101,which is only half the value of elm for the hydrogen atom: this,however, left it undecided whether the a-particle is a hydrogenmolecule (molecular weight 2) carrying the ionic charge, or a heliumatom (atomic weight 4), carrying twice the ionic charge.

In 1904 William Henry Bragg (1862-1942), at that time professorin the University of Adelaide, South Australia, showed 2 that thea-l?article, on account of its mass, has only a small probability ofbemg deviated when jassing through matter, and that in generalit continues in a fixe direction, gradually losing its energy, untilit comes to a stop: the distance traversed may be called the rangeof the a-l?article. He found in the case of radium definite rangesfor four kmds of a-particles, corresponding to emissions from radium,radium emanation, radium A and radium C: the a-particles fromany particular kind of atom are all shot out with the same velocity,but this velocity varies from one kind of atom to another, as mightbe expected from Rutherford's theory. ~-particles, on the otherhand, are easily deflected from their paths by collisions with gasmolecules, and their tracks in a Wilson cloud-chamber are zig-zag:they are scattered by passing through matter, so that a narrow pencilof ~-rays, after passing through a metal plate, emerges as an ill-defined beam.

In 1907 Rutherford was translated to the chair of physics in theUniversity of Manchester. Here he found a young graduate ofErlangen, Hans Geiger, with whom he devised 3 an electrical methodof counting the a-particles directly, the Geiger counter as it has sincebeen generally called. The a-rays were sent through a gas, exposedto an electric field so strong as to be near the breakdown value atwhich a discharge must pass. When a single a-particle passed andproduced a small ionisation, the ions were accelerated by the electricfield and the ionisation was magnified by collisions several thousandtimes. This made possible the passage of a momentary discharge,which could be registered. This counting of atoms one by one wasa great achievement: it was found that the number of a-particlesemitted by I gram of radium in one second is 3·4 X 1010: whenthis was combined with the value of the total charge (found in thesecond paper), it became clear that an a-particle carries double theelectron char~e, reversed in sign.

The questIon as to the possible connection of a-rays with heliumwas finally settled later in the same year. Rutherford placed a

1 Phys. Rev. xxii (Feb. 1906),p. 122; Phil. Mag. xii (Oct. 1906),p. 348• Phil. Mag. viii (Dec. 1904), p. 719; W. H. Bragg and R. Kleeman, Phil. Mag. x

(Sept. 1905), p. 318; Paper read before the Royal Society of South Australia, 6 June1904.

• Proc. R.S.(A), !xxxi (27 Aug. 1908),pp. 141, 1627


quantity of radium emanation in a glass tube, which was so thinthat the a-rays generated by the emanation would pass through itswalls: they were received on the walls of a surrounding glass tube,and, after diffusing out, were found to give the spectrum of helium.This proved definitely that the a-particles are helium atoms, carryingtwo unit positive charges, 1 a conclusion which he had also reached 2

a short time before by a different line of reasoning.An interesting corroboration of Rutherford's account of the

emission of a-rays was obtained somewhat later, when he andGeiger investigated the fluctuations 3 in the recorded numbers ofparticles emitted by a radio-active substance in successive equalmtervals of time. H. Bateman had shown that if the emission isa random one, then the probability that n particles will be observedin unit time is

xne-%n!

where x is the average number per unit time, and n is a whole number(n=O, 1, 2, ... (0). Rutherford and Geiger in 1910 verified thisformula experimentally. 4 Yet a further completion of the work ona-particles was a measurement, made with Boltwood,5 of the volumeof helium produced by a large quantity of radium. By combiningthe result now obtained with that of the counting experiment it waspossible to evaluate the number of molecules in a quantity of thesubstance whose weight in grams is equal to the molecular weightof the substance (the Avogadro number).

The determination of this constant had been the object of manyresearches in the years immediately preceding, beginning with anotable paper by Einstein.6 Albert Einstein was born at Dim inWtirttemberg on 14 March 1879. The circumstances of his father'sbusiness compelled the family to leave Germany; and after receivinga somewhat irregular education in Switzerland, he became anofficial in the Patent Office in Berne. It was in this situation thathe wrote, in six months, four papers, each of which attracted muchattention.7

The paper now to be considered was really a sequel to two earlierpapers 8 on the statistical-kinetic theory of heat, in which, however,Einstein had only obtained independently certain results which hadbeen published a year or two earlier by Willard Gibbs. He now

1 Rutherford and T. Royds, Mem. Manchester Lit. and Phil. Soc. liii (31 Dec. 1908),p. 1 ; Phil. Mag. xvii (Feb. 1909),p. 281

• Nature, !xxix (5 Nov. 1908),p. 12• That the emissionof a-particles is a random process, and so subject to the laws of

probability, seems to have been first clearly stated by E. von Schweidler, Premier CongolnUrnat. pour l'Etude de La Radiologic, Liege, 1905.

• Phil. Mag. xx (Oct. 1910),p. 6986 Rutherford and Boltwood, Phil. Mag. xxii (Oct. 1911),p. 586• Ann. d. Phys.(4) xvii (1905), p. 549; continued in Ann. d. Phys. xix (1906), p. 3717 Two of these will be referred to in Chapter II and one in Chapter II 1.• Ann. d. Phys. ix (1902), p. 417; xi (1903), p. 170

8

THE AGE OF RUTHERFORD

applied these results to the motion of very small particles suspendedin a liquid. The particles were supposed to be much larger thana molecule, but it was assumed that as a result of collisions with themolecules of the water, they require a random motion, like thatof the molecules of a gas. The average velocity of such a suspendedparticle, even in the case of particles large enough to be seen witha microscope, might be of observable magnitude: but the directionof its motion would change so rapidly, under the bombardment towhich it would be exposed, that it would not be directly measurable.However, as a statistical effect of these transient motions, there wouldbe a resultant motion which might be within the range of visibility.Einstein showed that in a finite interval of time t the mean squareof the displacement for a spherical particle of radius a is

RTt37TapN

where R is the gas-constant, T the temperature, N is Avogadro'snumber, and J1- is the coefficient of viscosity. Thus by this pheno-menon the thermal random motion, hitherto a matter of hypothesis,might actually be made a matter of visible demonstration.

The motion of small particles suspended in liquids had beenobserved as early as 1828 by Robert Brown 1 (1773-1858), a botanist,after whom it was called the Brownian motion. Einstein identified themotion studied by him with the Brownian motion, somewhat tenta-tively in his first paper, but without hesitation in the second.

The theory of the Brownian motion was investigated almost atthe same time by M. von Smoluchowski 2 (1872-1917), and it wasconfirmed experimentally by Th. Svedberg,3 M. Seddig," andP. Langevin.5 Particular mention might be made of the experi-mental studies made in 1908-9 by Jean-Baptiste Perrin (1870-1942)of Paris.6 These experiments yielded a value of the mean energyof a particle at a definite temperature, and thus enabled him todeduce the value of Avogadro's number, which is 6·06 x 1023.7 Thedirect confirmation of the kinetic theory provided by these researcheson the Brownian movement was the means of converting to it somenotable former opponents, such as Wilhelm Ostwald and Ernst Mach.

The statistical-kinetic theory of heat was confirmed experi-mentally in a different way in 1911 by L. Dunoyer,8 who obtaineda parallel beam of sodium molecules by allowing the vapour of

1 Phil. Mag. iv (1828), p. 161I Bull. Acad. Sci. Cracovie, vii (1906), p. 577; Ann. d. Phys. xxi (1906), p. 756I ZS. Elektrochern!. xii (1906), pp. 853, 909; ZS. phys. chern. lxv (1909), p. 624; !xvi

(1909), p. 752; lxVlI(1909)iY' 249; !xx (1910), p. 571• Phys. ZS. ix (1908), ? 4t>5 • Comptes Rendus, cx1vi(1908), p. 503• Comptes &ndus, cx1Vl(1908), p. 967; cx1vii (1908), pp. 415, 530; cx1ix (1909),

pp. 477, 549; Ann. Chim. Phys. xvii (1909), p. 5; cr. also R. Furth, Ann d. pAys. !iii(1917), p. 117

7 Another method of determining Avogadro's number is to study the diffusion ofions in a gas under the influence of an electric force.

I Comptes Rmdus, clii (1911), p. 592; Le Radium, viii (1911), p. 1429

AETHER ANn ELECTRICITY

heated sodium to pass through two diaphragms pierced with smallholes in an exhausted tube: the behaviour of the molecular beamswas entirely in agreement with the predictions of the kinetic theoryof gases.

In his statistical studies, Einstein also recognised that the thermalmotion of the carriers of electric charge in a conductor should giverise to random fluctuations of potential difference between the endsof the conductor. The effect was too small to be detected by themeans then available, but many years later, after the developmentof valve amplification, it was observed by J. B. Johnson,l and thetheory was studied by H. Nyquist.2 This phenomenon is one of thecauses of the disturbance that is called ' noise' in valve amplifiers.

We must now return to the consideration of the radio-activeelements themselves. In 1903-5 Rutherford 3 identified a numberof members of the radium sequence later than the emanation:radium A, Band C were known by the summer of 1903, and radiumD, E, F were discovered in the next two years. It was suspectedthat one of these later products was identical with the poloniumwhich had been the first new element found by the Curies, and infact polonium was shown to be radium F. One of the most remark-able discoveries was that of radium B: for at the time no radiationsof any kind could be found accompanying its translOrmation intoradium C, and there was therefore no direct evidence of its existence:the only reason for postulating it was, that to suppose an immediatederivation of radium C from radium A would have violated thelaws of radio-active change laid down in 1902-3; and it was there-fore necessary to assume the reality of an intermediate body.

As soon as the principle that radio-active elements are derivedfrom each other in series had been established in 1902-3, the suspicionwas formed that radium, which is found in nature in uranium ores,might be a descendant of uranium; this conjecture was supportedby the facts that uranium is one of the few elements having a higheratomic weight than radium, and that the proportion of radium inpitchblende corresponds roughly with the ratio of activity of radiumand uranium. Soddy 4 in 1904 described an experiment whichshowed that radium is not produced directly from uranium: if itis produced at all, it can only be by the agency of intermediatesubstances. Bertram B. Boltwood (1870-1927), of Yale University,5worked on this investigation for several years, and at last in 1907succeeded in showing that radium is the immediate descendant of

1 Nature, cxix (1927), p. 50; Phys. Rev. xxix (1927), p. 367; xxxii (1928), p. 97The possibility that under certain conditions the thermal motion of electrons in

conductors could create a measurable disturbance in amplifiers had been recognised ontheoretic grounds by W. Schottky, Ann. d. Phys. lvii (1918), p. 541.

• Phys. Rev. xxxii (1928), p. 110• Proc. R.S.(A), Ixxiii (22 June 1904), p. 493; Phil. Trans. cciv (Nov. 1904), p. 169

(Bakerian lecture); Phil. Mag. viii (Nov. 1904), p. 636; Nature, !xxi (Feb. 1905),p.341

• Nature, !xx (12 May 1904), p. 30 • Nature, !xx (26 May 1904), p. 8010


new radio-active element which he named ionium, and which istself descended from uranium.l

The great number of different radio-active atoms that had noween discovered raised questions concerning their atomic weights,

particularly with regard to their position in what was known asthe periodic table of the chemical elements. In a paper published in864, John A. R. Newlands 2 had pointed out that when the chemicallements are arranged according to the numerical values of their

fltomic weights, the eighth element starting from any given one is,in regard to its properties, closely akin to the first, , like the eighthnote of an octave in music.' This idea he developed in later papers,saIling the relationship the 'Law of Octaves.' 4 He read a paper

pn the subject before the Chemical Society on 1 March 1866; butt was rejected, on the ground that the Society had' made it a ruleot to publish papers of a purely theoretical nature, since it was

likely to lead to correspondence of a controversial character." 6

Newland's ideas were adopted and developed a few years latery Dmitri Ivanovich Mendeleev (1834-1907),6 who arranged thelements in a periodic table. From gaps in this he inferred the existence

and approximate atomic weights of three hitherto unknown elements,o which he gave the names eka-boron, eka-aluminium and eka-ilicon; when these were subsequently discovered (they are nownown as scandium, gallium and germanium), the importance ofe periodic table became universally recognised; and the inert

ases helium etc., when they were discovered still later, were foundo fit into it perfectly.

As new members of the radio-active sequences were discovered,it was found in some cases that two or more of the atoms in theserieshad exactly the same chemical properties, so that they belongedto the same place in Newland's and Mcndeleev's periodic table.For instance, in 1905, O. Hahn, working with Sir WIlliam Ramsayt University College, London, discovered 7 the parent of thorium X,

:which he called radio-thorium. This was found to be not separablechemically from thorium; and Boltwood found that his ioniumalso was not separable chemically from thorium. It was shownby A. S. Russell and R. Rossi,8working in Rutherford's laboratory,that the optical spectrum of ionium is indistinguishable from the

I Nature, !xxvi (26 Sept. 1907), p. 544; Amer. ]oum. Sci. xxiv (Oct. 1907), p. 370 ;xxv (May 1908), p. 365

• Chern. News, x (20 Aug. 1864), p. 94• Chern. News, xii (18 Aug. 1865), p. 83; xii (25 Aug. 1865), p. 94• The group of elements Helium, Neon, Argon, Krypton, Xenon and Niton was not

known at the time; when they are introduced into the table, it is the ninth elementstarting from any given one which is akin to the first. We leave aside the complicationsassociated with the rare earths, etc.

• J. A. R. Newlands, The Periodic Law; Londo!), E. and F. W. Spon, 1884, p. 23• ZS.j. Chern. v (1869), p.. 405; Deutsch. Chern. Gesell. Ber. iv (1871),p. 348; Ann.

d. Chern., Supplementband, VIii (1873), p. 133, Proc. R.S.(A), !xxvi (24 May 1905), p. 115; Chern. News, xcii (I Dec. 1905), p. 25:• Proc. R.S.(A), !xxxvii (Dec. 1912), p. 478

II


spectrum of thorium. The radio-active properties of the threesubstances are, however, totally different, since the half-value perio Iof thorium is of the order of 1010 years, that of ionium is of the ordenof 105 years, and that of radio-thorium is 1·9 years: and the atomi_weights are different: but chemically they are different forms 011the same element.

Curiously enough, the possibility of such a situation had beensuggested so far back as 1886 by Sir William Crookes.l 'I conceive,therefore,' he said, 'that when we say the atomic weight of, forinstance, calcium is 40, we really express the fact that, while themajority of calcium atoms have an actual atomic weight of 40,there are not a few which are represented by 39 or 41, a less numberby 38 or 42, and so on.'

As the investigation of the radio-active atoms progressed stillfurther, many other examples became known of atoms which areinseparable by chemical methods but have different radio-activeproperties and different atomic weights. Attention was drawn tothe matter in 1909 by the Swedish chemists D. Str6mholm andTh. Svedberg,2 and in 1910 by Soddy,Sand much experimental workrelating to it was done by Alexander Fleck.4

New light on the problem now came from an unexpected quarter.Sir Joseph Thomson (he had been knighted in 1908) took up workon the canal rays,5 or positive rays as he now called them, and deviseda method of 'positive-ray analysis' for findin~ the values of m/efor the positively charged particles which constltute the rays; themethod was to shoot the rays through a narrow tube, so as to obtaina small spot on a phosphorescent screen or a photographic plate,and to subject them between the tube and the screen to an electricfield and also a magnetic field, so as to deflect the beam of particles,the electrostatic deflection and the magnetic deflection being perpen-dicular to each other. He showed that all particles having the samevalue for mfe would be spread out by the two fields so as to strike thescreen in points lying on a parabola; thus, particles of different masswould give different parabolas. Parabolas were found correspondingto the atoms and molecules of various gases in the discharge-tube;and the atomic weights of the particles could be at once inferredfrom measures of the parabolas. On applying this method of positive-ray analysis to the gas neon, he found 6 in addition to a parabolabelonging to atomic weight 20, another corresponding to atomicweight 22. These proved to be, both of them, atoms of neon, butof different masses. Thomson had in fact discovered two ordinarynon-radio-active atoms having the same chemical behaviour butdifferent physical characteristics. This result, which was immediately

1 Brit. Ass. Rep., Birmingham, 1886, p. 569• ZS.f. Anorg. Chern. lxi (1909), p. 338 and !xiii (1909), p. 197• Chern. Soc. Ann. &p. (1910), p. 285, cf. Fleck, Brit. Ass. Rep., Birmingham, 1913, p. 447• cf. Vol. I, p. 363 • Proc. R.S.(A), !xxxix (I Aug. 1913), p. 1

12


confirmed by Francis William Aston 1 (1877-1945) showed that thephenomenon of a place in the Newlands-Mendeleev table beingoccupied by more than one element was not confined to the highestplaces in the table. Elements which are chemically inseparable,but have different atomic weights, were named by Soddy isotopes.

A striking example was furnished when the question as to theend-:product of radio-active changes was solved. Having becomeconVInced so early as 1905 that the end-product of the radiumseries was not helium, Rutherford sought for some other element tofill this position; and both he and Boltwood suggested that it mightbe lead,2 since lead appears :eersistently as a constituent of uranium-radium minerals: it was possIble indeed that lead might be radium G.This proved to be correct, and lead was found to be also the finalproduct of the thorium series. The atomic weights of these two kindsof lead are not, however, equal, that of radium lead being 206 andthat of thorium lead being 208. (The atomic weight of ordinarylead is 207'20).3

Atoms obtained by radio-active disintegrations occupy places inthe periodic table which are determined by what are called thedisplacement laws, first enunciated in 1913 by A. S. Russel},4K. Fajans 5

(another pupil of Rutherford's) and F. Soddy,8 which may be statedas follows; a disintegration with emission of an a-particle causesthe atom to descend two places in the Newlands-MendeIeev table(Le. the atomic weight is diminished) ; a disintegration with emissionof a ,B-particlecauses the atom to ascend one place in the table, butdoes not change the atomic weight.

In 1919-20 it was stated by F. W. Aston 7 that within the limitsof experimental accuracy the masses of all the isotopes examinedby him were expressed by whole numbers when oxygen was takenas 16: the only exception was hydrogen, whose mass was 1·008.

Possible methods for separating isotopes were indicated in 1919by F. A. Lindemann and F. W. Aston,S but for long no notablesuccess was attained in practice; in 1932-3, however, two isotopesof hydrogen were successfully separated by electrolytic methods.9The isotopes of neon have been separated by repeated diffusion byGustave Hertz.1o

1 Brit. Ass. Rep., Birmingham, 1913, p. 4032 Rutherford, RadiotU:tivity (second edn., May 1905), p. 484; Boltwood, Phil. Mag.(6)

ix (April 1905), p. 599• Soddy, Ann. Rep. Chern. Soc., 1913, p. 269; Chern. News, cvii (28 Feb. 1913), p. 97 ;

NaJure, xci (20 March 1913), p. 57; Nature, xcviii (15 Feb. 1917), p. 469• Chern. News, cvii (31 Jan. 1913), p. 49• Phys. ZS. xiv (15 Feb. 1913), pp. 131 and 136• Chern. News, cvii (28 Feb. 1913), p. 97• Nature, civ (18 Dec. 1919), p. 393; cv (4 March 1920), p. 8; Phil. Mag. xxxix

(April 1920), p. 449; ibid. (May 1920), p. 611 ; Nature, cv (1 July 1920), p. 547• Phil. Mag. xxxvii (May 1919), p. 523• E. W. Washburn and H. C. Vrey, Proc. Nat. Acad. Sci. xviii (July 1932), p. 496 ;

E. W. Washburn, E. R. Smith ~d M. Frandsen, Bureau Q/ Standards J..of Research, xi(Oct. 1933), p. 453; G. N. LeWlSand R. T. Macdonald, J. Chern. Phys. 1 (June 1933),p.341 •• ZS.f. Phys. !xxix (1932), p. 108

13

AETHER AND ELECTRICrrYA new phenomenon in radio-activity was described in 1908 b~

Rutherford,l and confirmed later by Fajans 2 and other workers,namely that in some cases (e.g. radium C, thorium C and actinium C)some of the atoms emitted an a-p<:.rticle,and in the next transforma-tion a ,$-particle, while the rest of the atoms reversed the order 0the transformations, emitting first a ,$-particle and afterwards ana-particle. This is known as a branching of the series. Rutherfordin his original paper expressed the belief that in this way uraniummight give rise to the actinium family as well as the radium family;a conjecture which was afterwards generally accepted as correct.

An account must now be given of some notable advances con-cerned with X-rays. Charles Glover Barkla (1877-1944), when aresearch student under J. J. Thomson at Cambridge, had becomeinterested in X-rays. In 1902 his work was transferred to LiverpoolUniversity, and there in 1904 he discovered that the rays may bepartly polarised.8 In the final dispostion 4 of his experiments, a massof carbon was subjected to a strong primary beam of X-rays, andso became a source of secondary radiatIOn. A beam of this secondaryradiation, propagated in a direction at right angles to that of theprimary, was studied. In this second beam was placed a second massof carbon, and the intensities of tertiary radiation proceeding indirections perpendicular to the direction of propagation of thesecondary beam were observed. The X-ray tube was turned roundthe axis of the secondary beam, while the rest of the apparatus wasfixed, and the intensities of the tertiary radiations were observed fordifferent positions of the tube. It was found that the intensity of thetertiary radiation was a maximum when the primary and tertiarybeams were parallel, and a minimum when they were at right anglestoeach other, which showed that the secondary radiation was polarised.This result told decidedly in favour of the hypothesis that X-rayswere transverse waves.

Continuing his work on X-rays, Barkla resolved to test a sugges-tion of J. J. Thomson's, that the number of electrons in an atommight be found by observing the amount of the scattering whenX-rays fall on the lighter chemical elements, and comparing itwith the scattering produced when they fall on a single electron.In 1903 Thomson had already given, in the first edition of hisConduction of Electricity through Gases, Ii a theoretical discussion, basedon classical electrodynamics, of the scatterin~ of a pulse of electro-magnetic force by an electron on which it 18 incident. He found

~ Nature, ~!i (5 March 1908), p.: 422Phys. ZS. Xll (1911), p. 369; Xlll (1912), p. 699

• Nature, !xix (17 March 1904), p. 463; Proc. R.S.{A),!xxiv (1905), p. 474; Phil.Trans.{A), cciv (1905), p.. 467

• Proc. R.S.{A), !xxvii (1906), p. 247• J. J. Thomson, CondUJ:tion of Electricity through Gases, 1st edn. (1903), p. 268;

2nd edn. (1906), p. 321; 3rd edn., Vol. II (1933), p. 256. cf. alsoJ. J. Thomson, Phil.Mag. xi (1906), p. 769, where he suggested three different methods of determining thenumber of electrons in an atom, based respectively on (1) the dispersion of light bygases, (2) the scattering of X-rays by gases, (3) the absorption of {J-rays.

14

THE AGE OF RUTHERFORDthat the energy radiated by the electron is 871ee13m' times theenergypassing through unit area of the wave-front of the primarybeam (when the charge e is measured in electromagnetic units).Thus it it is assumed that the electrons, in the chemical elementexposedto the X-rays, all scatter independently, the value of themass-scatteringcoefficient is

871 een3" m1p

wheren is the number of electrons per cms, and p is the density.Nowlet

N= number of molecules in one gram-moleculeZ = number of free electrons per atomA = atomic weight.

ThenNZ b fl' nA = num er 0 e ectrons In one gram = p

so the value of the mass-scattering coefficient is871 e'NZ3" mlA·

Barkla 1 found experimentally for the mass-scattering coefficient ofthe lighter elements (except for hydrogen) a value about 0·2, whichwould therefore give

3m'AZ = 4071eeN.

The values accepted at the time for the quantities on the right-handsideof this equation were inaccurate, and the result deduced, namelythat there were between 100 and 200 electrons per molecule of air,was replaced by Barkla in 1911 2 by a much better determinationbased on Bucherer's value for elm, Rutherford and Geiger's valuefor e, and Rutherford's value for N. This gave approximately

Z=iA,i.e. the number of scattering electrons per atom, for the lighterelements, is about half the atomic weight of the element, exceptin the case of hydrogen, for which Z =1. These results anticipatedlater discoveries in a remarkable way.

The secondary X-rays were destined to furnish other contribu-tions to atomic physics. In 1906 Barkla S found that in some casesthe secondary rays consisted mainly of a radiation which differedaltogether in 'hardness,' or penetrating power, from the primaryradiation, so that it could not be regarded as the result of' scattering.'

1 Phil. Mag. vii (May 1904), p. 543. cr. also J. A. Crowther, Phil. Mag. xiv (Nov.1907), p. 653

• Phil. Mag. xxi (May 1911), p. 648 • Phil. Mag. xi (June 1906), p. 81215


He pursued this matter further, with the help of one of hisstudents, C .A. Sadler, and in 1908 they 1 found that the secondaryX-rays emitted by a chemical element exposed to a primary beamof X-rays were of two distinct types :

(i) A scattered radiation, not of great amount, of the samequality as the primary beam.

(ii) A radiation characteristic of the exposed chemical element,and almost, if not quite, homogeneous, Le. all of the same degree ofhardness. It was, moreover, emitted uniformly in all directions,unlike the scattered radiation. This J;haracteristic radiation wasproduced only when the primary X-rays contained a constituentharder than the characteristic radiation that was to be excited. (Onthis account the characteristic X-rays were often spoken of at thetime as 'flourescent.') Barkla found also that the hardness of thecharacteristic radiation increased as the atomic weight of the emittingchemical element increased.

R. Whiddington I found that the primary rays from an X-raytube can excite the radiation characteristic of an element of atomicweight w only when the velocity of the parent cathode rays exceeds108 w cm/sec; when the velocity of the primary rays is less thanthis, only a truly 'scattered' radiation is emitted, resembling theprimary.

It was found 3 that the characteristic secondary radiations maybe divided into several groups, the radiation belonging to each groupbecomin~ more penetrating as the atomic weight of the radiatingelement mcreases; in other words, each chemical element emitsa line spectrum of X-rays, each line moving to the more penetratingend .of the spectrum as the atomic weight of the element increases.Two groups which were described in 1909 received the notation Kand L in 1911, and an M-group was found a little later.' The K-series, which is the most penetrating, was found together with theL-series for elements from zirconium (atomic wt. 90'6) to silver(atomic wt. 107·88). For elements heavier than silver, the K-serieswas difficult to excite, since very great velocities would be required inthe exciting cathode rays: and for elements lighter than zirconium,the L-series was difficult to observe because it was so easily absorbed.5

It was shown by G. W. C. Kaye 6 that the radiation characteristicof a chemical element can be excited not only by exposing it to a

1 Phil. Mag. xiv (Sept. 1907), p. 408; xvi (Oct. 1908), p. 550I Proc. R.S.(A), lxxxv (April 1911), p. 323I Proc. Camb. Phil. Soc. xv (1909), p. 257; Barkla and J. Nicol, Nature, !xxxiv (Aug.

1910), p. 139 • Barkla and V. Collier, Phil. Mag. xxiii (June 1912), p. 987: E. H. Kurth, Phys. Rev. xviii (1921), p. 461, found for the convergence wave-lengths

in Angstroms: K-series of carbon, 42,6, oxygen 23·8; L-series of carbon, 375, oxygen248, iron 16,3, copper 12·3: M-series of iron, 54'3, copper 41,6: N-series of iron 247,copper 116.

• Phil. Trans.<A), ccix (Nov. 1908), p. 123. Kaye found that the intensity of ~eneralX-radiation was nearly proportional to the atomic weight of the element forrrnng theanticathode; later, W. Duane and T. Schimizu, Phys. Rev. xiv (1919), p. 525, showedthat the intensity is proportional to the atomic nwnber.

16


beam of primary X-rays, but also by using it as the anticathode inan X-ray tube, so that it is bombarded by cathode rays. It wassuggested that this might be an indirect effect, produced by themediation of non-characteristic X-rays: but this suggestion wasdisproved by R. T. Beatty,! who proved beyond doubt that the char-acteristic X-rays are excited directly by the impact of cathode rays.In the following year Beatty 2 verified experimentally a result whichhad been reached theoretically by.T. J. Thomson 3 in 1907, namelythat the total intensity of general X-radiation is proportional to thefourth power of the velocity of the exciting electrons.

The question as to whether X-rays were corpuscles or waveswas still unsettled in 1910. In that year W. H. Bragg published apaper ( in which, interpreting his experiments by the light of thecorpuscular hypothesis, he arrived at conclusions which were in facttrue and of great significance. We have seen 5 that when X-raysare passed through a gas they render it a conductor of electricity,and that this property is due to the production of ions in the gas.Bragg now asserted that the X-rays do not ionise the gas directly;they act by ejecting, from a small proportion of the atoms of thegas, electrons (photo-electrons) of high speed, each of which actsas a ,8-particle and ionises the gas by detaching electrons in a succes-sion of collisions with molecules along its path. The speed of theejected electron depends only on the hardness or penetrating powerof the X-rays (which was later shown to be, in effect, their frequency),and not at all on their intensity, or on the nature of the atom fromwhich the electron is expelled. What Bragg emphasised as speciallyremarkable was that the energy of the electron was as great as thatof an electron in the beam of cathode rays by which the X-rays hadbeen excited originally: the X-ray pulse seemed to have the prop-erty of keeping its energy together In a small bundle, without anyof the spreading that might have been expected on the wave theory,and to be able to transfer the whole of this energy to a single electron.He enunciated 'the general :principle, that if one radiant entity(a-, ,8-, Y-, X- or cathode-ray) enters an atom, one and only oneentity emerges, carrying with it the energy of the entering entity.', One X-ray provides the energy for one ,8-ray, and similarly in theX-ray bulb, one ,8-ray excites one X-ray. No energy is lost in theinterchange of forms, ,8- to X-ray and back again; and the speedof the secondary ,8-ray is independent of the distance that the X-rayhas travelled: so the X-ray cannot diffuse its energy as it goes,that is to say, it is a corpuscle.' It was in fact now established thatthe X-ray behaves in some ways 6 as a wave and in other ways asa corpuscle.

1 Proc. R.S.(A), !xxxvii (Dec. 1912), p. 511• Proc. R.S.(A), !xxxix (1913), p. 314 • Phil. Mag. xiv (1907), p. 226• Phil. Mag. xx (Sept. 1910), p. 385; Brit. Ass. Rep. 1911, p. 340; W. H. Bragg and

H. L. Porter, Proc. R.S.(A), !xxxv (1911), p. 349 • cr. Vol. I, p. 359• cr. A. Joffe and N. Dobronrawov, ZS.f. P., xxxiv (1925), p. 889

17


Bragg's conclusions were fully confirmed in 1911-12 by C. T. R.Wilson,! using his method of cloud-chamber photographs. Thewhole of the region traversed by the primary X-ray beam was seento be filled with minute streaks and patches of cloud: examiningthe photographs more closely, the cloudlets were seen to be smallthread-like objects, consisting of droplets deposited on ions producedalong the paths of the ,B-particles,which were the actually effectiveionising agents.

In the early part of the twentieth century, many attempts weremade to test the hypothesis that X-rays are waves, by trying toobtain diffraction-effects with them. In 1899 and 1902 H. Hagaand C. H. Wind 2 of Groningen observed a broadening of the imageof a wedge-shaped slit, and inferred that the wave-length of thevibrations concerned was of the order of one Angstrom.3 However,in 1908, when B. Walter and R. Pohl 4 repeated the experiments,they found that different times of exposure gave different resultsas regards the image, and concluded that the effect was not con-firmed. In 1912 the question was re-opened when P. P. Koch,smaking a special study of the blackening of photographic plates ingeneral, re-examined Walter and Poh1's images, and decided thatthere was evidence of genuine diffraction. Thereupon ArnoldJ. W. Sommerfeld (1868-1951), Professor at Munich, compared theresults of theory with Koch's photometric measurements,6 anddeduced a value of 0·3 Angstroms for the wave-length of theX-rays.

At that time a young student, Peter Paul Ewald (b. 1888), whohad just taken his doctorate at Munich, was interested in thetransmission of light through the atomic lattice of a crystal. Somenotion of the dimensions of crystal-lattices could by this timebe formed; the Avogadro number (the number of molecules ina number of grams equal to the molecular weight) was known tobe approximately 6 x 1023; this together with a knowledge of thedensity and molecular weight of a crystal made it possible to estimatethat the distance apart of the atoms in a crystal was of the order of10-8 em. or one Angstrom. Ajunior lecturer in Munich, Max Laue 7

(b. 1879), who was in contact with Sommerfeld and Ewald, saw thatif the X-rays had a wave-length of the order suggested by Sommerfeld,then the crystal-lattice had the right dimensions for acting as athree-dimensional diffraction grating, so to speak, for the X-rays.He promptly arranged for an experimental test of this idea, whichwas carried out by W. Friedrich and P. Knipping; and a paper

1 Proc. R.S.(A), \xxxv (April 1911), p. 285; (A), \xxxvii (Sept. 1912),p. 277• Proc. Arnst. Ac. (25 March 1899) (English edn. i, p. 420) and 27 Sept. 1902(English

edn. v, p. 247). This work was discussed by Sommerfeld, Phys. ZS., i (1899), p. 105and ii (1900), p. 55. • cf. Vol. I, p. 367, note 5

• Ann. d. Phys. xxv (1908), p. 715; xxix (1909), p. 331• Ann. d. PhIS. xxxviii (1912), p. 507 • Ann. d. Phys. xxxviii (1912), p. 4737 About thIStime Laue's father, who was a general in the German Army, received

a title of nobility, so the son was known subsequently as Max von Laue.18

THE AGE OF RUTHERFORDwas published 1 in June-July 1912 in which it was completelyvindicated. A thin pencil of X-rays was allowed to fall on a crystalof zinc sulphide. A photographic plate, placed behind the crystalat right angles to this primary pencil, showed a strong central spot,where it was met by the primary rays, surrounded by a numberof other spots, in a regular arrangement: these were situated at theplaces where the plate was met by diffracted pencils, produced byreflection of the primary X-rays at sets of planes of atoms in thecrystal. The pOSItionsof the spots in the simplest imaginable caseare given by the following rule 2: Suppose that the atoms aredisposed so that the three rectangular co-ordinates of any atom areintegral multiples of a length a, every such place being filled, andsuppose that the incident rays are parallel to one of the axes. If thedistance from an atom A to another atom B is an integral multipleof a, then in the direction AB there will be one of the diffractedpencils that cause the spots. Clearly such directions correspond toall ways of expressing a square as the sum of three squares.

In this case the spots furnish no information regarding the wave-length of the radiation; and indeed the radiation used by Friedrichand Knipping had no definite wave-length, being a heterogeneousmixture of rays whose wave-lengths formed a continuous series.3A mathematical theory of the spots in more general cases was givenby Laue himself and by other writers.4

Laue's discovery was of the first importance, for not only didthe diffraction-patterns under suitable conditions serve to determinethe wave-length of the X-rays, but the idea was soon developedinto a regular method for determining the arrangement of the atomsin crystals; and its merit was fitly recognised by the award to Lauein 1914 of the Nobel Prize for physics. The question as to whetherX-rays were corpuscles or waves seemed to be settled in favour ofthe undulatory hypothesis; it was in fact found that the wave-lengthof high-frequency X-rays was about one Angstrom: but W. H.Bragg wrote 5 ' The problem becomes, it seems to me, not to decidebetween two theories of X-rays, but to find one theory which possess~~the capacity of both '-a remarkable anticipation of the vj_~·wthatwas made possible-many years later by the discover" of quantummechanics. -1

William Lawrence B.ragg (b.,1890)" sol.:'.ol·W. H. Bragg, in a paperread before the. Cambndge Phl.lo~~1JnicalSociety in the autumn ofthe same year,6 mtroduced c~':;sIderable simplifications in the theory.

i W. Friedrich,.P. ~ni';-;'~ng and M. Laue, Miinchen Ber. 8 June, p. 303 and 6 Julyp. 363, 1912; repn':'.~t:d Ann. d. Phys. xli (1913), p. 971

• W. H. B;:'agg, Nature, xc (24 Oct. 1912), p. 219• H. G. J. Moseley and C. G. Darwin, Phil. Mag. xxvi (July 1913), p. 210 found a

continuous spectrum, with maxima due to characteristic rays from the anticathode.• M. von Laue, loc. cit. : Ann. d. Phys. xlii (1913), p. 397 ; Phys. ZS. xiv (1913), p. 1075;

H. Moseley and C. G. Darwin, Nature, xc (14 Oct. 1912), p. 21.9; P. P. E~a1d, Phys. ZS.XlV (1913), p. 1038; xv (1914), p. 399; C. G. Darwm, PhIL. Mag. XXVll (Feb. 1914),p. 315; ibid. (April 1914), p. 675 • Nature, xc (28 Nov. 1912), p. 360

• Proc. Camb. Phil. Soc. xvii (Feb. 1913), p. 4319


His leading idea (which replaced Laue's assumption of scatteringat the points of a crystal-grating) was that parallel planes in thecrystal which are rich in atoms can be regarded, taken together,as a reflecting surface for X-rays; and experiments with a slip ofmica about a millimetre thick whose surface was a cleavage planeshowed him that the laws of reflection were obeyed when the rayswere incident at nearly glancing angles; reflection takes place onlywhen the wave-length A of the rays, the distance d between theparallel planes in the crystal, and the angle of incidence 4>, areconnected by the relation

nA=2d cos 4>

where n is a small whole number. This is known as the Bragg law.His father, continuing the work with him,l devised an X-ray spectro-meter, the principle of which is to allow monochromatic X-rays to fallin a fixed direction on a crystal, which is made to turn so that eachplane can be examined in detail: and with this instrument thearrangement of the atoms in many different crystals was determined.From this point the study of crystal-structure was developed by theBraggs with great success over an immense range: the Nobel Prizefor physics was awarded to them in 1915.

The discoveries regarding X-rays led to a better understandingof the y-rays from radio-active substance. J. A. Gray 2 establishedthe similarity in nature of y-rays and X-rays by showing that they-rays from RaE excite the characteristic X-radiations (K-series)of several elements, just as very penetrating X-rays would: andthat the y-rays behave similarly to X-rays (both qualitativelyand quantitatively) in regard to scattering. In 1914 Rutherford andE. N. da C. Andrade,3 by methods based on the same principle asthose used by the Braggs and by Moseley and Darwin, measuredthe wave-lengths of the y-rays from radium Band C. The wave-lengths of y-rays are usually less than those of X-rays, being generallybetween 0·01 and 0·1 Angstroms.

In the work of Rutherford and Geiger on counting a-particles bythe eie~tric method, carried out in 1908,4 some of the difficultiesthat had to !;:: overcome were due to the scattering of a-rays inpassing through mat~~r. ~eiger made .a special study of the scatter-mg for small angles of deh.:'~tlOn-band m 1909 Ruther~ord. suggestedto one of his research students, L'. Marsden, an exammatIOn of t~epossibility of scattering through large angles. ~ a result of thISsuggestion, experiments were carried out by' GelfSer and Ma!'~d~!!;which showed 6 that a-particles fired at a thin plat~ of matter canbe scattered inside the material to such an extent that soniCof thememerge again on the side of the plate at which they entered: and

1 Proc. R.S.(A), !xxxviii (July 1913), p. 428; Ixxxix (Sept. 1913), pp. 246, 248;ibid. (Feb. 1914), p. 468 • Proc. R.S.(A), !xxxvii (Dec. 1912), p. 489

• Phil. Mag. xxvii (May 1914), p. 854 • cf. p. 7• Proc. R.S.(A), !xxxii (July 1909), p. 495; Phil. Mag. xxv (1913), p. 604

20


calculation showed that some of the a-particles must have beendeflected at single encounters through angles greater than a rightangle.

Now at that time the atom was generally pictured in the formsuggested by J. J. Thomson in his Silliman lectures of 1903.1 He wasthen working out the consequences of supposing that the negativeelectrons occupy stationary positions in the atom. In order thatthe atom as a whole may be electrically neutral, there must be alsoa positive charge: and he saw that this could not be concentratedin positively charged corpuscles, since a mixed assemblage of negativeand positive corpuscular charges could not be in stable equilibrium.He therefore assumed that the positive electrification was uniformlydistributed throughout a sphere of radius equal to the radius of theatom as inferred from the kinetic theory of gases (about 10-8 cm.) :the negative electrons he supposed to be situated inside this sphere,their total charge being equal and opposite to that of the positiveelectrification.

In attempting to picture the way in which the negative electronswould dispose themselves, Thomson was guided by some experimentswith magnets which had been made many years earlier by AlfredMarshall Mayer of the Stevens Institute of Technology, Hoboken.2

Mayer magnetised a number of sewing-needles with their points ofthe same polarity, say south. Each needle was run into a smallcork, of such a size that it floated the needle in an upright position,the eye end of the needle just coming through the top of the cork.If three of these vertical magnetic needles are floated in a bowl ofwater, and the north pole of a large magnet is brought down overthem, the mutually repellent needles at once approach each other,and finally arrange themselves at the vertices of an equilateraltriangle. With four needles a square is obtained, with five either aregular pentagon or (a less stable configuration) a square with oneneedle at its centre, and so on. The under-water poles of thefloating needles, and the upper pole of the large magnet, wereregarded as too far away to exert any appreciable influence, so theproblem was practically equivalent to that of a number of southpoles in presence of a single large north pole.

Thomson examined theoretically the problem of the configura-tions assumed by a small number of negative electrons inside asphere of positive electrification, and found that when the numberof electrons was small, they disposed themselves in a regular arrange-ment, all being at the same distance from the centre; but whenthe number of electrons was increased, they tended to arrangethemselves in rings or spherical shells, and that the model imitatedmany of the known properties of atoms, particularly the periodicchanges with increase of atomic weight which are set forth in the

1 Published as Electricity and Matter in 1904. cr.].]. Thomson, Phil. Mag. vii (1904),p. 237, and for ~ earlier model, Lord Kelvin, Phil. Mag. iii (1902), p. 257.

I Phil. Mag.(5) \"(1878), p. 397; (5) vii (1879), p. 9821


Newlands-MendeIeev table. If one of the electrons were displacedslightly from its position of equilibrium, it would be acted on by arestitutive force proportional to the displacement. This was a mostdesirable property, since it was just what was required for anelectronic theory of optical dispersion and absorption; and, more-over, it would explain the monochromatic character of spectral lines :but in no way could Thomson's model be made to give an accountof spectral series.1

A model atom alternative to Thomson's had been proposed inthe same year (1903) by Philipp Lenard 2 (1862-1947) of Kiel, whoobserved that since cathode-ray particles can penetrate matter,most of the atomic volume must offer no obstacle to their penetration,and who designed his model to exhibit this property. In it therewere no electrons and no positive charge separate from the electrons:the atom was constituted entirely of particles which Lenard calleddynamides, each of which was an electric doublet possessing mass.All the dynamides were supposed to be identical, and an atomcontained as many of them as were required to make up its mass.They were distributed throughout the volume of the atom, but theirradius was so small «0·3 x 10-11 cm.) compared with the radiusof the atom, that most of the atomic volume was actually empty.Lenard's atom, however, never obtained much acceptance, as noevidence could be found for the existence of the dynamides.

The deflection of an a-particle through an angle greater than aright angle was clearly not explicable on the assumption of eitherThomson's or Lenard's atom; and Rutherford in December 1910came to the conclusion that the phenomenon could be explainedonly by supposing that an a-particle occasionally (but rarely)passed through a very strong electric field, due to a charged nucleus 3

of very small dimensions in the centre of the atom. This was con-firmed a year later by C. T. R. Wilson's photographs 4 of cloud-chamber tracks of a-particles which showed violent sudden deflectionsat encounters with smgle atoms.

Thus Rutherford was led to what was perhaps the greatest ofall his discoveries, that of the structure of the atom; the first accountof his theory was published in May 1911.5 He found that if amodel atom were imagined with a central charge concentrated withina sphere of less than 3 x 10-12 cm. radius, surrounded by electricityof the opposite sign distributed throughout the rest of the volumeof the atom (about 10-8 cm. radius), then this atom would satisfyall the known laws of scattering of a- or ,B-particles, as found byGeiger and Marsden. The central charge necessary would be Ne,

1 See, however, an attempt by K. F. Herzfeld, Wien Ber. cxxi, 2a (1912), p. 593, Ann. d. Phys. xii (1903), p. 714, at p. 736I The term nuJ:leusfor the central charge seems to have been used first in Rutherford's

book Radioactive Substances and their Radiations, which was published in 1912.• Proc. R.S.(A), lxxxvii (Sept. 1912), p. 279• Phil. Mag. xxi (May 1911), p. 669; xxvii (1914), p. 488. For an anticipation that

the atom might prove to be of this type, cf. H. Nagaoka, Phil. Mag. vii (1904), p. 445.22

THE AGE OF RUTHERFORDwhere e is the electronic charge, and N is a number equal to abouthalf the atomic weight. This fitted in perfectly with the discoveryalready made by Barkla,1 that the number of scattering electronsper atom is (for the lighter elements, except hydrogen) about halfthe atomic weight: for the positive central charge, and the negativecharges on the electrons in the space around it, must exactly neutraliseeach other.

Thus the Rutherford atom is like the solar system, a smallpositively charged nucleus in the centre, which contains most ofthe mass of the atom, being surrounded by negative electronsmoving around it like planets, at distances of the order of 10-8 cm.Occasionally an a-particle passes near enough to unbind and detachan electron and thus ionise the atom: still more infrequently (onlyabout one a-particle in ten thousand, even in the case of heavyelements) the a-particle may come so close to the nucleus as toexperience a violent deflection, due to the electrical repulsion betweenthem. The encounters were studied mathematically by C. G. Darwin 2

(b. 1887), who found a satisfactory agreement between theory andexperiment, and showed that Geiger and Marsden's results couldnot be reconciled with any law of force except the electrostatic lawof the inverse square, which is obeyed to within 3 x 10-18 cm. of thecentre of the atom.

Rutherford now laid down the principle 3 that the positivecharge on the nucleus (or the number of negative electrons) is thefundamental constant which determines the chemical propertiesof the atom: this fact explains the existence of isotopes, whichhave the same nuclear charge but different nuclear masses, andwhich have the same chemical properties. He pointed out alsothat gravitation and radio-activity, being unaffected by chemicalchanges, must depend on the nucleus. His old discovery, that thea-particle is a doubly ionised atom of helium, was now reformulatedin the statement, that the a-particle, at the end of its track, capturestwo electrons (one at a time), and thus becomes a neutral heliumatom; he suggested, moreover, that the nucleus of the hydrogenatom might actually be the' positive electron.' It was seen that thehydrogen nucleus differed from the negative electron not only in thereversal of sign of its charge, but also in having a much greatermass-in fact, almost all the mass of the hydrogen atom'; and atthe Cardiff meeting of the British Association in 1920, Rutherfordproposed for it the name proton, which has been universally accepted.

A proposal for removing the uncertainty which still remained asto the precise amount of the nuclear charge was made in 1913 by

1 cf. p. 15 • Phil. Mag. xxvii (March 1914), p. 499I Nature, xcii (Dec. 1913), p. 423; Phil. Mag. xxvi (Oct. 1913), p. 702; xxvii (March

1914),p. 488• Poincare in his St. Louis lecture of 1904had said (Bull. des Sci. Math. xxviii (1904),

p. 302) 'The mass of a body would be the sum of the masses of its positive electrorul,the negative electrons not counting.'


A. van der Broek 1 of Utrecht. He remarked that when a-particlesare scattered by a nucleus, the amount of scattering per atom,divided by the square of the charge in the nucleus, must be constant.As Geiger and Marsden had shown, this condition is roughly satisfiedif the nuclear charge is assumed to be proportional to the atomicweight; but van der Broek now pointed out that it would be satisfiedwith far greater accuracy if the nuclear charge were assumed to beproportional to the number representing the place of the elementIn the Newlands-Mendeleev periodic table. He suggested, therefore,that the nuclear charge should be taken to be Ze, where e is theelectronic charge (taken positively) and Z is the ordinal numberof the element in the periodic table.

This suggestion received a complete confirmation from experi-ments performed in Rutherford's laboratory at Manchester by HenryGwyn-Jeffreys Moseley 2 (b. 1887, killed at the Suvla Bay landingin the Dardanelles, 10 August 1915) in continuation of the workwhich he and Darwin had been carrying on together. Moseleyexposed the chemical elements, from calcium to nickel, as anti-cathodes in an X-ray tube, so that under the bombardment ofcathode-rays they emitted their characteristic X-ray spectra, con-sisting essentially of two strong lines (the K- and L-lines) 3; andthe wave-lengths of these lines were determined by the crystalmethod. Taking either of these lines and following it from elementto element, he found that the square root of its frequency increasedby a constant quantity as the transition was made from any elementto the next higher element in the periodic table; so that the fre-quency was expressible in the form k(N -a) 2, where k was an absoluteconstant, N was the 'atomic number' or place in the periodictable, and a was a constant which had different values for the K- andL-lines. So there must be in the atom a fundamental number, whichincreases by unity as we pass from one element to the next in theperiodic table; and, having regard to the results of Rutherford,Geiger, Marsden and van der Broek, this quantity can only be theamount of the nuclear charge, ex.(>ressedIn electron-units. Thusthe number of negative electrons which czrculate round the nucleus of an atomof a chemical element is equal to the ordinal number of the element in theperiodic table.

Two incidental results of Moseley's work on X-ray spectra mustbe mentioned. It now became clear that the atomic numbers ofiron, cobalt and nickel must be respectively 26, 27, 28, thus con-firming the opinion, already suggested by chemical considerations,that cobalt should have a lower place in the periodic table than

1 Phys. ZS. xiv (1913), E. 32 ; Nature, xcii (27 Nov. 1913),p. 372; xcii (25 Dec. 1913),p. 476; Phil. Mag. xxvii (March 1914), p. 455

• Phil. Mag. xxvi (Dec. 1913), p. 1024; xxvii (April 1914), p. 703. Moseley leftManchester for Oxford at the end of 1913,and completed his work there. cf. his obituarynotice in Prot. R.S.(A), xciii (1917), p. xxii.

I cf. p. 16. Actually each of these lines is a multiplet.24

THE AGE OF RUTHERFORDnickel, although it has a higher atomic weight; and the vexedquestions of the number of elements in the group of the rare earths,and of missing elements in the periodic table, could also be settled,since it was now known what the X-ray spectra of these elementsmust be.l Some predictions which had been made by the Danishchemist, Julius Thomsen (1826-1909) were now verified in a remark-able way.

Thus Rutherford, with the help of the young men in his researchschool-Geiger, Marsden, Moseley and Darwin 2-created a definitequantitative theory of the atom, lending itself to mathematicaltreatment, and satisfying every comparison with experiment. Ithas been the foundation of all later work.

During the years 1914-18 Rutherford was occupied chiefly withmatters connected with the war: but in 1919 he made a contribu-tion 3 of the highest importance to atomic physics. I t originatedfrom an observation made by Marsden, who had shown 4 that whenan a-particle collides with an atom of hydrogen, the hydrogen atommay be set in such swift motion that it travels (nearly in the directionof the impinging particle) four times as far as the colliding a-particle,and that it may be detected by a scintillation produced on a zincsulphide screen. Rutherford now showed, by measurements ofdeflections in magnetic and electric fields, that these scintillationswere due to hydrogen atoms carrying unit positive charge, in otherwords, to hydrogen nuclei, or protons as they soon came to becalled.

He next bombarded dry air, and nitrogen, with a-particles, andagain found scintillations at long range. The similarity in behaviourof the particles obtained from nitrogen to those previously obtainedfrom the hydrogen led him to suspect that they were identical, i.e.that the long-range particles obtained by bombarding nitrogen witha-particles were actually hydrogen nuclei. The general idea nowpresented itself, that some of the lighter atoms might be actuallydisintegrated by a collision with a swift a-particle: going beyondthe earlier discovery that an a-particle might be deflected througha large angle by a close collision with a nucleus, he now came tothe conclusion that on still more rare occasions (say one a-particlein half a million) it might break up the nucleus. The phenomenonwas found to occur markedly with nitrogen, but not with dry oxygen.

In the summer of 1919 Rutherford succeeded J. J. Thomson asthe Cavendish Professor of Physics at Cambridge. Continuing hisexperiments there, he succeeded in. proving definitely that the

1 For individual elements the nuclear charges were found directly by J. Chadwick(Phil. Mag. xl (1920), p. 734) by experiments on the scattering of pencils of a-rays. Hisresults agreed with those deduced from Moseley's law of X-ray spectra. Practically allof the elements which have been discovered since Moseley's day, and which fill thegaps that then existed in the periodic table, have been identified by the study of theircharacteristic X-ray spectra.

I And Bohr, whose work will be described in a later chapter• Phil. Mag. xxxvii (June 1919), p. 537 • Phil. Mag. xxvii (May 1914), p. 824

25


nitrogen atom can be disintegrated by bombarding it with a-particles.lAs P. M. S. Blackett (b. 1897) showed, the tracks of the particlescould be seen in the Wilson cloud-chamber. Since the nitrogennucleus, of charge 7 electronic units, captures the a-particle, ofcharge 2, and expels the proton, of charge 1, the particle obtainedby the transformation must have charge 8, that is, it must be thenucleus of an isotope of oxygen. Since the nitrogen nucleus hasmass 14, the captured a-particle has mass 4, and the expelledproton has mass 1, it follows that the oxygen isotope must havemass 17.

In 1921 Rutherford and J. Chadwick (b. 1891) found 2 thatsimilar transformations could be produced in boron, fluorine,sodium, aluminium and phosphorus: and other elements werelater added to the list. In each case the a-particle was capturedand a swift proton was ejected, while a new nucleus of mass threeunits greater and charge one unit higher was formed. Thus themedieval alchemist's dream of the transmutation of matter wasrealised at last.

Rutherford died at Cambridge on 19 October 1937, and wasburied in Westminster Abbey near the graves of Newton and Kelvin.He was survived by his old teacher J. J. Thomson, who had in 1918been elected Master of the great foundation of which he had beena member uninterruptedly since 1875.

, How fortunate I have been throughout my life !' Thomsonwrote, near the end of it, , I have had good parents, good teachers,good colleagues, good pupils, good friends, great opportunities, goodluck and good health.' He lived to be eighty-three, dying at TrinityLodge on 30 August 1940, and was buried on 4 September in theAbbey.

1 Proc. R.S.(A), xcvii (July 1920), p. 374; Engineering, cx (17 Sept. 1920), p. 382(a paper read to the British Association at its Cardiff meeting); Proc. Rhys. Soc. xxxiii(Aug. 1921), p. 389

• Nature, cvii (10 March 1921), p. 41

Chapter IITHE RELATIVITY THEORY OF POINCARE

AND LORENTZ

AT the end of the nineteenth century, one of the most perplexingunsolved problems of natural philosophy was that of determiningthe relative motion of the earth and the aether. Let us try to presentthe matter as it appeared to the physicists of that time.

According to Newton's First Law of Motion, any particle whichis free from the action of impressed forces moves, if it moves at all,with uniform velocity in a straight line. But in order that this state-ment may have a meaning, it is necessary to define the terms straightline and uniform velocity; for a particle which is said to be ' movingin a straight line' in a terrestrial laboratory would not appear tobe moving in a straight line to an observer on the sun, since he wouldperceive its motion compounded with the earth's diurnal rotationand her annual revolution in her orbit. We can, however, definea straight line with reference to a system ofaxes Oxyz as the geometricalfigure defined by a pair of linear equations between x, y, z; andwe can assert as a fact of experience that certain systems of £LxesOxyz exist such that free particles move in straight lines with referenceto them. Moreover, we can assert that there exist certain waysof measuring time such that the velocity of free particles along theirrectilinear paths is uniform. A set of axes in space and a systemof time-measurement, which possess these properties, may be calledan inertial system of reference.

In Newtonian mechanics, if S is an inertial system of reference,and ifS'isanothersystem such that the axes O'x'y'z' of S' have anyuniform motion of pure translation with respect to the axes Oxyzof S, and if the system of time-measurement is the same in the twocases, then S' is also an inertial system of reference: the Newtonianlaws of motion are valid with respect to S' just as with respect to S.No one inertial system of reference could be regarded as havinga privileged status, in the sense that it could properly be said tobe fixed while the others were moving. Newtonian mechanicsdoes not involve the notion of the absolute fixity of a point inspace.

The laws of Newtonian dynamics thus presuppose the knowledgeof a certain set of systems of reference, which is necessary if the lawsare to have any meaning. In the nineteenth century many physicistsinquired how this set of systems of reference should be described anddefined. When Carl Neumann (1832-1925) was appointed professorof mathematics at Leipzig in 1869, he devoted his inaugural

27


changes in the relations of the molecules to the aether, that anisotropic substance does not lose its simply refracting character.

Even before the end of the nineteenth century, the failure of somany promising attempts to measure the velocity of the earthrelative to the aether had suggested to the penetrating and originalmind of Poincare a new possibility. In his lectures at the Sorbonnein 1899,1after describing the experiments so far made, which hadyielded no effects involving either the first or the second powers ofthe coefficient of aberration (i.e. the ratio of the earth's velocity tothe velocity oflight), he went on to say,2 , I regard it as very probablethat optical phenomena depend only on the relative motions of thematerial bodIes, luminous sources, and optical apJ?aratus concerned,and that this is true not merely as far as quantItIes of the order ofthe square of the aberration, but rigorously.' In other words, Poincarebelieved in 1899 that absolute motion is indetectible in principle, whetherby dynamical, optical, or electrical means.

In the following year, at an International Congress of Physicsheld at Paris, he asserted the same doctrine.3 'Our aether,' he said,, does it really exist? I do not believe that more precise observationscould ever reveal anything more than relative displacements.' Afterreferring to the circumstance that the explanations then current forthe negative results regarding terms of the first order in (wfc) weredifferent from the explanations regarding the second order terms,he went on, 'It is necessary to find the same explanation for thenegative results obtained regarding terms of these two orders:and there is every reason to suppose that this explanation will thenapply equally to terms of higher orders, and that the mutual destruc-tion of the terms will be rigorous and absolute.' A new principlewould thus be introduced into physics, which would resemble theSecond Law of Thermodynamics in as much as it asserted theimpossibility of doing something: in this case, the impossibility ofdetermining the velocity of the earth relative to the aether.'

In a lecture to a Congress of Arts and Science at St Louis, U.S.A.,on 24 September 1904, Poincare gave to a generalised form of thisprinciple the name, The Principle of Relativity.6 'According to thePrinciple of Relativity,' he said, 'the laws of physical phenomenamust be the same for a " fixed" observer as for an observer whohas a uniform motion of translation relative to him: so that we havenot, and cannot possibly have, any means of discerning whether weare, or are not, carried along in such a motion.' After examiningthe records of observation in the light of this principle, he declared,

1 Edited by E. Neculcea, and printed in 1901 under the title Eketricite et Optique,Paris, Carre et Naud. • loc. cit., p. 536

• Rapports presentis au Congres International de Physique riuni d Paris en Igoo (Paris,Gauthier-Villars, 1900),Tome I, p. 1, at pp. 21, 22

• In April 1904 Lorentz asserted the same general principle: cr. Versl. Kon. Akad.v. Wet., Amsterdam, Dl. xii (1904), p. 986; English edn. (Arnst. Proe.), vi (1904), p. 809.

• This address appeared in Bull. des Sc. Math.(2) xxviii (1904), p. 302; an Englishtranslation by G. B. Halsted was published in The Monist forJanuary 1905.

30

THE RELATIVITY THEORY OF POINCARE AND LORENTZ

'From all these results there must arise an entirely new kind ofdynamics, which will be characterised above all by the rule, that no velocitycan exceed the velocity of light.'

We have now to see how an analytical scheme was devised whichenabled the whole science of physics to be reformulated in accordancewith Poincare's Principle of Relativity.

That Principle, as its author had pointed out, required thatobservers who have uniform motions of translation relatlve to eachother should express the laws of nature in the same form. Let usconsider in partIcular the laws of the electromagnetic field.

Lorentz, as we have seen,1had obtained the equations of a movingelectric system by applying a transformation to the fundamentalequations of the aether. In the original form of this transformation,quantities of order higher than the first in (wlc) were neglected.But in 1900 Larmor 2 extended the analysis so as to include quantitiesof the second order. Lorentz in 1903 went further still,3and obtainedthe transformation in a form which is exact to all orders of thesmall quantity (wlc). In this form we shall now consider it.

The fundamental equations of the aether in empty space are

div d=O, ahc curl d= -- at

div h=O, c curl h= adat·

I t is desired to find a transformation from the variables t, x, y, z,d, h, to new variables th Xl, Yh Zh dI, hI, such that the equations interms of these new variables may take the same form as the originalequations, namely

divi dl = 0, C curli dl = _ ahlat1

Evidently one particular class of such transformation is that whichcorresponds to rotations of the axes of co-ordinates about the origin.These may be described as the linear homogeneous transformationsof determinant unity which transform the expression (x· +y. +z.)into itself. It had, however, already become clear from Lorentz'searlier work that some of the transformations must involve not only

1 cf. Vol. I, p. 406. cr. also Lorentz, Proc. Arnst. Acad. (English edn.), i (1899),p.427

• Larmor, Aether and Malter (1900), p. 173• Proc. Arnst. Acad. (English edn.), VI (1903), p. 809

31

(1)


X, y, z, but also the variable t.1 So (guided by the approximateformulae already obtained) he now replaced the condition oftransforming (x2 +y2 + Z2) into itself, by the condition of transform-ing the expression (Xl +y2 + Z2 - e2t2) into itself; and, as we shallnow show, he succeeded in proving that the transformations soobtained have the property of transforming the differential equationsof the aether in the manner required.

We shall first consider a transformation of this class in whichthe variables y and Z are unchanged. The equations of this trans-formation may easily be derived by considering that the equationof the rectangular hyperbola

x2_ (et) 2 = 1

(in the plane of the variable x, et) is unaltered when any pair ofconjugate diameters are taken as new axes, and a new unit of lengthis taken proportional to the length of either of these diameters.The equations of transformation thus obtained are

et = etl cosh a + Xl sinh aX=Xl cosh a+etl sinh ay=YlZ=ZI

where a denotes a constant parameter. The simpler equationspreviously given by Lorentz 2 may evidently be derived from theseby writing w=e tanh a, and neglecting powers of (wle) above thefirst. It will be observed that not only is the system of measuringthe abscissa X changed, but also the system of measuring the time t :the necessity for this had been recognised in Lorentz's originalmemoir by his introduction of ' local time.'

Let us find the physical interpretation of this transformation (1).If we consider the point in the (tl, Xl, Yl, Zl) system for whichXl, Yl> Zl are all zero, its co-ordinates in the other system are givenby the equations

x=et tanh a,so

t= t1 cosh a, X = etl sinh a,

y=o,y=o, Z=o,

Thus if we regard the axes of (Xl> Yl> Zl) and the axes of (x, y, z) astwo rectangular co-ordinate systems in space, then the origin of the(Xl' Yl> Zl) system has the co-ordinates (et tanh a, 0, 0), that is to

1 Larmor, Aether and Matter (1900), in commenting on the FitzGerald contraction,had recognised that clocks, as well as rods, are affected by motion: a clock movingwith velocity v relative to the aether must run slower, in the ratio

• cf. Vol. I, p. 40732


say, the origin of the (Xl,Yl, Zl) system moves with a uniform velocityc tanh a along the x-axis of the (x, Y, z) system. Thus if w is therelative velocity of the two systems, we have

( w')-~cosh a = 1- C2 2,w ( w1)-J..sinh a = - 1_ _ 2,C cl

and Lorentz's transformation between their co-ordinates may bewritten

Xl +wtlx= (-'W')-'V 1- cl

In this transformation the variable x plays a privileged part, ascompared withy or z. We can of course at once write down similartransformations in which y or Z plays the privileged part; and wecan combine any number of these transformations by performingthem in succession. The aggregate of all the transformations soobtained, combined with the aggregate of all the rotations in ordinaryspace, constitutes a group, to which Poincare 1 gave the name thegroup of Lorentz transformations.

By a natural extension of the equations formerly given byLorentz for the electric and magnetic forces, it is seen that theequations for transforming these, when (i, x, v, z) are transformedby equations (1), are

d:r=dx1

dy = dy, cosh a + hz, sinh a

d.= d" cosh a - hYI sinh a

hx=h:n

hy = hYI cosh a - d" sinh a

hz =h%l cosh a + dYI sinh a.

(2)

When the original variables are by direct substitution replacedby the new variables defined by (1) and (2) in the fundamentaldifferential equations of the aether, the latter take the form

divl dl =0, C curl I dl = _ ah}atladlat;

that is to say, the fundamental equations of the aether retain their formunaltered, when the variables Ct, x, y, z) are subjected to the Lorentz

I Compl.s Rendus, exl (5.Tune 1905), p. 1504. It should be added that these trans-formations had been applied to the equation of vibratory motions many years before byW. Voigt, Giiu. Nach. (1887), p. 41.

33


transformation (1), and at the same time the electric and magnetic intensitiesare subjected ot the transformation (2).

The fact that the electric and magnetic intensities undergo thetransformation (2) when the co-ordinates undergo the transformation(1), raises the question as to whether the transformation (2) is familiarto us in other connections. That this is so may be seen as follows.

In 1868-9 J. Plucker and A. Cayley introduced into geometrythe notion of line co-ordinates; if (xo, Xl> X2, Xa) and (Yo, Yl> Y2, ya) arethe tetrahedral co-ordinates of two points of a straight line P, andif we write

then the six quantities

POI,are called the line-eo-ordinates of p.

Now suppose that the transformationxo=x'o cosh a+x'i sinh aXI = X'1 cosh a + x' 0 sinh aX2 =X' 2

Xa =x'a

(3)

is performed on the co-ordinates (xo, XI, X2, Xa), and the same trans-formation is performed on the co-ordina,tes (yo, YI, Y2, Ya). Thenwe have

POI = XOYI - XIYO= (x' 0 cosh a + X'I sinh a) (Y'l cosh a +y' 0 sinh a)

- (X'1 cosh a + x' 0 sinh a) (y'o cosh a +Y'I sinh a)= X' oy' 1 - x' IY' 0

=P'OI

and in the same way we findP02 =p' 02 cosh a +p' 12 sinh aPoa =p' oa cosh a - p' al sinh apaa = p' 2aP3l =p' al cosh a - p' oa sinh aP12 =p' 12 cosh a +p' 02 sinh a.

But these equations of transformation of the p's are precisely thesame as the equations of transformation (2) of the electric andmagnetic intensities, provided we write

POI = dx, P02 = dy, poa = dz, P23 = hx, pal = Izy, P12 = hz.

The line-eo-ordinates of a line have this property of transforminglike the six components of the electric and magnetic intensities notonly for the particular Lorentz transformation (l) but for the mostgeneral Lorentz transformation. A set of six quantities which trans-

34


form like the line-co-ordinates of a line when the co-ordinates aresubjected to any Lorentz transformation whatever, is called a six-vector. Thus we may say that the quantities (dx, dy, dz, ke, hy, hz)constitute a six-vector.! In the older physics, d was regarded as a vector,and h as a distinct vector: but if an electrostatic system (in whichd exists but h is zero) is referred to axes which are in motion withrespect to it, then the magnetic force with respect to these axes willnot be zero. The six-vector transformation takes account of thisfact, and furnishes the value of the magnetic force which thus appears.

We see, therefore, that in electromagnetic theory, as in Newtoniandynamics, there are inertial systems of co-ordinate axes with associatedsystems of measurement of time, such that the path of a free materialparticle relative to an inertial system is a straight line described withuniform velocity, and also that the equations of the electromagneticfield relative to the inertial system are Maxwell's equations, andany system if" axes which moves with a uniform motion of translation, relativeto any given inertial system of axes, is itself an inertial system of axes, themeasurement if" time and distance in the two systems being connected by aLorentz transformation. All the laws if" nature have the same form in the co-ordinates belonging to one inertial system as in the co-ordinates belonging to anyother inertial system. No inertial system of reference can be regardedas having a privileged status, in the sense that it should be regardedas fixed while the others are moving: the notion of absolute fixityin space, which in the latter part of the nineteenth century wasthought to be required by the theory of aether and electrons wasshown in 1900-4 by the Poincare-Lorentz theory of relativity to bewithout foundation.

Suppose that an inertial system of reference (t, x, y, .<:) is knownon earth: and imagine a distant star which is moving with auniform velocity relative to this framework (t, x,y, z). The theoremof relativity shows that there exists another framework (t1) Xl, Yl, '<:1)with respect to which the star is at rest, and in which, moreover, aluminous disturbance generated at time tl at any point (Xl' Yl, '<:1)will spread outwards in the form of a sphere

(X, - Xl)2 + (Yl - Yl)2 + (Zl-Zl)2 =c2(Tl- tl)2,

the centre of this sphere occupying for all subsequent time anunchanged position in the co-ordinate system (Xl,Yl, '<:1)' This frame-work is peculiarly fitted for the representation of phenomena whichhappen on the star, whose inhabitants would therefore naturallyadopt it as their system of space and time. Beings, on the otherhand, who dwell on a body which is at rest with respect to the axes(t, x, y, z), would prefer to use the latter system; and from thepoint of view of the universe at large, either of these systems is asgood as the other. The electromagnetic equations are the samewith respect to both sets of co-ordinates, and therefore neither can

1 Raurn-Zeit- Vektor IJ Art of H. Minkowski, Gott. Nach. 1908, p. 5335


claim to possess the only property which could confer a primacy-namely, a special relation to the aether.

Some of the consequences of the new theory seemed to contem-porary physicists very strange. Suppose, for example, that twomertial sets of axes A and B are in motion relative to each other,and that at a certain instant their origins coincide: and supposethat at this instant a flash of light is generated at the commonorigin. Then, by what has been said in the subsequent propagation,the wave-fronts of the light, as observed in A and in B, are sphereswhose centres are the origins of A and B respectively, and thereforedi.fferent spheres. How can this be ?

The paradox is explained when it is remembered that a wave-front is defined to be the locus of points which are simultaneously inthe same phase of disturbance. Now events taking place at differentpoints, which are simultaneous according to A's system of measuringtime, are not in general simultaneous according to B's way ofmeasuring: and therefore what A calls a wave-front is not the samething as what B calls a wave-front. Moreover, since the systemof measuring space is different in the two inertial systems, what Acalls a sphere is not the same thing as what B calls a sphere. Thusthere is no contradiction in the statement that the wave-frontsfor A are spheres with A's origin as centre, while the wave-frontsfor B are spheres with B's origin as centre.

In common language we speak of events which happen at differentpoints of space as happening' at the same instant of time,' and wealso speak of events which happen at different instants of time ashappening 'at the same point of space.' We now see that suchexpressions can have a meaning only by virtue of artificial conven-tions; they do not correspond to any essential physical realities.

It is usual to regard Poincare as primarily a mathematician, andLorentz as primarily a theoretical physicist: but as regards theircontributions to relativity theory, the positions were reversed: it wasPoincare who proposed the general physical principle, and Lorentzwho supplied much of the mathematical embodiment. Indeed,Lorentz was for many years doubtful about the physical theory :in a lecture which he gave in October 19101 he spoke of' die Vorstel-lung (die auch Redner nur ungern aufgeben wurde), dass Raumund Zeit etwas vollig Verschiedenes seien und dass es eine " wahreZeit" gebe (die Gleichzeitigkeit wurde denn unabhangig vomOrte bestehen).' 2

A distinguished physicist who visited Lorentz in Holland shortlybefore his death found that his opinions on this question wereunchanged.

We are now in a position to show the connection between the

1 Printed in Phys. ZS. xi (1910), p. 1234• • The concept (which the present author would disliKe to abandon) that space

and time are something completely distinct and that a " true time" exists (simultaneitywould then have a meaning independent of position).'

36


Lorentz transformation and FitzGerald's hypothesis of contraction;this connection was first established by Larmor 1 for his approximateform of the Lorentz transformation, which is accurate only to thesecond order in (w/c) , but the extension to the full Lorentz trans-formation is easy.

Suppose that a rod is moving along the axis of x with uniformvelocity w; let the co-ordinates of its ends at the instant t be Xiand x2. Take a system of axes O'x'y'z' which move with the rodj

the axis O'x' being in the same line as the axis Ox, and the axes O'y'and 0' z' being constantly parallel to the axes Oyand Oz respectively.In this system the length of the rod will be x'2 -x' 1, where, ofcourse, x' 2 and X'l do not vary with the time. The Lorentz trans·formation gives

X'2=X2 cosh a-ct sinh aX'l = Xl cosh a - ct sinh a

where tanh a= (w/c). Subtracting, we have

X'2-X'l=(X2-Xl) cosh a=(x2-xl){1-(w/c)2}-,

or

This equation shows that the distance between the ends of the rod,in the system of measurement furnished by the original axes, withreference to which the rod is moving with velocity w, bears theratio (l-w2/c2)~ : I to their distance in the system of measurementfurnished by the transformed axes, with reference to which the rodis at rest: and this is precisely FitzGerald's hypothesis of contraction.The hypothesis of FitzGerald may evidently be expressed by thestatement, that the equations of the figures of material bodies are covariantwith respect to those transformations for which the fundamental equations ofthe aether are covariant: that is, for all Lorentz transformations.

Now let us look into Poincare's remark 2 that the Principle ofRelativity requires the creation of a new mechanics in which novelocity can exceed the velocity of light.

Suppose that an inertial system B is being translated relativeto an inertial system A with velocity w along the axis of x. Leta point P moving along the axis of x have the co-ordinates (t, x, 0, 0)in system A and (t', x', 0, 0) in system B. Denote the components ofvelocity dx/dt and dx'/dt' by Vx, v'x, respectively, and let w=c tanh a.Then the Lorentz transformation gives at once

dx c(dx' cosh a+cdt' sinh a) v'x+wVx= - = ---------~ =---.

dt c dt' cosh a + dx' sinh a I + vxwc2

1 Aether and Matter (1900), p. 173 • cr. p. 3137


Now, vx being the velocity of P relative to A, v'X the velocityof P relative to B, and w the velocity of B relative to A, inNewtonian kinematics we should have vx = v'x + w. The denominator(l + vxw/cl) in the relativist formula expresses the difference betweenNewtonian theory and relativity theory, so far as concerns thecomposition of velocities. We see that if v' x = c, then Vx = c; thatis to say, any velocity compounded with c gives as the resultant coveragain, and therefore that no velocity can exceed the velocity oflight.

This result enables us to solve a problem which had perplexedmany generations of physicists. It had been supposed that if thecorrect theory of light is the corpuscular theory, then the corpusclesemitted by a moving star should have a velocity which is compoundedof the velocity of the star and the velocity of light relative to a sourceat rest, just as an object thrown from a carriage window in a movingrailway train has a velocity which is obtained by compounding itsvelocity relative to the carriage with the velocity of the train (theballistic theory); whereas, if the correct theory of light is the wave-theory, the velocity of the light emitted by the star should be un-affected by the velocity of the star, just as the waves created bythrowing a stone into a pond move outwards from the point wherethe stone entered the water, without being affected by the velocityof the stone. The new relativist theory led to the surprising con-clusion that the velocity of light would be unaffected by the velocityof its source even on the corpuscular theory.

An attempt to explain the Michelson-Morley experiment, andthe other evidence which had given rise to relativity theory, withoutassuming that the'velocity of light is independent of the velocity ofits source, was made in 1908 by W. Ritz,! who postulated that thevelocity of light and the velocity of the source are additive, as inthe old physics. It is, however, now known certainly that the velocityoflight IS independent of the motion of the source. The astronomicalevidence for this statement has been marshalled by several writers,2and further confirmation has been furnished by Majorana by directexperiment.3 It should be remarked that since in purely terrestrialexperiments the light rays always describe closed paths, the resultsto be expected from' ballistic' and non-ballistic theories can differonly by quantities of the second order,4 but the performance of the

1 Ann. de chim. et phys. xiii (1908), p. 145; Arch. de Gtneve, xxvi (1908), p. 232; cf. acareful discussionof it by R. C. Tolman, Phys. Rev. xxxv (1912), p. 136

• Particularly by R. C. Tolman, Phys. Rev. xxxi (1910), p. 26; W. de Sitter,Amsterdam Proc. xv (1913), p. 1297; xvi (1913), p. 395; Phys. ZS. xiv (1913), pp. 429,1267; Bull. of the Astron. Inst. of the Netherlands, ii (1924), pp. 121, 163; R. S. Capon,Month. Not. R...A.s.lxxiv (1914),pp. 507, 65~; H. C. Plummer, ibid., p. 660; H. Thirring,ZS.f. P. XXXI (1925), p. 133; G. Wataghm, ZS.f. P. xl (1926), p. 378

• Comptes Rmdus, c1xv (1917), p. 424; clxvii (1918), p. 71; clxix (1919), p. 719;Phys. Rev. xi (1918), p. 411; Phil. Mag. xxxvii (1919), p. 145; xxxix (1920), p. 488 ;cf. alsoJeans, Nature, cvii (1921), p.p. 42,169

, cr. P. Ehrenfest, Phys. ZS. xiIi (1912), p. 317; F. Michaud, Comptes Rendus, c1xviii(1919), p. 507

THE RELATIV1TY THEORY OF POINCARE AND LORENTZ

Michelson-Morley experiment with light from astronomical sourcesby R. Tomaschek 1 in 1924 definitely disproved the ballistic hypo-thesis.

A further result in harmony with the new theory was obtainedwhen Michelson 2 showed experimentally that the velocity of amoving mirror is without influence on the velocity of light reflectedat its surface.

It was now recognised that these observational findings, whichin the nineteenth century might have been supposed to tell in favoutof the wave-theory, were actually without significance one way orthe other in the dispute between the wave and corpuscular theoriesof light. For, according to relativity theory, even on the corpuscularhypothesis, a corpuscle which had a velocity c relative to its sourcewould have the same velocity relative to any observer, whether heshared in the motion of the source or not.

In 1905 Poincare 3 completed the theorem of Lorentz" on thecovariance of Maxwell's equations with respect to the Lorentztransformation, by obtaining the formulae of transformation of theelectric density p and current pv. The fundamental equations are

div d =47rp ;

div h=O;

8hc curl d= -- atadc curl h = at +47rpv

and it is desired to find a transformation from the variable t, x, y,z, p, d, h, v to new variables t1, Xl, Y1, Zl, P1, d1, h1, v1, such thatthe equations in terms of these new variables may have the sameform as the original equations. The transformations of t, X, y, z,d, h have already been found. Poincare now showed that

p = pi cosh a + (p.vx,/ c) sinh aPVx = p. Vx. cosh a + Cpt sinh aPVy=p. VYIPVz =p. Vz1•

When the original variables are by direct substitution replaced bythe new variables in the differential equations, the latter take theform

I Ann. d. Phys. lxxiii (1924), p. 105I Comples Rendus, ex! (June 1905), p. 1504-

39

• Aslrophys. J. xxxvii (1913), p. 190• cf.p.33

or


that is to say, the fundamental equations of aether and electronsretain their form unaltered, when the variables are subjected to thetransformation which has been specified.

In the autumn of the same year, in the same volume of theAnnalen der Physik as his paper on the Brownian motion, 1 Einsteinpublished a paper which set forth the relativity theory of Poincareand Lorentz with some amplifications, and which attracted muchattention. He asserted as a fundamental principle the constancyof the velocity of light, i.e. that the velocity of light in vacuo is the samein all systems of reference which are moving relatively to each other :an assertion which at the time was widely accepted, but has beenseverely criticised by later writers. 2 In this paper Einstein gave themodifications which must now be introduced mto the formulae foraberration and the Doppler effect.3

Consider a star, which is observed from the earth on two occasions.The distance of the star is assumed to be so great that its apparentproper motion in the interval between the observations is negligible.Denote an inertial system of axes at the earth at the time of thefirst observation by K, and an inertial system of axes at the earth atthe time of the second observation by K': and choose these axesso that the x-axis has the direction of the velocity w (= c tanh a)of K' relative to K. Let ifJ be the angle which the ray of lightarriving at the earth from the star makes with the x-axis as measuredin K, and ifJ' the corresponding angle in the system K/. Then theLorentz transformation gives for the co-ordinates of the star in thetwo systems

{ct' = ct cosh a - x sinh ax' =x cosh a - ct sinh ay'=y

(taking the plane of xy to contain the star): and since light ispropagated with velocity c in both systems, we have ct= Y(X2+yl),ct'= Y(XII+y'I). Thus

I x' x' x cosh a - ct sinh a cos ifJcosh a - sinh acos ifJ = Y(X'2 +y'2) = ct' - ct cosh a - x sinh a - cosh a - cos ifJ sinh a

./.' C cos ifJ- wcOS'f' = .f ••c-w cos 'f'

This is the relativist formula for aberration: it may be written

sin ifJ'_ifJ -tanh ~ sin ifJ'+ifJ.222

1 Ann. d. Phys. xvii (Sept. 1905), p. 891• e.g. H. E. Ives, Proc. Amer. Phil. Soc. xcv (1951), p. 125; Sc. Proc. R.D.S. xxvi (1952),

p. 9, at pp. 21-2 • if. Vol. I, pp. 368 and 38940

which is the aberration-formula of classical physics.To find the relativist formula for the Doppler effect, we suppose

that K' is an inertial system with respect to which the star IS atrest, and K is an inertial system in which the earth is at rest: andchoose the axes so that the system K' is moving with velocityw (= c tanh a) parallel to the axis of x in the system K. Let ifi bethe angle which the line joining the star to the observer makes withthe x-axis in the system K, and let ifi' be the corresponding anglein the system K'. Then the phase in the system K is determined by

v (t+x cos ifi/ y sinifi)

where v is the frequency of the light as observed by the terrestrialobserver; and as the phase is a physical invariant, we must have

v (t + x cos ~ +y sin ifi) = v' (t' + x' cos ifi; +y' sin ifi')

when v' is the frequency of the light as measured by an observer onthe star. Thus

v {t' cosh a + Z' sinh a + ~ [(x' cosh a + ct' sinh a) cos ifi+y' sin ifiJ}

=v' (t' + x' cos ifi' ; y' sin ifi)

Equating coefficients of t', we have

v (cosh a+sinh a cosifi) =v'

or

w1+- COS",v' C 'f'

~-= -v'(1-- W-cz

)'

This is the relativist formula for the Doppler effect. When only first-order terms in (wlc) are retained, it gives

v=v' (1-~r)41


way. If two events (I) and (2) are considered, and if in an inertialsystem A these events happen at different points of space, whereasin an inertial system B (moving with velocity w relative to A), thetwo events happen at the same point of space, then the Lorentztransformation gives

Since XlB=xsB, these equations give

(ts - tl)A = (ts - tl)Bv(1-~2S)

so the time between the events, measured in system A, is (I _ws/c2)-ltimes greater than the time between the events, measured in system B.

Now certain particles called cosmic-ray mesons, discovered obser-vationally in 1937, disintegrate spontaneously; and it may beassumed that the rate of disintegratIOn depends on time as measuredby an observer travelling with the meson. Thus to an observer whois stationary with respect to the earth, the rate of disintegrationshould appear to be slower, the faster the meson is moving. Thiswas found in 1941 to be actually the case.l

The study of relativist dynamics was begun in 1906, when MaxPlanck 2 found the equations which, according to the new theory,should replace the Newtonian equations of motion of a materialparticle. Considering first the one-dimensional case, let a particleof mass m and charge e be moving along the axis of x with velocityw( =c tanh a) in the system Oxyz, in a field of electric force parallelto Ox. Let 0'x'y' z' be axes parallel to these, whose origin 0' moveswith the particle. The relations between (t, x,y, z) and (t', x',y', z')are

{

ct' =ct cosh a-X sinh a

x' = x cosh a -ct sinh ay'=yz'=z.

The Newtonian equation of motion is assumed to be valid withrespect to the axes O'x'y'z', so the equation of motion of the particleis

d2x'm-=ed':Z'=edxdt'21 B. Rossi and D. B. Hall, Phys. Rev. lix (1941), p. 223• Verh. d. Deutsch, Phys. Ges. viii (1906), p. 136

44


where d~' and dx denote the electric force in the two systems.l Now

dx' dx/dt cosh a - e sinh aedt' = e cosh a - dx/dt sinh a

so, } {(dx/dt) cosh a-e sinh a} dS~cosh a

d2~_dt e cosh a- (dx/dt) sinh a _ dt2~-------.,-e

2dt'2- e cosh a _ ~ sinh a - {e cosh a _ ~; sinh a}I ,

remembering that ~: cosh a - sinh a = O. But

dx . e sinh2a cc cosh a--- smh a=e cosh a----- =-_. --dt cosh a cosh a

and therefore

d2x' d2x (W2)-~ dw d { w }dt'2 = Jt2cosh3 a = 1- C2 dt = di yO _w2/eS) •

Thus the equation of motion is (writing X for the moving force onthe particle, namely ed.),

and extending the investigation to three dimensions, we can showthat if the components of velocity are dx/dt, dy/dt, dz/dt, and if theirresultant is w, then the general equations of motion of a particle acted onby aforee (X, Y, Z) are

d { m dx/dt } _ XJi yO - w2/e2) - ,

d { m dy/ dt } _ YJi y(l - w2/e2) - ,

d { m dz/dt .} _ Zdi y(l - w2/e2) - •

When e -+00, these evidently reduce to the Newtonian equations

(1)

day _m dtl - Y,

1 d. p. 3345


For from (4) we have in this case

few) = (w W2)'

yl--e2

d h k" J mwdw me2 Can t emetic energy = ( w2)! = ( w2) + onstant1-- y 1--

e2 e2

in agreement with (2).Equations (2) and (4) fulfil the prediction made by Poincare i.n

his St Louis lecture of 24 September 1904, that there would be ' anew mechanics, where, the inertia increasing with the velocity, thevelocity of light would become a limit that could not be exceeded.'

The arguments by which Planck derived his expressions for thekinetic energy and momentum of a material particle in relativitytheory were felt to be perhaps not completely cogent. However,three years afterwards, Gilbert N. Lewis (1875-1946) and Richard C.Tolman (1881-1948)1 gave a proof of a very different character.

Consider two systems of reference (A) and (B), in relative motionwith velocity w parallel to the axes of x and x'. Let a ball P havecomponents of velocity (0, -u, 0) in (A), and let an exactly similarball Q have components of velocity (0, u, 0) in (B). Let the ballsbe smooth and perfectly elastic. The experiment is so planned thatthe balls collide and rebound. From the relativist formulae

v'z+wVz=---

1+ v'zwe2

we see that the velocity ofQas estimated by (A) before the collision is

The collision is perfectly symmetrical. But as estimated by (A),the y-component of Q:s velocity changes from uy(l- w2/e2

) to-uy(l-w2/e2), and they-component ofP's velocity changes from-u to u.

We assume that there exists a vector quantity called the momentumdepending on the mass and velocity, which is such that the momen-tum gained by one of the spheres in a collision is equal to themomentum lost by the sphere which collides with it. We assumefurther that this momentum approximates to the ordinary Newtonian

1 Phil. Mag. xviii (1909), p. 51748

j(v)Vz, j(v)vr, j(v) v"

where v= (v2z+V'y+v2.)', and the functionj(v) reduces to the massm when v ->- O. From the law of conservation of momentum, (A)assumes that the ball P experiences the same change of momentumas the ball Q Therefore

where VQ and Vp are the total velocities of Q and Pin (A)'s system.Divide by u.

Now make u tend to zero. Thus

( WI)1..j(w) l-CT 2=j(O)=m

or

so the momentum oj a particle whose mass is m, and which is moving withveloci{y (vz, Vy, v.) is

where v· = vz· + v/ + vz".

Next consider a collision between two elastic spheres, whosemasses are m1 and m2 respectively, and which are moving along theaxis of x with velocities (u1, us) before the collision, and with veiocities(U'l, u' 2)' after the collision. The condition of conservation ofmomentum gives the equation:

Now consider another set ot ~:xes, which are moving relatively tothe first set with velocity c tanh a pai~!!e! tou'l:: ;ucis of x. Let the

49


velocities relative to this second set of axes be denoted by graveaccents placed over the letters, so that for anyone of the u's we have

, u cosh a - C sinh aU = --------,cosh a- (u/c) sinh a

J( 1- ~~)=cosh a- (~/c) sinh aJ (l-~).

Substituting from (2) in the equation

(2)

we obtain

+ mlJ2

V(l- u;~) + m2u'2

v(l-U~:),

ml(ul cosh a - c sinh a) + m2(u2 cosh a - c sinh a)

V( 1 - :~2) V( 1_ U;22)

-= ml(u'l cosh a-C sinh a) + m2(u'2 cosh a-C sinh a)

(U' 2) ( U' 2) .V 1 - ci- V 1 - -C=

Subtracting this equation from equation (1) multiplied by cosh a,and dividing the resulting equation by C sinh a, we have

(3)

This equation shows that if the quantity m (1 - U2/c2)-i be calculatedfor each of the colliding spheres, then the sum of these quantitiesfor the two spheres is unaltered by the impact. We have thereforeobtained a new invariant property. Let us see what correspondsto this in Newtonian dynamics. Supposing that (ul/c) and (U2/C)are small, and expanding by the binomial theorem, we have

... )

or2 3 Ul' 2 Us't mlUl +"B"m1"2 + . . . + t m2U2 + i m2 2" + . . .c c

, 2 U't' , 2 U'.'=tmlU 1 +iml-. + ... +im2U 2 + i m2 -.-+c c

When c-+oo, this equation becomes the ordinary equation of con-servation of kinetic energy in the collision. We therefore describe(3) as the equation of conservation of energy in the relativist theory ()f theimpact, and we call

(save for an additive constant) the kinetic energy of a particle, whosemass at rest is m, which is moving with velocity v. The c2 is insertedin the numerator in order to make the expansion in ascendingpowers of (ulc) begin with the terms [Constant + t mu2] and thus beassimilated to the Newtonian kinetic energy.

Thus Planck's expressions for the momentum and kinetic energyof a material particle were verified. The quantity m is called theproper mass.

We have now to trace the gradual emergence of one of thegreatest discoveries of the twentieth century, namely, the connectionof mass with energy.

As we have seen,! J. J. Thomson in 1881 arrived at the resultthat a charged spherical conductor moving in a straight line behavesas if it had an additional mass of amount (4/3c2) times the energyof its electrostatic field.2 In 1900 Poincare,3 referring to the factthat in free aether the electromagnetic momentum is (1leI) timesthe Poynting flux of energy, suggested that electromagnetic energymight possess mass density equal to (1Ic2) times the energy density:that is to say, E = mc2 where E is energy and m is mass: and heremarked that if this were so, then a Hertz oscillator, which sendsout electromagnetic energy preponderantly in one direction, shouldrecoil as a gun does when it is fired. In 1904 F. Hasen6hrl 4 (1874-1915) considered a hollow box with perfectly reflecting walls filledwith radiation, and found that when it is in motion there is an

l Vol. 1, pp. 306-310, It was shown long afterwards by E. Fermi, Lincei Rend. xxxi, (1922), pp. 184,306,

that the transport of the stress system set up in the material of the sphere should betaken into account, and that when this is done, Thomson's result becomes .

Addi tional mass - ~ x Energy of field.c

The same result was obtained in a different way by W. Wilson, Proc. Phys. Soc. xlviii(1936), p. 736. 8 Archives Neerland. v (1900), p. 252

• Ann. d. Phys. xv (1904), p. 344; Wien Sitz. cxiii, 2a (1904), p. 103951


apparent addition to its mass, of amount (8/3e2) times the energypossessed by the radiation when the box is at rest: in the followingyear 1 he corrected this to (4/3e2) times the energy possessed by theradiation when the box is at rest 2; that is, he agreed withJ. J. Thomson's E=!me2 rather than with Poincare's E=mc2•

In 1905 A. Einstein 3 asserted that when a body is losing energy inthe form of radiation its mass is diminished approximately (i.e.neglecting quantities of the fourth order) by (1/e2) times the amountof energy lost. He remarked that it is not essential that the energylost by the body should consist of radiation, and suggested thegeneral conclusion, in agreement with Poincare, that the mass ofa body is a measure of its energy content: if the energy changesby E ergs, the mass changes in the same sense by (E/eS) grams.In the following year he claimed 4 that this law is the necessary andsufficient condition that the law of conservation of motion of thecentre of gravity should be valid for systems in which electromagneticas well as mechanical processes are taking place.

In 1908G. N. Lewis 5 proved, by means of the theory of radiation-pressure, that a body which absorbs radiant energy increases in massaccording to the equation

dE = e2dm

and affirmed that the mass of a body is a direct measure of its totalenergy, according to the equation 6

E=mc2•

As we have seen, Poincare had suggested this equation but hadgiven practically no proof, while Einstein, who had also suggestedit, had given a proof (which, however, was put forward only asapproximate) for a particular case: Lewis regarded it as an exactequation, but his proof also was not of a general character. Lewis,however, pointed out that if this principle is accepted, then inPlanck's equation of 1906

(Kinetic energy of ~ particle whose) = me2

_ me2

mass when at rest IS m v(1_ ~:)

, Ann. d. Ph,s. xvi (1905), p. 589• The movmg hollow box filled with radiation was discussed further by K. von

Mosengeil (a pupil of Planck), Ann. d. Phys. xxii (1907), p. 867, and M. Planck, &rlinSite. (1907), p. 542, whose formulae essentially involve the general law E=mc!.

• Ann. d. Phys. xviii (1905), p. 639; his reasoning has, however, been criticised; d.H. E. Ives, J. Opt. Soc. Amer. xli, (1952), p. 540

• Ann. d. Phys. xx (1906), p. 627; cr. a further paper in Ann. d. Phys. xxiii (1907),p.371

• Phil. Mag. xvi (1908), p. 705; cf. however the above note on Planck's paper of 1907• A little earlier D. F. Comstock, Phil. Mag. xv (1908)", p. I, had obtained E-imc'

in accordance with the formulae of J. J. Thomson and Hasen6hrl, and had remarkedthat' assuming the loss of mass accompanying the dissipation of energy, the sun's massmust have decreased steadily through millions of years.'

52


the last term, me2, must be interpreted to mean the energy of theparticle when at rest, whereas the difference

represents the additional energy which it possesses when in motion;and therefore the total energy of the particle when in motion mustbe simply 1

For confirmation of this, Lewis referred to experiments byW. Kaufmann 2 and A. H. Bucherer,3 who studied the magnetic andelectric deviations of the f3-rays for radio-active substances. Theoriginal experiments of Kaufmann 4 showed only that for greatvelocities the 'mass' of the electron increases with its velocity ingeneral qualitative agreement with the formula m/ -yI(1 - wi/eS) : butBucherer showed that the formula is accurate to a high degree ofprecision for values of (wle) ranging from 0·38 to 0·69.

The mass of a system can therefore be calculated from its totalenergy by the equation

and the researches that have been described show that in calculatingE, we must include energy resident in the aether. In 1911 Lorentz 5

showed that every kind of energy must be included-masses, stretched

1 Lorentz in 1904 (Arnst. Proc. vi (1904), p. 809] had given the formula

mom~ -----'\1'(1- (w/c)')

for the mass of an electron whose mass when at rest is mo, and which is moving withvelocity w, on the assumption that electrons in their motion experience the FitzGeraldcontraction .

• Gott. N(J(;h. (1901), p. 143; (1902), p. 291 ; (1903), p. 90; Phys. ZS. iv (1902), p. 54·BeTlin Sit •.. xlv (1905), p. 949; Ann. d. Phys. xix (1906), p. 487; cr. also Planck, Verh:d. Deutsch, Phys. Ges. ix (1907), p. 301 ; Kaufmann, ibid. p. 607; Stark, ibid. x (1908),p.14

• BeTl. Phys. Ges. vi (1908), p. 688; Ann. d. Phys. xxviii (1909), p. 513; Phys. ZS. ix(1908), p. 755; cr. also C. Schaefer and G. Neumann, Phys. ZS. xiv (1913), p. 1117;G. Neumann, Ann. d. Phys. xlv (1914), p. 529; C. Guye and Ch. Lavanchy, Arch. des Sc.(Geneva) xlii (1916), p. 286

• It may be mentioned that in pre-relativity days the interpretation placed onKaufmann's experiments was that by means of them it would be possible to find for theelectron the proportion of proper mass (which was independent of velocity) to electro-magnetic mass (which increased with velocity).

• Arnst. Vers/. xx (1911), p. 87

53


strings, light rays, etc. For example, if a system consisting of twoelectrically charged spheres, of charges el and e2, at distance a apart,is considered, then when we calculate the value of E for the system,we do not obtain simply the sum of the values of E for the twospheres separately (as calculated when they are infinitely remotefrom each other), but we must include also a term representingthe electrostatic mutual potential energy of the two charges, namely,(elel/a) : and therefore the mass of the system must include a term 1

(elel/e2a) .Similarly, the mass of a system of gravitating bodies is not the

sum of their masses taken separately, but includes a term representing(l/e·) times their mutual potential energy. 2 Thus, if two Newtoniangravitating particles ml and m. are at rest at a distance a apart,their mass is

where y is the Newtonian constant of gravitation.The equivalence of mass and energy was expressed by Planck in

1908 3 in the form of a unified definition of momentum. The flux ofenergy, he said, is a vector, which when divided by e2 is the densityof momentum. This had long been known in the case of electro-magnetic energy, by the relation between the Poynting vector andthe momentum densitv resident in the aether. But Planck nowasserted that it was u~iversally true, e.g. in the cases of radiation,or of conduction or convection of heat. In the case of a singleparticle of proper-mass m and velocity v, the energy is me2/ y(l- v2/e2),

the streammg of energy is me2v/ y(l - V2/C2), and this divided by c· ismv/ y(1 - v2/e2), which is precisely the momentum of the particle.The unified definition of momentum is a more general expression ofthe equivalence of mass and energy than the equation E = me2, for theconcept of mass becomes more difficult to define when, e.g. momen-tum and velocity are no longer parallel to each other.

Planck's new conception of momentum was soon found to becapable of explaining some paradoxical consequences which could

1 A value not agreeing with this was found by L. Silberstein in 1911 [Plrys. ZS. xii(1911), p. 87], but an error in his method was pointed out by E. Fermi [Rend. Lincei,xxxii (1922), pp. 184, 306], whose work led to the correct value.

• On this problem cf. A. S. Eddington and G. L. Clark, Proc. R.S.(A), clxvi (1938),p. 465; Eddington [Proc. R.S.(A), clxxiv (1940), p. 16] proposed to define the massof a system to be that of a point-particle which would produce the same gravitationalfield as the system at very great distances. This and other definitions were discussedby G. L. Clark, Proc. R.S. E. !xii (1949), p. 412. It was shown hy Josephine M. Gillochand W. H. McCrea, Proc. Camb. ph. Soc. xlvii (1951), p. 190, that in the case of a cylinderrotating freely on its axis, the gravitational mass IS (to a first approximation) the sumof the proper-mass and (I/C') times the kinetic energy; as was to be expected accordingto the general principle of the equivalence of mass and energy.

• Verh. d. Deutsch. Plrys. Ges. x (1908), p. 728; Phys. ZS. ix (1908), p. 828. Thisstatement had been to some extent anticipated (in connection with the moving boxcontaining radiation) by Planck, Herlin Sitz. (1907), p. 542, and by F. Hasenohrl, WienSi!z. cxvi 2a (1907), p. 1391.

54


apparently be deduced from the theory of relativity. One of these,due to Lewis and Tolman,l may be described as follows. Considera rigid bent lever abc at rest, pivoted at b, whose arms ba and be areequal and perpendicular, and suppose that forces Fx and Fy, eachequal to Fo, are applied at a and e in directions parallel to be andba respectively. The system is thus in equilibrium.

Now let the whole system be referred to axes with respect towhich it is moving with velocity w in the direction be. Obviouslyit will still be in equilibrium. But according to the theory of relativity,with reference to the new axes the arm be should eXJ?erience theFitzGerald contraction, and so should be shortened In the ratiov/(l- w2/e2) to 1, while ab has the same length as at rest. Moreover,if force is defined as the rate of communication of momentum withrespect to the time used in the inertial system concerned, we canshow that the values of the forces referred to the new axes are

Thus the forces produce a moment

or

tending to turn the system round b; so apparently it would not bein equilibrium.

The paradox is resolved by the following explanation, which isdue to Sommerfeld and Laue.2 At the point a the force Fx furnishesthe work at the rate wF o. An energy current of this strength entersthe lever at a, travels to b and then passes into the axis of the lever,since the axis does work at the rate - wFo on the lever. Correspond-ing to this flux of energy there is, by Planck's principle, a momentumparallel to ab, of amount (1/e2) times the volume integral of theenergy flux, or (1/e2) • ab . wF o. Due to the existence of this momen-tum there is an angular momentum about a fixed origin 0, lyingin the prolongation of ab, of amount (1/e2) • ab . Ob . wFo, and itsrate of increase with respect to the time is

1.. ab . ~(dOb) . wFo or (1/e2). Fow2 • aboe2 t

Thus we see that the couple (1/e2) • Fow2 • ba, produced by the twoforces F% and Fy, is needed in order to account for the rate of increase

Phil. Mag. xviii (1909), p. SID. Relativity statics is treated fully by P. S. Epstein.Ann. d. Phys. xxxvi (1911), p. 779.

I Laue, Verh. Deutsch, Phys. CeseUs. (1911), p. 513

55


(l/c2) • FOWl. ab of the angular momentum of the lever, and thedifficulty is satisfactorily explained.

It may be remarked that if the lever is contained in a case, whichsupports the axis b of the lever, and also (e.g. by elastic strings attachedto points of the case) provides the forces F% and Fy which act ata and c, then the energy current after leaving the lever at b entersthe case there, and after travelling in the case re-enters the lever bythe elastic string which is attached to a. The energy current is there-fore closed, so the system consisting of case and' lever together hasnot a variable angular momentum. The case and lever in fact exertequal and opposite couples on each other.

This may be regarded as a model of the Trouton-Noble experi-ment,l the electric field bein~ compared to the lever and the materialcondenser to the case. NeIther the elctromagnetic momentum ofthe field nor the mechanical momentum of the condenser is parallelto the velocity, and both therefore need couples in order to preservetheir orientation in translatory motion, but these couples are equaland opposite, and the system condenser plus field requires no couple.

Not long after the }?ublication of Planck's paper of 1906 writerson the theory of relativIty began to take advantage of some develop-ments in pure mathematics, of which an account must now be given.

It was Felix Klein (1849-1925) in his famous Erlanger Programmof 1872 2 who first clearly indicated the essential nature of a vector.Let (P, q, r) be the components of a'vector with respect to the rect-angular axes 0 xy z. Then px + qy+ rz is the product of the lengthsof the vectors (p, q, r) and (x,y, z) into the cosine of the angle betweenthem, and is therefore invariant if the axes of reference are changedby a rotation about the origin to any other set of rectangular axes.Klein regarded all geometry as the invariant theory of some definitegroup, and following him, we can take the property just mentionedas the definition of a vector: that is, a set of three numbers (p, q, r)will be called a vector if px + qy + rz is invariant under the group ofrotations of orthogonal axes. This definition suffices to furnish thelaws according to which (p, q, r) are transformed when the axes ofreference are changed. Since {(x. x) + (y .y) + (z. z)} or (x2 +y2 + Z2)is invariant under a rotation of the axes, we see that (x,y, z) is aparticular vector. And since all vectors are transformed in thesame way, we may say that (P, q, r) is a vector if its components(P, q, r) are transformed like (x,y, z).

Vectors are not the only physical quantities that are related todirection: another class is represented by elastic stresses. If wedenote by (X%,Yx, Z%)the components of traction across theyz-planeat a given point P, by (Xy, Yy, Zy) the components of traction acrossthe zx-plane at P, and by (X:, Y:, Z:) the components of traction

1 cr. p. 29• Programm zum Eintritt in die philosophische Fakultiit d. Univ. zu Erlangen, Erlangen,

A. Deichert, 1872. Reprinted in 1893 in Math. Ann. xliii, and in Klein's Ges. Math.Abhandl. i, p. 460.

56


across the xy-plane at P, then, as is known, we have Zy = Yz, Xz = Zx,Yx= Xy, so we can write

Xx=a, Yy=b, Zz=c, Zy= Yy=f, Xz=Zx=g, Yx=Xy=h,

and the stress can be represented by the six numbers (a, /I, c,f, g, h).Now let the axes of reference be changed by any rotation about theorigin. Then, as is known, if the components of stress at P withrespect to the new axes Ox'y'z' are denoted by (a', b', c',f', g', h'),the expression

ax2 + by2+ cz2 + 2fyz + 2gzx + 2hxy

is transformed into the expressiona' X'2+ b'y'2 + a' Z'2 + 2j'y' z' + 2g' z' x' + 2h' x'y'.

Any set of six quantities (a, b, c, f, g, h) which, when the axes arechanged by a rotation about the origin, changes in this way, thatis, in the same way as the coefficients of a quadric surface, is saidto constitute a symmetrical tensor 1 of rank 2. The analogy with thedefinition of a vector is obvious, and a vector may be called a tensorof rank 1. A quantity which is invariant under all rotations of theaxes of co-ordinates is called a scalar or tensor of rank zero.

Sincex2. X2+y2 .y2+Z2. Z2+2yz .yz+2zx. zx+2xy. xv= (X2+y2+ZI)2

is an invariant for rotations of the system of co-ordinate axes, itfollows that

(x2, y2, Z2, yz, ZX, xy)

is a particular symmetric tensor of rank 2, and since all symmetrictensors of rank 2 are transformed in the same way, we see thata set of6 quantities (a, b, c,f, g, h) constitutes a symmetric tensor of rank 2,if (a, b, c,f, g, h) are transformed tn the same way as (X2,y2, z·,yz, ZX, xy).It is easily shown, for example, that if A, B, C, F, G, H denote themoments and products of inertia of a system of masses with respectto the co-ordinate axes, then (A, B, C, - F, - G, - H) is a symmetrictensor of rank 2.

The definition just given can be generalised, so as to furnisha definition of a tensor of rank 2 which is not necessarily symmetrical.Let (PI' ql, rl) and (P2' q2, r2) be two different vectors. Then a setof nine numbers

I Attention was drawn to the properties of sets of quantities obeying these laws oftransformation by C. Niven, Trans. R. S. E. xxvii (1874), p. 473; cf. also W. Thomson(Kelvin), Phil. Trans. cxlvi (1856), p. 481 and W. J. M. Rankine, ibid. p. 261. Thename tensor (with this meaning) is due to J. Willard Gibbs, Vector Analysis, New Haven(1881-4), p. 57.

57


will be called a tensor of rank 2, if they transform in the same way as

P1P2, q1q2, r1r2, q1r2, r1q2, r1P2, Ptr2, P1q2, QIP2'

So far we have considered only tensors which have invariantproperties with respect to the rotations of a system of orthogonalco-ordinate axes in three-dimensional space. This theory wasgeneralised into a tensor-calculus applicable to transformations incurved space of any number of dimensions by Gregorio Ricci-Curbastro (1853-1925) of Padua, from 1887 onwards: it firstbecame widely known when a celebrated memoir describing it waspublished in 1900 by Ricci and Levi-Civita.1

Let Xl, X2, ••• Xn be any' generalised co-ordinates' specifyingthe position of a point in space of n dimensions. Let n new variablesXl, X2, ••• Xn be introduced by arbitrary equations

Xr=fr(XI' X2, ••• Xn) (r= 1,2, ... n). (1)

Then the differentials of the co-ordinates are transformed accordingto the equations

(r= 1, 2, ... n).

At a point P of the n-dimensional space we can consider various typesof quantities analogous to the scalars, vectors and tensors that wehave already considered.

Firstly, there may be a function of position whose value is un-changed when we perform the transformation (1). Such a functionis called a scalar, or tensor of rank zero.

Secondly, we consider a set of n numbers (VI, V2, ... vn),which are defined wIth respect to all co-ordinate systems and which,when we perform the transformation (1), are transformed in thesame way as the dXr, so that

Vr='; aXrve (1 2 )L r= , , .•. ne=l aXe

whence

(r= 1, 2, ... n).

Such a set of n numbers is called a contravariant tensor of rank 1, orcontravariant vector, and the numbers are called its components.

Next, consider sets of n numbers (Xl' X2, ••• Xn), whichare such that if (VI, V2, ... Vn) is any contravariant tensor ofrank 1, the sum Xl VI + X2 V2 + . . . + Xn vn is a scalar. Such

1 Math. Ann. liv (1900), p. 125; cr. J. A. Schouten, )ahresb. d. Deutsch. Math.-Verein.xxxii (1923), p. 91


a set of n numbers is called a covariant tensor if rank 1 or covariantvector.

Since

we have- '" ox, Xwhence Xk = L <l- r.

r UXk

The covariant or contravariant character is indicated by placing theindex in the lower or upper position respectively. In Euclideanspace, for rotations of rectangular axes, there is no distinctionbetween contravariant and covariant tensors.

If, at the point P of the n-dimensional space, we have n2 numbers(Vll, V12, ••• vnn

) which, when we perform the transformationof co-ordinates, are transformed like (PIQ;, plQ2, ... pnQn),where (Pl, ... pn) and (Q\ ... Qn) are two different contra-variant tensors of rank 1, then (Vl1, va, ... vnn

) are said tobe the components of a contravariant tensor if rank 2. Similarly n2

numbers (Xu, X12, ••• Xnn) which transform like (XIYI, XIYh

•.. XnYn) where (Xl' .. Xn) and (YI, ••• Yn) are two differentcovariant tensors of rank 1, are said to be the components of acovariant tensor if rank 2 ; while n2 numbers (WlI, W12, W\, ... W")which transform like (PIXI, PlX2, P2XI, ... P"X,,) where (Pl, P2,... pn) is a contravariant tensor of rank 1 and (Xl' X2, ... X,,)is a covariant tensor of rank 1, is called a mixed tensor if rank 2.Tensors of rank greater than 2 are defined in a similar way. Atensor whose typical component is, say, X;', is often denoted by

(X;').Consider a tensor such that any two of its components, which

may be obtained from each other by a simple interchange of twoindices, are equal to each other; thus, Vpq =Vqp

• If this propertyholds for anyone system of co-ordinates, it will still hold after anychange of the co-ordinate system, as is evident from the equationsof transformation. Such a tensor is said to be symmetric. If a tensoris such that two components which may be derived from each otherby a simple interchange of two indices are equal in magnitude butopposite in sign, thus Vpq = - Vqp

, the tensor is said to be skew.This property also holds in all systems of co-ordinates, provided itholds in anyone system.

Two tensors of the same kind (contravariant, covariant or mixed)and of the same rank, are said to be equal if their correspondingcomponents are equal in all co-ordinate systems. This is the caseif the corresponding components are equal in anyone co-ordinatesystem.

Consider the transformation of tensors when the co-ordinates59

Lorentz transformation (writing

dX1= dxo sinh a+ ax1 cosh a,dx. = ax •.

X.o = XO! = X02 cosh a + Xu sinh aXso = Xos = Xos cosh a + X1Ssinh aXu = Xu = X02 sinh a + Xu cosh aXu = Xu = Xos sinh a + X1S cosh a

for any covariant skew tensor of rank 2 :X01= XOh X01= X02 cosh a + Xu sinh a,

Xu = X2S' X31 = X31 cosh a + Xso sinh a,


are subjected to the particularct=xo, X=X1, y=xJ, z=xs)

dxo = axo cosh a + dX1 sinh a,dXJ= dxz,

It is found at once that:for any contravariant vector :

Jo=Jo cosh a+ J1 sinh a, Y =]0 sinh a+ J1 cosh a, J2 =j2, Y =Js.

for any covariant vector:

J 0=J0 cosh a - J1 sinh a, J 1= - J0 sinh a + 11 cosh a, J J =Jz, J s=J•.for any covariant symmetric tensor of rank 2 :

Xoo=Xoo coshJ a+2X01 cosh a sinh a+Xn sinhJ aXll=XOO sinh! a+2X01 sinh a cosh a+Xn cosh! aXu = Xu, Xss = Xss, XSJ = Xu = X23

X10= X01= Xoo cosh a sinh a+ X01 (cosh! a+ sinhJ a) + Xnsinh a cosh a

Xu = Xos cosh a_ +Xu sinh aXu = Xu cosh a

+Xos sinh a

It is evident from these last equations that a six-vector,! such as isconstituted by the electric and magnetic intensities in vacuo, is askew tensor of rank 2. We can write

L=~ L=~ L=~ L=~ L=~ L=~

From the definition of a tensor, it is evident that if two tensorsof the same type are taken, say (X~),and (V:'), then the quantitiesformed by adding corresponding components of these tensors

Z~=X;' +V;'are the components of a tensor of the same type, which is called thesum of the tensors (X~) and (Vr~).

1 cr. p. 3560


Moreover it is evident from the definitions that if two tensors,say of rank A and rank fl-, are given in n-dimensional space, and ifwe multiply each of the nA components of one by each of the n"components of the other, then the nH" products so formed are thecomponents of a new tensor of rank (A + fl-), thus:

The tensor (U.;.) is called the outer product of the tensors (X.;)and (Y~). It may properly be called a product, since the distri-butive law

X(Y +Z) =XY +XZ

holds. We can form in this way the outer product of any numberof tensors.

An arbitrary tensor cannot in general be expressed as an outerproduct of tensors of rank 1, since there would not be enoughquantities at our disposal to satisfy all the conditions. Thus, ifa tensor (X~) is given, we cannot in general find tensors of rank1 (Yr), (Z.), and (VP), such that

X~ = Yr Z. VP (p, r, s = 1, 2 . . . n) :

but the sum of any number of outer products of this type will be atensor of the type (X~) ; and by taking the number of such productssufficiently great, we shall have enough quantities at our disposalto represent any tensor (X~) in the form

X~ = Yr Z. VP + Hr K. LP + Er F. GP + : ••

Next consider a tensor which has both contravariant and covariantindices, e.g. (X;'). Make one of the upper or contravariant indicesidentical with one of the lower or covariant indices, and sum withrespect to this index, thus:

n

2:1'=1

Then we can show that the numbers thus obtained, when k, q, r = 1, 2 ... n,are the components of a new tensor (Y;), which is two units lower in

( Ik) . ( Ik)rank than Xpqr' To prove thIS, we remark that Xpqr can beexpressed as a sum of outer products of tensors of rank 1, and thetheorem will therefore evidently be true in general if it is true for

61

62


the case when (Xpqll:r) • • l d f f kIS a Stng e outer pro uct 0 tensors 0 ran1, say

Then we have

i X:; = i (Y"U") ZqTrVl::p-l 1'=1

R

and since L T"U" is a scalar, these quantities are the components,,-1of a tensor of type (Yq~); which establishes the theorem. Thisprocess is called contraction.

By forming the outer product of any number of tensors, and thencontracting (once or oftener) the tensor thereby obtained, we obtainresults such as

L (abe rst . .. af3y ••• Imn •• ') = Zr st. • • l m n ••.

abc, ••. af3y . .. X af3y pen ••• Yabc ••• jhk . • • pen ••• jhk •••

This process is called transvection.The spaces we consider will generally be supposed each to possess

a metric, that is to say, there will be an equation expressing an elementds of arc-length at any point of the space in terms of the infinitesimaldifferences of the co-ordinates between the ends of the arc-element:thus in ordinary Euclidean three-dimensional space with rectangularco-ordinates (x,y, z), we have

(ds)2 = (dx) 2 + (dy)2 + (dZ)2,

and with spherical-polar co-ordinates (r, 8, c/», we have

(ds)2 = (dr) 2 + r2(d8) 2 + r2 sin2 8(dc/» 2.

We assume generally that the square of the line-element ds isa homogeneous quadratic form in the differentials of the co-ordinates.These differentials will be written (dxl, dx2, ... dxR

), the indexbeing placed above since (dxl, ..• dxR

) is a contravariant vector:thus

(ds)2 = L gpqdx"dxq•

1',q

Since (dS)2 is a scalar, it is obvious from this equation that thenumbers g1'q (P, q= 1, 2, ... n) must be the components of a co-variant symmetric tensor of rank 2, (g"q); this is called the covariantfundamental tensor.


Let g denote the determinant I/gpqil of the coefficients gpg, andlet gpq denote (l/g) times the co-factor of gpq in g, so that L gprgP'l = Srq,

1'_1where Srq is equal to 1 or 0 according as q is equal to, or differentfrom, r. Then

L gq.gprgpq = L gq.S/ = g •••p. '1 '1

Now if Xp and Yp are two arbitrary covariant vectors, and if

so that Xr is a contravariant vector, we have

"'" gpqX Y _ "'" 1''1 r ,L I' q-Lgprgq.g Xypq we

= Lg ••Xry,TJ

which is a scalar: and therefore the gpq are the components of acontravariant tensor of rank 2. It is called the contravariant funda-mental tensor.

Moreover, if UP is a contravariant vector, and Xq is any covariantvector, we have

L S/UPXq = L UPXp = a scalarpq I'

and therefore (Spq) is a tensor of rank 2, covariant with respect tothe index p and contravariant with respect to the index q. It iscalled the mixed fundamental tensor.

By aid of the fundamental tensor (gpq) we can derive from anycovariant tensor (Xp1"1 ••. Pm) a contravariant tensor of the samerank by writing

'1'1 '1 "'" 1''1 pq ••• 1''1X 1 I··· m = L g 1 1 g I I g m m Xp p ••• Pm1 I •

P., P~1 .•. Pm

It is easily shown that this equation is equivalent to

"'" Xq'1 ••• '1

XI' P •• , I' = L gp,q, gp,ql ••• gPmqm 1 I m1 I m 1111 2't ... fJm

Thus to every contravariant tensor we can correlate a definitecovariant tensor; and we may say that the distinction betweencovariant and contravariant tensors loses most of its importance

63


when the fundamental tensor is given, i.e. in a metrical space, sinceit is not the tensors that are essentially different, but only theirmode of expression, i.e. their components. For example, we regard(gP2), (gP2)' and (op2), as essentially the same tensor.

1f two vectors (X) and (Y) are such that when (X) is expressedin covariant form (Xp) and the other in contravariant form (Y2),we have

then the two vectors are said to be orthogonal.After this rather long excursus on Ricci's tensor calculus, we can

return to physics. A contribution of great importance to relativitytheory was made in 1908 by Hermann Minkowski (1864-1909).1Its ostensible purpose, as indicated in its title, was to show that thedifferential equations of the electromagnetic field in moving ponder-able bodies under the most general conditions (e.g. of magnetisation)can be derived from the differential equations for the same systemof bodies at rest, by the principle of relativity: and to criticise someof the formulae that had been given by Lorentz. But these werenot actually the most important elements in the paper; the greatadvances made by Minkowski 2 were connected with his formulationof physics in terms of a four-dimensional manifold, the use of tensorsin this manifold, and the discovery of some of the more importantof these tensors.3

The phenomena studied in natural philosophy take place eachat a definite location at a definite moment, the whole constitutinga four-dimensional world of space and time. The theory of relativityhad now made it clear that the separation of this four-dimensionalworld into a three-dimensional world of space and an independentone-dimensional world of time may be effected in an infinite numberof ways, each of which is distinguished from the others only bycharacteristics that are merely arbitrary and accidental. Inorder to represent natural phenomena without introducing thiscontingent element, it is necessary to abandon the customarythree-dimensional system of co-ordinates, and to operate in fourdimensions.

If (tl, Xl' Yl' ZI) and (t2, X2, Y2, Z2) are the time-and-space co-

1 GOtt. Ntu:h. (1908), p. 53; cr. also Malh. Ann. Ixviii (1910), p. 472I Minkowski had been to some extent anticipated by Poincare, who had substantially

introduced the metric

•ds' = e"dl' - dx' - dy' - dz" - - :E dxr"r=1

(wherex,=x, xl-y, XI=Z, x,-ct v-I) in Rend. eire. Palermo, xxi (1906), p. 129., The principle of treating the time co-ordinate on the same level as the other

co-ordinates was introduced and developed simultaneously with Minkowski's paper byR. Hargreaves, (Camb. Phil. Trans. xxi (1908), p. 107]: his work suggests the use ofspace-time vectors just as Minkowski's does. For comments on this point, cr. H. Bateman,Phys. Rev. xii (1918), p. 459.

is invariant under all Lorentz transformations, and therefore hasthe same value whatever be the inertial framework of reference. Thisquantity is therefore an invariant of the two point-events, whichis the same for all observers: and we can make our four-dimensionalspace-time suited to describe nature when we impose a metric on it,which we do by taking the interval (the four-dimensional analogue oflength) between the two events (tl, Xl,Yl, Zl) and (/2' Xh)/2, Z2) to be 1

Taking any point in the four-dimensional manifold as origin, the cone

x2+ y2 + Z2 _ c2t2 = 0

which is called the null cone, partitions space-time into two regions,of which one is defined by the inequality

and includes the hyperplane t =0: the directions at the ongmsatisfying this inequality are said to be spatial: directions at theorigin in the other region are said to be temporal. Lorentz trans-formations are simply the rotations and translations in this manifold.

Now consider tensors in the manifold.Minkowski had not properly assimilated the Ricci tensor-

calculus as applied to non-Euclidean manifolds, and in order tobe able to work with a space of Euclidean type, he used the deviceof writing x, for cty - I (the space-eo-ordinates being denoted byXl) X2, xs), so that the expression

which is invariant under all Lorentz transformations, became

1 The metric of space-time thus introduced is that of a four-dimensional Cayley-Klein manifold which has for absolute (in homogeneous co-ordinates)

x'+y'+z'-c't' - O}w' - 0

a double hyperplane at infinity containing a quadric hypersurface, which is real but withimaginary generators, like an ordinary sphere.

65


this enabled him to take as his metric(ds) 2 = (dXl) 2 + (dX2)2 + (dX3) 2 + (dx() ,

which defines a four-dimensional Euclidean manifold.lIt is, however, simpler to work with the real value of the time,

and to express Minkowski's results in terms of tensors which existin the non-Euclidean four-dimensional manifold we have introduced,whose metric is specified by

(ds) 2 = c2(dt)2 _ (dX)2 _ (dy)2 - (dz) 2

which we may write(ds) 2 = (dXO)2 _ (dX1)2 _ (dX2)2 _ (dx3) 2

sogOO= 1, gl1=g22=g33= -1,gOO= I, gll = g22 = g33 = _ 1.

His greatest discovery 2 was that at any point in the electromagneticfield in vacuo there exists a tensor of rank 2 of outstanding physicalimportance, which in its mixed form (Epq) may be defined by theequation

where Xpq is the electromagnetic six-vector, that is to say, if(dx, dy, dz) and (hx, hy, hz) are the electric and magnetic intensitiesrespectively, then 3

dX=X01= -X01, dy=X02= -XJ2, dz=X03= -X03

hx = X23= X23,

Substituting in the equation which defines Ep\ we find the values ofthe components of this tensor, namely,

Eoo = i;. (dx2 + dy2 + dz2 + hx2 + hy2 + h2z) :

this represents the density of electromagnetic energy, discovered byW. Thomson (Kelvin) in 18534 ;

E01= i;(dyhz-dzhy), E02= 411T(dzhx-dxhz), E03= ;};.(dxhy-dyhx):

1 Minkowski's use of x. ~ctv - I led some philosophers to an outpouring ot' meta-physical nonsense about time being an imaginary fourth dimension of space.

sloe. cit., equation (74)• Xpq is immediately derived from X'q by the formula

Xpq =:E gps gq' X·'.I.

• cf. Vol. I, pp. 222, 22466

( - Elo, - Ezo, - EsO) represents c times the density of electro-magnetic momentum, discovered by J. J. Thomson in 1893 a;

Ell =8~(dz' - dy'- dz' + kr' -hy' - hz'),

and similarly for E; and E,";

Ess = Ess =4-l(dydli+ hyhli),• 1T

and similarly for Esl, EIs, EIs, Eal.

The nine quantitiesEll Esl EslEla Esz Es!Els Ezs Ess

represent the components of stress in the aether, discovered byMaxwell in 1873.3 Thus, each component if the tensor E/ has a physicalinterpretation, which in every case had been discovered many yearsbefore Minkowski showed that these 16 components constitute atensor of rank 2. The tensor Epq is called the energy tensor of theelectromagnetic field.

Since Eqp = L gprE/ = gppEl for this metric, we have

Eop = EoP, E1P = - EIP, Ezp = - El, Esp = - EsP,

and hence we find EOl = Elo and generally Epq = Eqp, that is, Epq isa symmetric tensor.

Moreover we can show that if p is the density of electricity andv its velocity, then

oEoo oEol oEoz oEos p-- +-- + --+ -- = - -(uxdx+ uydy+ uzdz)OXO ox I OXZ Oxs C

(A)

oEsO+oEsl +q~sZ +oEs~ __ p (d h h )- Ii+ Ux y - Uy x •OXO Oxl OXS Oxs C •

1 cf. Vol. I. pp. 313-4 I cf. Vol. I. p. 317 I cf. Vol. I, pp. 271-267


The first of these equations is

~ 81 (d,,' + d,,'+dz' + k;:' +hr' +hz') +:- 41 (dyh. - d.hy)Cut 1T uX 1T

+~_-l (d.h:-d:h.) + ~ _!- (dxh"-d,,h:)Oy 41T 0<; 41T

or

a/at (density of electromagnetic energy) + a/ox (x-component of fluxof electromagnetic energy + 0/ oy (y-component of flux of electro-magnetic energy) + 0/0<; (<;-component of flux of electromagneticenergy)

or

rate of increase of electromagnetic energy in unit volume + rate atwhich energy is leaving unit volume= - (work done by the electromagnetic forces on electric chargeswithin the unit volume)and this is clearly nothing but the equation of conservation of energy.Similarly the other three of the equations (A) are the equationsof conservation of x-momentum, y-momentum and <;-momentumrespectively.1

In an appendix to his paper,2 Minkowski threw a new light onthe equations of the relativistic dynamics of a material particle,which had been discovered by Planck two years earlier. 3 Denotingby (x, y, z) the co-ordinates of the particle at the instant t, he intro-duced the notion of the proper-time T of the particle, whose differentialis defined by the equation

It is evident from this equation that dT is invariant under all Lorentztransformations of (t, x,y, z), i.e. it is, in the language of the tensor-calculus, a scalar. Now writing xO=ct, x1=x, x2=y, x3=z, weknow that

(dxO, dx\ dx', dx3)

1 On the energy tensor cf. also A. Sommerfeld, Ann. tf.: Phys. xxxii (1910), p. 749 ;xxJciii(1910), p. 649; and M. Abraham, Palermo Rend. xxx (1910), p. 33

• loc. cit. • cf. p. 44-68


is a contravariant vector, and therefore

(dxO qXl dx2 dX3)

dr' dr' dr' dr

is a contravariant vector. But

where v denotes the velocity of the particle. Therefore

e Vx Vy

( V3)'V I-Ci

are the components of a contravariant vector.Now Planck had shown that if m is the mass of the particle, its

energy E is me2(1 - vl/e2)-i, and its components of momentum are

mv.p.= ( VI)'V 1--

cl

Thus(Efc, pz, py, p.)

is a contravariant vector. This is called the energy-momentumvector.

The Newtonian and relativist definitions offorce may be comparedas follows. In Newtonian physics the momentum (px,Pr,p.) is a vectorand the time t is a scalar, so dpz/dt, dpy/dt, dp./dt is a vector, namely,the Newtonianforce. In relativist physics, as we have seen, insteadof a momentum vector (px, Pr, p.), we have the contravariantenergy-momentum vector (Efe, px, Pr, p.), or in its covariant form(Efe, - pz, - Pr, - p.), and the scalar which takes the place of thetime is the interval ot proper-time,

Thus it is natural to represent a force in relativity by the covariantvector

dP.)d-r •

Comparing these results, we see that we must take the last threecomponents of the relativist force to be

( v2)-l (v2)-t (v2)-!Fl= 1-(2 X, F2= 1-(2 Y, Fs= 1-2- Z,


Now the equations of motion of a particle, as found by Planck, were

d f ( V2)-t dx} _ d {( V

2)-! dy}_m-l1-- - -x m- 1-- --ydt c2 dt' dt c2 dt '

m_d f(l _ ~)-l~<:}= Zdt L c2 dt '

dh= Ydt '

#:=Xdt '

or

and then the last three relativist equations of motion will be

d2x d?y d2zm dT2 = Fl, mdT2 = F2) mdT2 = Fa.

Since

is a covariant vector, the first relativist equation of motion mustevidently be

This completes Minkowski's set of equations of motion. The last equationmay be written

2 d2t dEmc dT2 =dt '

where E is the energy; which is evidently true, since E=mc· dt/dT.

Since

we havedE=Xdx+ Ydy+ Zdz,

or

( V2)-J..- edT I - e2 2 F 0 = FIdx + F2dy + Fadz

70

or


foedt+ F1dx+ Fsdy+ Fsdz=O

an equation which may be expressed geometrically by the statementthat the vector (F0, F I, Fs, Fs) is orthogonal to the vector which representsthe veloei!)' of the particle, namely,

(dt

e (h.'dxdT'

dydT'

dZ)dT •

We can now obtain a simple expression for the ponderomotiveforce on a particle of charge e and velocity v in electromagnetic theory.In Newtonian physics the three components are

e (d,,+~ dz_!!:! dx)c dt c dt '

e (dt+ '!x dx _ ~ dy).c dt c dt

The corresponding force in relativity theory will have for its lastthree components these quantities multiplied by dt/dT. So if therelativist force is (Fk), we have

F1 =~ ( dx ~~ + h,;~ - h"~~) =~ (XlO Vo + Xu vs+ XIS VS)

where dxdT'

dy(l-:;"

is the contravariant vector representing the relativist velocity ofthe particle. This may be written

and similarly we have

These are the last three components of the covariant vector whichis obtained by transvecting the electromagnetic six-vector X"q withthe particle's velocity Vq• Clearly the first component of the forcemust be the first component of this transvectant: and therefore therelativist force on a partzcle of charge e and ve/oei!)' V'l is .

Fk=~2: Xk" V'l.q

The fact that energy-density occurs as the component Eooof Minkowski's energy-tensor, while energy occurs as the first

71

(1)


component of his energy-momentum vector, leads naturally to aninquiry into the connection between these two tensors. This canbe investigated as follows.

Consider a system occupying a finite volume and involving energyof any kind (e.g. electromagnetic energy, or stress-energy, or gravita-tional energy) for which we can define an energy tensor T pq suchthat Too is the energy-density, (T 01, T os, T 03) is (llc) times the fluxof the energy, (T1°, Tso, T3°) is (-c) times the density of momentum,and (Til, Tss, T33, TS3, T3s, T13, T3l, Tsl, TIS) are the componentsof flux of momentum, just as in the case of Minkowski's energy-tensor of the electromagnetic field in vacuo: and suppose that thefollowing conditions are satisfied :

(i) the system is rigidly-connected, and is considered in the firstPlace as being at rest:

(ii) Its state does no vary with the time:(iii) there is conservation of momentum, so

a~,l + aT,S + a1!3 = 0 (r= 1, 2, 3)CJX By oz

(iv) there is no flux of energy in the state of rest, so

Tor =0 (7= 1,2,3). (2)

From (1) we have

HIT1ldx dy dz= HI {:X(XT11) + ~(XT1S) + ~ (XT13) }.dx dy dz

where the integration is taken over the whole volume occupied bythe system and therefore

HIT1ldx dy dz= Hx(lTl1+mT,s+nT13)dS

where the last integral is taken over a surface S enclosing the wholesystem, and (l, m, n) are the direction-cosines of the outward-drawnnormal to S.

If we suppose the surface S so large that it includes the wholeof the space in which there are any sensible effects due to the system,then T pq is zero on S, and therefore the last integral vanishes: sowe have

(3)

Now suppose that (t, x, y, z) is the frame of reference relative towhich the system is at rest, and let (1,x,}, z) be a frame of referencesuch that relative to it, the system is in motion parallel to the axis

72


of X with velocity w = c tanh a. The axes of X, ji, z, are taken to beparallel to the axes of x, y, Z respectively. Then the two sets ofco-ordinates are connected by the equations

t=lcosh a-(xJc) sinh a.x = X cosh a - ci sinh a.y=j, z=z,

and the equations of transformation of the mixed tensor T,/ give

Too= cosh2 a ToO - c sinh a cosha T 1 0 +~cosha sinh a.To1 - sinh 2 a T 11

= cosh2 a Too- sinh 2 a Til

by (2). Thus

HJToo dXdj dz = JH (cosh2a Too- sinh2a T11)dx dj dz

= H J (cosh2 a Too- sinh2 a.T 11) sech a.dx dy dz,

since o(x,y, z)Jo('X,y, z) = cosh a, it being understood that x,Y; Z aremeasured over the field at a constant value of t. So by (3),

JHToo dx dj dz= cosh aHJToo dx dy dz.

Now let U = HJToo dx dy dz, so U represents the total energy of the

system when at rest. Then since cosh a.= (1 - w2 /c2)-t, the result nowbecomes:

Total energy associated with the moving system =U (1 - Wl/Cl)-t.

This may be regarded as an extension, to systems of finite size, ofthe formula that in relativity theory the energy of a particle ofproper-mass m, moving with velocity w, is

Now consider the momentum. The equation of transformationof a mixed tensor of rank 2 gives

73


since Tol and Tlo are zero. Therefore

HJTIO dx dJ dz= - ~ sinh a cosh aJH (ToO-TIl) ax dj di.

= -~ sinh aJH(ToO-TII) dx dy rk

= - ~sinh aJHToo dx dy dz, by (3)

Now - T10 represents the density of x-momentum. Therefore thetotal momentum of the moving system parallel to the x-axis is

w U

v(1-~:)ca·This may be regarded as an extension, to systems of finite size, ofthe formula for the x-component of momentum of a particle.

The above analysis shows how the components of the energy-momentum vector (now no longer restricted by the condition thatit is to apply only to a single particle) can be derived from thoseof the energy-tensor. It is evident that whereas the energy-tensoris localised (i.e. each of its components is a function of position inspace), the energy-momentum vector is not localised.l

Before the discovery of relativity theory, physicists were accus-tomed to think of energy not as a component of a tensor, but as ascalar: and indeed even in relativity theory, energy as observed bya particular observer is a scalar. For let an observer be moving alongthe axis of x with velocity v, so that the covariant vector representinghis velocity, namely,

(c dt _ dx _ dy _ dz) (l:. I: I: 1:)dT' dT' dT' dT or !oO, !ol, !os, !os

is given by

( V2)-J..go=c 1-- 2c2 ,

1 On the relation of the energy-momentum vector of a particle to the energy-tensorof a continuous field, see further H. P. Robertson, Proc. Edin. Math. Soc.(2) v (1937),p. 63, and M. Mathisson, Proc. Camb. Phil. Soc. xxxvi (1940), p. 331.

74

(A)

epa = -a.,ep1= -ax, ep2= -ay,75

epo= ep,

v-WI_vW

c2

Now the relative velocity of the particle and the observer is, by theelativist formula,

Now if ep is the electric potential, and (ax, ay, ax) the vector-potential,we have

oep oaxdx=----ox cot'

rr'he transvectant of these vectors, namely, go'1}° + g1'1}1+ g2'1}2+ gs'1}s,

is a scalar: its value is

and the energy of a particle moving with this velocity relative tothe axes of reference is

mc2 {I _ (v - W) 2 } -!c2(1- VW/c2)2


l.et a particle of proper-mass m be moving in the same straight lineith velocity w, so that its contravariant energy-momentum vector is

which reduces at once to the expression (A). Thus we see that theenergy of an observed particle may properly be regarded as a scalar, being thetransvectant of the particle's energy-momentum vector and the observer'svelocity.

A vector which is of importance in electromagnetic theory maybe introduced in the following way. We have seen that the electricintensity (dx, dy, dx) and the magnetic intensity (hx, hy, hx), at a pointin free aethcr, are parts of a six-vector

and four similar equations. The question therefore suggests itself,what is the character of the potential ep, ax, ay, ax, from the pointof view of the tensor-calculus? The answer, which is easily verifiedby examining the effects of Lorentz transformations, is that if


then (cPO' cPh cPI' cPa) is a covariant vector: its connection with the six-vector is given by the equations

It was discovered in 1915 by D. Hilbert 1 that the energy-tensorof a system can be expressed in terms of the Lagrangean function ofthe system. This theorem was developed further by E. Schrodinger I

and H. Bateman 3 in 1927: the rule was given by Schrodinger asfollows:

Let (ao, aI, ai, as) be one of the four-vectors on which the LagrangeanL depends (as e.g. the Lagrangean in electromagnetic theory dependson the electromagnetic potential-vector), and let apq denote the derivativeof ap with respect to the co-ordinate Xq: then the components of the energy-tensor are given by

(3 oL 3 oL oL)E/= 2: 2:a,p ~ + 2:apl -~-+ap ~ -spq L,

'_0 ua Iq 1-0 uaql uaq

where spq = 0 or 1 according as q is, or is not, different from p, and thesummation is taken over all the four-vectors a.

For example, consider the electromagnetic field in free aether,for which the Lagrangean function is

L = 8~ (dz'+ d,/+ d,' -hz' - h,t' -h.'),

or, if (ao, aI, aI, as) denotes the covariant electromagnetic potentialvector,

1L=--817 }

from this we have

8L =~ Xd

8a1l:n 417

1 GOtt. Ntuh. (1915), p. 395; cr. also F. Klein, GOtt. Nach. (1917), p. 469 and Hilbert,Math. Ann. xcii (1924), p. I

I Ann. d. Phys. !xxxii (1927j, p. 265I Proc. Nat. Acad. Sci. xiii (1927), p. 326

76


hus Schrodinger's formula gives

which is the usual formula for Minkowski's energy tensor.

77

Chapter IIITHE BEGINNINGS OF QUANTUM THEORY

At the end of the nineteenth century the theory of radiation wasin a most unsatisfactory state. For the energy per cm.S of pure-temperature or black-body radiation, in the range of wave-lengthsfrom Ato A+ dA, two different formulae had been proposed. Firstly,that of Wien,l

E= CA-5e-b/ATdA

where Ais wave-length, T is absolute temperature, and band Careconstants. This formula is asymptotically correct in the region ofshort waves (more precisely, when AT is small) ; but, as O. Lummerand E. Pringsheim showed,2 is irreconcilable with the observationalresults for long waves. Secondly, that of Rayleigh and Jeans S

E=81TkTA-4dA

where k is Boltzmann's constant; which, as shown by the experimentsof Rubens and Kurlbaum,4 is asymptotically correct for the longwaves, but is inapplicable at the other end of the spectrum. Whatwas wanted was a formula which for the extreme limits A..•.0 andA..•.00 would tend asymptotically to Wien's and Rayleigh's formulaerespectively, and which would agree with the experimental valuesover the whole range of wave-lengths.

In the spring and summer of 1900 attempts were made to con-struct such a formula empirically by M. Thiesen,6 by O. Lummerand E. Jahnke,6 and by O. Lummer and E. Pringsheim.7 Theseformulae were of the type

E = CT5-PA -Pe-b/(AT)"

which for 11-=5, v= 1, gives Wien's law, and for 11-=4, v= 1, b=O,gives Rayleigh's.

The correct law was first given by Max Karl Ernst Ludwig Planck(1858-1947) in a communication which was read on 19 October1900 before the German Physical Society.8 Planck was the son of

(1900), p. 65(1900), p. 163(1900), p. 202

78

• Verh. d. deutschphys. Ges. i (1899), p. 215; ii (1900), p. 163• loc. cit.• Ann. d. Phys. iii (1900), p. 283

1 Vol. I, p. 381• Vol. I, p. 384• Verh. d. deutsch. phys. Ges.7 Verh. d. deutsch. phys. Ges.• Verh. d. deutsch. phys. Ges.

THE BEGINNINGS OF Q.UANTUM THEORY

professor of law at Kiel, later translated to Munich; he waseducated at the University of Munich, but for one year attended

e lectures of Helmholtz and Kirchhoff at Berlin. After four yearsas professor extraordinarius at Kiel, he was called in 1889 to succeedKirchhoff at Berlin, where the rest of his academic life was spent.

In the study of pure-temperature radiation, his starting-pointas the known fact that in a hollow chamber at a given temperature,

the distribution of radiant energy among wave-lengths is altogetherindependent of the material of which the chamber is composed;and he was therefore free to suppose the walls of the chamber toave any constitution which was convenient for the calculations,

so long as they were capable of absorbing and eInitting radiation,and thereby making possible the exchange of energy between matternd aether. He chose them to be of the simplest type imaginable,

namely an aggregate of Hertzian vibrators,l each wIth one properre9.uency. Each vibrator absorbs energy from any surroundmgradIation which is nearly of its own proper frequency, and acts as

resonator, eInitting radiant energy.He first calculated (by classical electrodynamics) the average

absorption and emission of a vibrator of frequency 2 v which is:mmersed in, and statistically in equilibrium with, a field of radia-tion, and found that if the average energy-density of the radiation,U~ the interval of frequency v to v +dv, is E, then 3

(1)

where U is the average energy of the vibrator.While most of the other workers on radiation were attempting

to find the relations between energy, wave-length and temperature,by direct methods, Planck, who was a master of thermodynaInics,felt that the concept of entropy must playa fundamental part: and

e exaInined the relation between the energy of a vibrator and itsentropy S, showing that if S is known as a function of U, then theaw of distribution of energy in the spectrum of pure-temperature

radiation can be determined.We have, from thermodynamics, for a system of constant volume,

dS=dUT

or dS 1dU=T' (2)

while Wien's law of radiation, namely

E = av3e-fMTdv

1 This, of course, does not mean (as it has sometimes been wrongly interpreted tomean) that actual matter necessarily has this character.

• It will be remembered that v is the number of oscillations in one second, that is,, multiplied by the wave-number 1/>..

• Ann. d. Phys. i (1900), p. 69, equation (34) : Phys. ZS. ii (1901), p. 530

79

d2S ConstantdU·- U·

which again is a case of (4). From (4) we have, by (2),

1 _ dS _ C t I (Const. + U)T - dU - ons. og U

This equation does not give the way in which the frequency ventersinto the formula for U. But as Wien had shown in 1893,2 E must beof the form T6ef>(T"A)d"A, or vst/J(v/T)dv, so by (1), U must be ofthe form

(5)

(4)

(3)

U = Const .•eConst•rr - 1

d2S ConstantdU·= U

or


requires by (1) that we should have

U=yve-fJ-rr

where a and f3 are constants. This is the simplest of all the expres-sionswhich give dS/dU as a logarithmic function ofU (as suggested bythe probability theory of entropy), and which for small values ofU agrees with equation (3). Moreover, if Rayleigh's law of radiationhad been taken instead of Wien's, we should have obtained

whence

Planck had earlier attempted 1 to give a proof of Wien's law ofradiation based on this equation (3), which he obtained indepen-dently by thermodynamical reasoning: but when confronted byLummer and Pringsheim's experimental results he realised thatWien's could not be the true law of radiation; and he now proposedto modify (3), which he did by writing

d2S adUB= -U~(f3-+-U-)

where y is a constant: so by (2)

dS 1 UdU = - f3v log yv

(6)

1 Berlin Sit". xxv (1899), p. 44080

• cr. Vol. I, p. 380

(7)

THE BEGINNINGS OF QUANTUM THEORY

Thus equation (5) must have the form

U = Const.el':_l

and therefore by (1) the average energy-density of the radiationin the frequency-range v to v + dv is

E= gv3dvel.,T - 1

where g and I are constants. This is Planck's formula which agreedwith the experimental determinations of Lummer and Pringsheim,and also of H. Rubens and F. Kurlbaum,1 and F. Paschen,2 so wellthat it soon displaced all other suggested laws of radiation.

It was, however, as yet hardly more than an empirical formula,since equation (4) had no complete theoretical justification. Thisdefect was remedied on 14 December of the same year (1900), whenPlanck read to the German Physical Society a paper 3 which placedhis new law on a sound foundation, and in so doing created a newbranch of physics, the quantum theory.

He considered a system consisting of a large number of simpleHertzian vibrators, in a hollow chamber enclosed by reflecting walls:let N of the vibrators have the frequency v, N' the frequency v'and so on. Suppose that an amount A of energy is in the vibratorsof frequency v. Planck assumed that this energy is constituted ofequal discrete elements, each of amount E, and that there arealtogether P such elements in the N vibrators, so that

A=PE.

Thus he assumed that the emission and absorption of radiation bythese vibrators takes place not continuously, but by jumps ofamount E.

Any distribution of these P elements among the N vibrators maybe called a complexion. The number of possible complexions is thenumber of possible ways of distributing P objects among N containers,when we do not take account of which particular objects lie inparticular containers, but only of the number contained in each.This number is, by the ordinary theory of permutations and com-binations,

(N+P-l)!(N-I)! P!·

1 Berlin Sitz., 25 Oct. 1900, p. 929 • Ann. d. Phys. iv (1901), p. 277• Verh. deutsch. phys. Ges, ii (1900), p. 237. This and the J(aper of 19 October were

re-edited and printed in the new form in Ann. d. Phys. iv (1901), p. 553.81


As Nand P are very large numbers, we can use Stirling's approximatevalue for the factorials, namely,

log {(z-I)!)= (z-t) log z-z+t log (217)

so the number of complexions is approximately

We assume that all complexions have equal probability, so theprobability W of any state of the system of N vibrators is pro-portional to the number of complexions corresponding to it; thatis, with sufficient approximation for our present purpose, we have

log W = (N + P) log (N + P) - N log N - P log P.

Now the entropy in any state of a system depends on the inequalityof the distribution of the total energy among the individual membersof the system: and Boltzmann had shown by his work on the kinetictheory of gases 1 that the entropy SN in any state of a systemsuch as these vibrators is closely connected with the probabilityW of the state. Planck developed this discovery into the equation

(8)

where the thermodynamic probability W is always an integer, and kdenotes the gas-constant for one molecule, or Boltzmann constant.2

ThusSN= k{(N + P) log (N + P) - N log N - P log Pl.

Now P=NU/€, where U is the average, taken over the N oscillators,of the energy of one of them. Thus, retaining only the mostimportant terms, and ignoring terms which do not involve U, wehave

SN= kN { ( I+ Y) log (I+ ¥) - ¥ log ¥}1 cf. L. Boltzmann, Vorlesungen iiber Gastheorie, i (1896), § 6. This is essentially Boltz·

mann's' H-theorem.'• cf. Vol. I, p. 382. With Boltzmann the factor k did not occur, since his calculations

referred not to individual molecules but to gramme-molecules, and with him the entropywas undetermined as regards an additive constant (i.e. there was an undetermined factorof proportionality in the probability W), whereas with Planck the entropy had a definiteabsolute value. This was a step of fundamental importance, and, as we shall see, leddirectly to the hypothesis of • quanta.' The occurrence of the logarithm in the formulais explained by the circumstance that in compound systems a multiplication of proba-bilities corresponds to an addition of entropies.

82

S = ~ = k{ (1 +¥) log ( 1+ ¥) - ~ log ¥}.Thus from equation (2) above,

1 dS k E+ U- =-=-log--T dU E Uor

(9)

But by equation (5) U must be of the form vrf;(v/T). This conditioncan be satisfied only if

E=hv (10)

where h is a constant independent of v. Thus the average energy of airysimple-harmonic Hertzian vibrator of frequency v must be an integral multipleof hv, and the smallest amount of energy that can be emitted or absorbedby it is hv.

From (9) and (10) we have

hvU = ehr{t'r:"'T

and therefore by (1) the average energy-density of black-body radiationin the interval of jrequency between v and v + dv is

8 Trch"A.-5d"A.or E = ech/kAT _ 1 ' (11)

which is Planck's fOrmula. This agrees with his earlier result (7),but the constants which were unknown in (7) are now replaced byhand k, which are important constants of nature and appear in manyother connections.

When v - 0, the formu1<•.gives

which is Rayleigh's law; and when v _ co it gives

which is Wien's law, now expressed in terms of the constants hand k.83


To obtain Wien's displacement law,1 we proceed as follows: LetAmdenote the wave-length corresponding to the maximum ordinateof a graph in which energy-density of radiation is plotted againstwave-length. Then by Planck's formula (11), Amis the value of Agiven by

oreh/kAT

0= - 5 + 1_ echl/cAT·

Let q be the root of the equation_x_=5l-rx

soq=4·965ll4.

Thenhe

kAmT=q,or

T- (he/k) - 1·4384 d -028971 dAm - q - - 4.965114 em. egree - . em. egree,

which is Wien's displacement law.Planck determined the values of the constants hand k by com-

paring his formula (11) with the measurements of F. Kurlbaum 2

and O. Lummer and E. Pringsheim,3 the results obtained beingh = 6·55 X 10-27 erg. sec., k = 1·346 X 10-16 ergs per degree.

He used this determination of k in order to calculate the numberof molecules in a gramme-molecule (Avogadro's number) ': fromthe equation

S=k log W

we can calculate the entropy of one gramme-molecule of an idealgas, and from this can derive thermodynamicalJy the relation

p=kNTV

where p denotes the pressure of the gas, V its volume, and N denotesAvogadro's number: this shows that ifR is the absolute gas-constant,then

R=kN.1 d. Vol. I, p. 380I Verh. d. deutsch phys. Gu. ii (1900), p. 176

84• Ann. d. Phys. !xv (1898), p. 759• cf. pp. 8, 18


From the known values of Rand k, Planck found

N= 6·175 X 1013;

this agreed satisfactorily with the value 6·40 x 1023which had beengiven by O. E. Meyer.1

Moreover, the knowledge ofN so obtained leads to a new methodof finding the charge of an electron. For the charge which is carriedin electrolysis by one gramme-ion, that is by N ions, was known, beingat that time believed to be 9,658 electromagnetic units. Thus if eis the charge of an electron in electrostatic units, we have

Ne =9,658 x 3 X 1010

which givese =4·69 x 10-10 e.s.u.

r. J. Thomson had found e =6 x 10-10 e.s.u. two years earlier 2 ;

Planck's value was actually much nearer to the later determinations,which gave approximately 4·77 x 10-10 e.s.u.

Planck's law made it possible to give a more accurate formulationof the Stefan-Boltzmann law 3 for the total radiation per second fromunit surface of a black body at temperature T. For 4 the elementof this radiation in the range of wave-lengths "Ato "A+ d"A is, byPlanck's law

2 Trhc2"A-6d"Aehe/kAT - 1

so the total radiation for all wave-lengths is

or

Now if Bn is the nth Bernoullian number, we have

J""t2n-1dtBn=4n ~1'oe1T

-

1 Die Kinetische Theone der Case, 2 Auf!. (1899), p. 337; Planck's actual result, Ann. d.Phys. IV (1901), p. 564, is that the number of oxygen molecules in I em' at 760 mm.pressureand 15°C., is 2·76. 10".

, d. Vol. I, pp. 364-5 ' cf. Vol. I, p. 374 • Vol. I, p. 37385


whence, remembering that B2= 1/30, we have

Putting p = h/27TkT, we see that the total radiation per second from unitsurface oj a black body at temperature T is

27T6k4 T4 .l5cz,iii ,

this is the precise expression of the Stefan-Boltzmann formula interms of the universal constants c, hand k.

A deeper insight into the physical conceptions underlying Planck'slaw of radiation was furnished by a later proof of it.l As is wellknown, in the kinetic theory of gases, it is shown that the probabilitythat for a particular molecule the x-component of velocity will liebetween u and u + du, its y-component of velocity between v andv + dv and its z-component between wand w + dw, is

where m is the mass of the molecule, U its kinetic energy, k is Boltz-mann's constant, and T the absolute temperature. This result wasgeneralised by Josiah Willard Gibbs 2 (1839-1903) into the followingtheorem: if we consider a large number of similar dynamicalsystems (which for simplicity we shall suppose to be linear oscillators),which are in statistical equilibrium with a large reservoir of heat attemperature T, and if q is the co-ordinate in an oscillator (e.g. theelongation of a vibrating electron) and p the momentum (definedas oL/o(oq/ot) where L is the kinetic potential), then the probabilitythat for any particular oscillator the co-ordinate lies between q andq + dq and the momentum lies between p and p + dp is a

e-Uffldq dp

Je-U/k'rdq dp

where U is the energy of the oscillator, and the integration is to betaken over all possible values of q and p.

The theorem corresponding to this in the quantum theory is that

1 cf. Lorentz, Phys. ZS. xi (1910), p. 1234; F. Reiche, Die Quantentheori4 (Berlin 1921),Note 48

• Elerruntary Plinciples in Statistical Mechanics (New York, 1902)• This is Gibbs's canonical distribution

86


if the energy of an oscillator can take only the discrete set of valuesVo, VI, V2, Vs, .. " then the probability that the energy of aparticular oscillator is V. is 1

e-u,wr/ i e-u./ItT,_0

Thus ifV. =shll for s=O, 1,2" .. , the probability is

The mean energy of an oscillator is therefore00

hv( 1- e-hv/1'r) L se-·hv/ItT,.=0

which has the valuehv

eholIT - I'

This leads at once, as before, to Planck's formula that the energy-density of black-body radiation in the frequency-interval fromv to v+ dll is

Other derivations of the law, based on many different assumptions,were given by various writers.2 Some of them will be discussedlater.

The next important advance in quantum theory was made byEinstein,3 in the same volume of the Annalen deT Physik as his papers

1 If to an energy-level Us there belongs a number gs of permissible states, then thelevel Us is said to be degenerate, and gs is called the we~f?h/ of the state. Taking thepossibility of degenerate states into account, the above formula should be written

g.e-U,lkT

00 -u.wr·Lg,e.1=0

• Special reference may be made to the following:J. Larmor, PrO(. R.S.(A), lxxxiii (1909), p. 82; P. Debye, Ann. d. Phys. xxxiii (1910),

p. f4~?, completed by A. Rub!no,,:icz, Phys. ZS. xviii (1917), p. 96; P. Franck, Phys.ZS. X111 (1912), p. 506; A. Einstein and O. Stern, Ann. d. Phys. xl (1913), p. 551 ;M. Wolfke, Verh. ~. deu./sch. phys. Gu..~ (1913), pp. 1123, 1215: J:'hys. ZS. xv (1914),pp. 308,463; A. Einstein, Phys. ZS. XVIII (1917), p. 121 ; C. G. Darwin and R. H. Fowler,Phil. Mag. xliv (1922), pp. 450, 823: Proc. Comb. Phil. Soc xxi (1922), p. 262; S. N. Bose,ZS./. P. xxvi (1924), p. 178, xxvii (1924), p. 384; A. S. Eddington, Phil. Mag. I (1925),p.803

• Ann. d. Phys. xvii (1905), p. 132: ef. also Ann. d. Phys. xx (1906), p. 19987


On the Brownian motion 1 and Relativity.2 Einstein supposedmonochromatic radiation of frequency v and of small density (withinthe range of values of vlT for which Wien's formula of radiation isapplicable) to be contained in a hollow chamber of volume Vo withperfectly-reflecting walls, its total energy being E : and, investigatingby use of Wien's formula the dependence of the entropy on thevolume, he found for the difference of the entropies when the radia-tion occupies the volume Vo and when it occupies a smaller volumev, the equation

S_So=Ek log!!....hv Vo

Now by inverting the Boltzmann-Planck relation

entropy = k x logarithm of probability

he calculated the relative probability from the difference of entropies,and found that the rrobability that at an arbitrarily-chosen instantof time, the whole 0 the energy of the radiation should be containedwithin a part v of the volume Vo, is

(~ytThis formula he studied in the light of a known result in the kinetictheory of gases, namely that if a gas contained in a volume Vo consistsof n molecules, the probability that at an arbitrarily-chosen instantof time, all the n molecules should be collected together within aa part v of the volume, is

Comparing these formulae, he inferred that the radiation behavesas if it consisted of Elhv quanta of energy or photons,3 each of amounthv. The probability that all the photons are found at an arbitraryinstant in the part 11 of the volume Vo is the product of the prob-abilities (vivo) that a single one of them is in the part v: which showsthat they are completely independent of each other.

Now it will be remembered that according to Planck's theory,a vibrator of frequency v can emit or absorb energy only in multiplesof hv. Planck regarded the quantum property as belonging essen-tially to the interaction between radiation and matter: free radiationhe supposed to consist of electromagnetic waves, in accordance with

1 cr. p. 9 • cr. p. 40• The word photon was actually introduced much later, namely, by G. N. Lewis,

Na/url, 18 Dec., 1926; but it is so convenient that we shall adopt it now.88


Maxwell's theory. Einstein in this paper put forward the hypothesisthat parcels of radiant energy of frequency v and amount hv occurnot only in emission and absorption, but that they have an inde-pendent existence in the aether.

It was shown by P. Ehrenfest 1 of Leiden, by A.Joffe 2 of St Pet~rs-burg, by L. Natanson 3 of Cracow and by G. Krutkow' of Leidenthat Einstein's hypothesis leads not to Planck's law of radiation butto Wien's, at any rate if we assume that each of the light-quantaor photons of frequency v has energy hv and that they are completelyindependent of each other. In order to obtain Planck's formula itis necessary to assume that the elementary photons of energy hvform aggregates, or photo-molecules as we may call them, of energies2hv, 3hv, ... , respectively, and that the total energy of radiationis distributed, on the average, in a regular manner between thephotons and the different kinds of photo-molecules. This will bediscussed more fully later.

Einstein applied his ideas in order to construct a theory ofphoto-electricity.s As we have seen,6 in 1899 J. J. Thomson andP. Lenard showed independently that the emission from a metalirradiated by ultra-violet light consists of negative electrons: andin 1902 Lenard,? continuing his researches, showed that the numberof electrons liberated is proportional to the intensity of the incidentlight, so long as its frequency remains the same: and that theinitial velocity of the electrons is altogether independent of theintensity of the light, but depends on its frequency.

Knowledge regarding photo-electricity had reached this stagewhen in 1905 Einstein's paper appeared. Considering a metalsurface illumined by radiation of frequency v, he asserted that theradiation consists of parcels of energy; when one such parcel orphoton falls on the metal, it may be absorbed and liberate a photo-electron: and that the maximum kinetic energy of the photo-electronat emission is (hv - eep), where eep is the energy lost by the electronin escaping from its original location to outside the surface. Thisof course implies that no photo-electrons will be generated unlessthe frequency of the light exceeds a certain' threshold' value eepfh.

Einstein's equation was verified in 1912 by O. W. Richardsonand K. T. Compton 8 and by A. L. Hughes,9 and with great carein 1916 by R. A. Millikan.1o For many metals, the threshold fre-quency is in the ultra-violet: but for the electro-positive metals, suchas the alkali metals, it is in the visible spectrum: for sodium, it isin the green.

1 Ann. d. Phys. xxxvi (1911), p. 91 I ibid, p. 534I Phys. '<:'S.xii (1911), p. 659 • Phys. '<:'S.xv (1914), p. 133I cr. Vol. I, pp. 356-7 • cr. Vol. I, p. 3657 Ann. d. Phys. viii (1902), p. 149; also E. R. Ladenburg (1878-1908), Ann. d. Phys.

xii (1903), p. 558• Phil. Mag. xxiv (1912), p. 575 • Phil. Trans. ccxii (1912), p. 205

10 Phys. Rev. vii (1916), p. 355. cf. also M. de Broglie, .1. de Phys. ii (1921), p. 265,and J. Thibaud, Campus ReMus, clxxix (1924), pp. 165, 1053, 1322

89


The function 4> is closely connected with the thermionic work-function measured at the same temperature: 1 in fact, the thermionicwork-function is equal to h times the least frequency which willeject an electron from the metal: 2 and 4> is therefore connectedwith the contact potential-differences between two metallic sur-faces : 3 the difference of the functions 4> for the two metals is equalto the contact difference of potential (reversing the order of themetals) together with the (small) coefficient of the Peltier effect atthe junction between them.

Gases and vapours also exhibit the photo-electric effect, if thefrequency of the incident radiation is sufficiently great: and thephenomenon can be observed for individual atoms by use of X-rayswith the Wilson chamber. This effect is simply ionisation: andthe law regarding the threshold frequency becomes the assertionthat for ionisation to take place, the energy of the incident photonmust be not less than the ionisation energy of the atom or moleculeconcerned. The electrons chiefly affected photo-electrically are thestrongly-bound ones in the K-shell: the electrons in the outershells, being more feebly-bound, do not absorb radiation to the samedegree. The function 4> in the equation

maximum kinetic energy of electron at emission = hv - e4>

IS now no longer connected with thermo-electric phenomenaor contact differences of potential, but has different values de-pending on the shell in the atom from which the electron hascome.

If a photo-electron is liberated from the K-shell, it may happenthat the vacant place is filled by an electron from an outer shell,creating a photon whose energy is equal to the difference of theenergies of the electron in the two shells: and this photon mayin its turn be absorbed in another shell, giving rise to a secondphoto-electron, so that two electrons are ejected together. Thiseffect, which was discovered by P. Auger,4 is called the compoundphoto-electric e.ffect.

The photo-electric effect cannot be explained classically, becausethe time-lag required by the classical theory, due to the necessityfor accumulating sufficient energy from the radiation, is foundnot to occur.5

A hypothesis closely allied to Einstein's light-quantum explana-tion of the photo-electric effect was put forward in 1908 byJ. Stark 6 :

namely, that the frequency of the violet edge of the band-spectrum

1 cr. Vol. I, pp. 426-8I O. W. Richardson and K. T. Compton, Phil. Mag. xxiv (1912), p. 595I O. W. Richardson, Phil. Mag. xxiii (1912), pp. 263, 594• Comptes Rendus, clxxxii (1926), p. 1215• On the time-lag, cr. E. Meyer and W. Gerlach, Archives des sc. phys. et nat. xxxvi

(1914), p. 253. • Phys. ZS. ix (1908), p. 8590


ofa gas is connected with the ionisation-potential of the gas (measuredby the potential-fall necessary to give sufficient kinetic energy to theionising electron) by the formula

1= v.

Experiments in agreement with this relation were published in thefollowing year by W. Steubing.1

Another hypothesis of the same type, also proposed by Stark 2

in 1908, and elaborated by Einstein 3 in 1912, related to photo-chemical decomposition: it asserted that when a molecule is dis-sociated as a result of absorbing radiation of frequency v, theamount of energy absorbed by the molecule is Izv. There must,therefore, be a lower limit to the frequency of light capable ofproducing a given chemical reaction, and a relation between theamount of reaction and the amount of light absorbed. The law isapplicable only within the range of validity of Wien's law andwhen the decomposition is purely a thermal effect. Experimentsdesigned to test this hypothesis were made by E. Warburg,4 withresults on the whole favourable.

From Einstein's doctrine that the energy of a photon offrequency v is Izv, combined with Planck's principle 6 that fluxof energy is momentum, it follows at once that in free aether,where the velocity of the photon is c, its momentum 6 must be Izvjc,in the direction of propagation of the light. Long afterwardsit was shown experimentally by R. Frisch 7 that when an atomabsorbs or emits a photon, the atom experiences a change of momen-tum of the magmtude and direction attributed to the photon byEinstein: but there had never been any doubt about the matter,since Einstein's value was assumed in the theory of many pheno-mena, and predictions based on the theory were experimentallyverified.

It may be remarked that the above relation between the energyand the momentum of a photon is in agreement with the classicalelectromagnetic theory of light: for if a beam of light is propagatedin free aether in a certain direction, the electric vector E and themagnetic vector H are equal, and at right angles to each otherand to the direction of propagation; and therefore Kelvin's energy-density8 1/87T(E2+ H2) is E2/47T,while J. J. Thomson's momentum-density 9 1/47TC [E.HJ is E2/47Tc: and the latter is equal to the formerdivided by c.

1 Phys. ZS. x (1909), p. 7872 Phys. ZS. ix (1908), p. 889: Ann. d. Phys. xxxviii (1912), p. 467; cf. E. Warburg,

Verh. d. deutsch.phys. Ces. ix (1907) p. 753• Ann. d. Phys. xxxvii (1912), p. 832 : xxxviii (1912), p. 881• Berlin Sit;;., 1911,p. 746: 1912,p. 216: 1913,p. 644: 1914,p. 872: 1915,p. 230 :

1916,p. 314: 1918,pp. 300, 1228 • cf. p. 54• cf. A. Einstein, Phys. ZS. x (1909), pp. 185, 817 : J. Stark, ibid. pp. 579, 902• ZS.f. P. Ixxxvi (1933), p. 42 • cf. Vol. I, p. 222 • cf. Vol. I, p. 317

91


The formula for the momentum of a photon is also in agreementwith the equation

momentum =mass x velocity :

for since the energy is hv, the mass is hvlc'; and the velocity is c ;so the momentum is hvlc.

The corpuscular theory of light thus formulated by Einsteinleads at once to the relativist formulation of the Doppler effect.lFor suppose that a star is moving with velocity w relative to anobserver P, and that a quantum oflight emitted towards the observerP has a frequency v' as measured by an observer pi on the star, anda frequency v as measured by the observer P.

Let the direction-cosines of the line PP', referred to rectangularaxes in P's system of measurement, of which the x-axis is in thedirection of the velocity w, be (I, m, n). Then the energy andmomentum of the light-quant as observed by P, that is, in a syst~of reference in which P is at rest, are (hv, - hvllc, - hvmlc, - hvnlc) :and the energy of the light-quant as observed by P', that is, in asystem of reference in which P' is at rest, is hv'. But the (energy)lcand the three components of momentum form a contravanant four-vector which transforms according to the Lorentz transformation, inwhich

and therefore

or

Vi

v

I+~ Ic

which is the relativist formula for the Doppler effect. Thus theDoppler effect is simply the Lorentz transformation if the energy-momentumfour-vector if the light-quant.

In the earlier years of the development of quantum-theory, muchattention was given to the relation of light-propagation to space,

I cf. p. 41

92


the aim being to find a conception of the mode of propagation whichwould account both for those experiments which were most naturallyexplained by the wave-theory, and also for those which seemed torequire a corpuscular theory. J. Stark 1 and A. Einstein 2 discusseda fact which seemed very difficult to reconcile with the wave-theory,namely, that when cathode rays fall on a metal plate, and the X-raysthere generated fall on a second metal plate, they generate cathoderays whose velocity is of the same order of magnitude as that of theprimary cathode rays.

More precisely, let the X-rays be excited by a stream of electronsstriking an anticathode (this is, of course, the process inverse to thephoto-electric effect). Suppose that the energy of the electrons iswhat would be obtained by a fall through a potential-difference V,so that the kinetic energy of the electrons is eV. When the electronsare stopped, they give rise to the X-rays, whose frequencies forma continuous 3 spectrum with a limit Vmax on the side of high fre-quencies given by the equation 4

Izv = eV.max

That is, X-rays of frequency v are not produced unless energy hv isavailable. It is reasonable to suppose that the maximum value of thefrequency is obtained when the whole of the energy eV of the electronis converted into energy of radiation (X-radiation of lower frequencyis also obtained, because the incident electron may spend part of itsenergy in causing changes in the atoms of the anticathode). Sincethe energy of an electron ejected by the X-ray in the photo-electriceffect is (save for differences due to other circumstances which neednot be considered at the moment) equal to hvmax, we see that it isequal to the energy of the electrons in the cathode rays which hadoriginally excited the X-ray: so that no energy is lost in the changesfrom electron to X-ray and back to electron again. The X-ray musttherefore carry its energy over its whole track in a compact bundle,without any diminution due to spreading: as had been asserted in1910 by W. H. Bragg (cf. p. 17).

On the other hand, X-rays are certainly of the nature of ordinarylight, and can be diffracted: so one would expect them to showthe spreading characteristic of waves. The apparent contradictionbetween the wave-properties of radiation and some of its otherproperties had been considered by J. J. Thomson in his Sillimanlectures of 19035; 'Rontgen rays,' he said, , are able to pass very

1 Phys. ZS. x (1909), pp. 579, 902: xi (1910), pp. 24,179I Phys. ZS. x (1909), p. 817; cr. H. A. Lorentz, Phys. ZS. xi (1910), p. 1234I Regarding the discontinuous spectrum of characteristic X-rays, cf. D. L. Webster,

Phys. &v. vii (1916), p. 599• cf. W. Duane and F. L. Hunt, Phys. Rev. vi (1915), p. 166. The value of h was

calculated on the basis of this property by F. C. Blake and W. Duane, Phys. &v. ix (1917),p. 568: x (1917), pp. 93, 624.

• J. .J. Thomson, Electricity and Matter (190-1), pp. 63-5, cf. his Conduction of Electricitythrough Gases (1903), p. 258

93


long distances through gases, and as they pass through the gas theyionise it: the number of molecules so split up is, however, anexceedingly small fraction, less than one-billionth, even for strongrays, of the number of molecules in the gas. Now, if the conditionsin the front of the wave are uniform, all the molecules of the gasare exposed to the same conditions: how is it, then, that so smalla proportion of them are split up ?' His answer was: 'The difficultyin explaining the small ionisatlOn is removed if, instead of supposingthe front of the Rontgen ray to be uniform, we suppose that It con-sists of specks of great intensity separated by considerable intervalswhere the intensity is very small.'

In this passage Thomson originated the conception of needleradiation, 1 i.e. that in the elementary process of light-emission, the radiationsfrom a source are not distributed equally in all azimuths, but are concentratedin certain directions. This hypothesis was now adopted by Einstein,2who, as we shall see, developed it further in 1916.

When, however, the phenomena of interference were taken intoaccount, the conception of needle radiation, and indeed the wholequantum principle of radiation, met with difficulties which were notresolved for many years. It was shown experimentally 3 that whena classical interference-experiment was performed with light so faintthat only a single photon was travelling through the apparatus atanyone time, the interference-effects were still produced. Thiswas interpreted at first to mean that a single photon obeys the lawsof partial transmission and reflexion at a half-silvered mirror andof subsequent re-combination with the phase-difference required bythe wave-theory of light. It is evident, however, that such anexplanation would be irreconcilable with the fundamental principleof the quantum theory, according to which interaction betweenthe light and matter at a particular point on the screen cantake place only by the absorption or emission of whole quanta oflight.

A further objection to the view that coherent beams of light,which are capable of yielding interference-phenomena, could beidentical with single photons, appeared when it was found that thevolume of a beam of light over which coherence can extend, wasmuch greater than the nineteenth-century physicists had supposed.In 1902 O. Lummer and E. Gehrcke,' using green rays from amercury lamp, obtained interference-phenomena with a phase-difference of 2,600,000 wave-Iengths-a distance of the order of one

1 cf. Thomson's further papers in Proc. Camb. Phil. Soc. xiv (1907), p. 417; Phil.Mag. (6) xix (1910), p. 301; and N. R. Campbell, Proc. Camb. Phil. Soc. xv (l910),p. 310. For an interesting application of the relativity equations to needle radiation,cf. H. Bateman, Proc. Lond. Math. Soc.(2) viii (1910), p. 469.

• loc. cit.I G. I. Taylor, Proc. Camb. Phil. Soc. xv (1909), p. 114; most completely by A. J.

Dempster and H. F. Batho, Phys. Rev. xxx (1927),p. 644; cf. E. H. Kennard, J. FranklinInst., ccvii (1929), p. 47

• Verh. deulsch. phys. Ges. iv (1902), p. 337; cf. M. von Laue, Ann. d. Phys.(4) xiii(1904), p. 163,§6

94


metre. Regarding the lateral extension of a coherent beam, sinceall the light from a star that enters a telescope-objective takes partin the formation of the image, it is evident that this light must becoherent: and a still greater estimate was obtained in 1920, wheninterference methods were used at Mount Wilson Observatory todetermine the angular diameter of Betelgeuse, and interference wasobtained between beams which arrived from the star twenty feetapart. It seemed impossible that these very large coherent beamscould be single photons.

The alternative hypothesis seemed to be, that the photons ina coherent beam form a regular aggregate, possessing a qualityequivalent to the coherence.l One supposition was that the motionof the photons is subject to a system of probability correspondingto the wave-theory explanation of interference, so that a largenumber of them is directed to the bright places of the interference-pattern, and few or none are directed to the dark places. Thisexplanation was, however, unsatisfactory: for it is not until thetwo interfering beams of light have actually met that the interference-pattern is determined, and therefore the guiding of the motion ofthe photons cannot take place during the propagation of the inter-fering beams, but must happen later, perhaps at the screen itself-a process difficult to imagine. It was therefore suggested that whilethe photons are being propagated in the beams which are laterdestined to interfere, they are characterised by a quality correspond-ing to what in the wave-theory is called phase. In order to constructa definite theory based on this idea, however, it would be necessaryfirst to consider whether the photons are to be regarded as points(in which case any particular photon would always retain the samephase, since it travels with the velocity of light, but different photonswould have different phases) or whether the photons are to be re-garded as extending over finite regions (in which case the phasewould presumably vary from one J?oint of a photon to another).In order to account for interference, It would be necessary to supposethat photons, or parts of photons, in opposite phases, neutralise eachother. But if the photons are regarded as points, the mutual annul-ment of two complete photons would be incompatible with theprinciple of conservation of energy, while if the photons are regardedas extending over finite regions, the annulment of part of one bypart of another would seem to be incompatible with the integralcharacter of photons. There is, moreover, a difficulty created bythe observation that interference can take place when only a singlephoton is travelling through the apparatus at anyone instant, forthis seems to require that the effect of a quantum on an atom persistsfor some time.

These various attempts-none of them entirely satisfactory-tocombine the new and the old conceptions of light, created a doubt

I This was first put forward by J. Stark, loco cit.

95


as to whether it was possible to construct, within the frameworkof space and time, a picture or model which would be capable ofrepresenting every known phenomenon in optics.l

In December 1906 Einstein 2 initiated a new development ofquantum theory, by carrying its principles outside the domain ofradiation, to which they had hitherto been confined, and applyingthem to the study of the specific heats of solids. We have seen 3

that according to the classical law of equipartition of energy, in astate of statistical equilibrium at absolute temperature T, with everydegree of freedom of a dynamical system there is associated on theaverage a kinetic ener~ tkT, where k is Boltzmann's constant.Now the thermal mooons of a crystal are constituted by theelastic vibrations of its atoms about their positions of equilibrium:one of these atoms, since it has three kinetic degrees of freedomand also three potential-energy degrees of freedom, will have amean kinetic energy !kT and a mean potential energy ikT, or atotal mean energy 3kT. Thus a gramme-atom of the crystal willhave a mean energy 3kNT, where N is Avogadro's number';or 3RT, where R is the gas-constant per gramme-atom. The atomicheat of the crystal (i.e. the amount of heat required to raise thetemperature of a gramme-atom by one degree) is therefore 3R.Now

R=8·3l36 X 107 ergs= 1·986 cal.

so the atomic heat (at constant volume) is 5·958 cal. This law hadbeen discovered empirically by P. L. Dulong and A. T. Petit f>

in 1819.While Dulong and Petit's law is approximately true for a great

many elements at ordinary temperatures, exceptions to it hadlong been known, particularly in the case of elements of lowatomic weight, such as C, Bo, Si, for which at ordinary temperaturesthe atomic heats are much smaller than 5·958 cal.: and shortlybefore this time it had been shown, particularly by W. Nernstand his pupils, that at very low temperatures all bodies havesmall atomic heats, while at sufficiently high temperatures eventhe elements of low atomic weight obey the normal Dulong-Petitrule, as was shown e.g. by experiments with graphite at hightemperatures.

1 The state of the coherence problem twenty years after Einstein's paper of 1905may be gathered from G. P. Thomson, Proc. R.S.(A), civ (1923), p. 115; A. Lande,ZS.f. P. xxxiii (1925), p. 571; E. C. Stoner, Proc. Camb. Phil. Soc. xxii (1925), p. 577;W. Gerlach and A. Lande,ZS.f. P. xxxvi (1926),p. 169; E. O. Lawrenceandj. W. Beams,Proc. JI.A.S. xiii (1927), p. 207.

I Ann. d. Phys. xxii (1907), pp. 180, 800I Vol. I, p. 382. The argument given here is substantially due to Boltzmann, Wien

SiLt;. !xiii (Abth. 2) (1871), p. 712.• cr. pp. 8, 18• Ann. de chim. et de. phys. x (1819), p. 395; Phil. Mag. liv (1819), p. 267

96


Einstein now pointed out that if we write Planck's formula inthe form:

Energy-density of radiation in the frequency-range v to v + dv

= 87T"-'d,, ehV'::-1'

then on comparing this with Rayleigh's derivation of his law ofradiation, 1 we see that in order to obtain Planck's formula, a modeof vibration of frequency v must be counted as possessing the averageenergy

tkT~l"e -

where x=hv/kT, instead of (as Rayleigh assumed) tkT. If then inthe above J?roof of Dulong and Petit's law we replace tkT bytkT x/(eX

- I), we find that a gramme-atom of the crystal will havea mean energy

3RT-~eX -1

(if for simplicity we assume that all the atomic vibrators have thesame frequency v) and therefore the atomic heat is

or or

This is Einstein's formula. As the temperature falls, x increases andx2ex/(eX_I)2 decreases, so the decrease of atomic heat with tem-perature is accounted for. As the absolute zero of temperature isapproached, the atomic heats of all solid bodies tend to zero.2

The determination of the quantity x, i.e. the determination ofY, was studied by E. Madelung,3 W. Sutherland,' F. A. Lindemann,5A. Einstein,8 W. Nernst,7 E. Griineisen,8 C. E. Blom 9 and H. S.Allen.1o: relations were found connecting v approximately with thecubical compressiqility of the crystal, with its melting-point andwith the ' residual rays' which are strongly reflected from it.

I Vol. 1, pp. 383-4, This is a special case of a more general theorem discovered and developed by

W. Nemst in 1910 and the following years, namely, that all the properties of solids whichdepend on the average behaviour of the atoms (including the thermodynamic functions)become independent of the temperature at very low temperatures. Thus, at the absolutezero of temperature, the entropy of every chemically homogeneous body is zero; cr.Nernst, Die tlUioretischenund experimentellen Grundlagen des neuen Wiirmesatzes (Halle, 1918).

• Gott. Nach. (1909), p. 100; Phys. ZS. xi (1910), p. 898• Phil. Mag,(6) xx (1910), !. 657 • Phys. ZS. xi (1910), p. 609• Ann. d. Phys. xxxiv (1911, p. 170; xxxv (1911), p. 679, Ann. d. Phys. xxxvi (1911 , p. 395 • Ann. d. Phys. xxxix (1912), p. 257• Ann. d. Phys. xlii (1913), p. 1397

'0 Proc. R.S.(A), xciv (1917), p. 100; Phil. Mag. xxxiv (1917), pp. 478, 488

97

AETHER AND ELECTRlc:rrv

There was, however, one obvious imperfection in all this work,which was pointed out by Einstein in the second of the papers justreferred to; namely, that the vibrations of the atoms in a crystaldo not all have the same frequency v. The mean energy of agramme-atom of a crystal will therefore not be

where x has a single definite value, but

when the summation is taken over all the frequencies (three foreach atom), and k is Boltzmann's constant. The atomic heat,obtained from this by differentiating with respect to T (remember-ing that x = hv/ kT), is therefore

%

'" kX,2 e 'L )2', (/r _ 1

In order to determine the frequencies of the natural vibrations ofthe atoms of a body, and so to be able to evaluate these expressions,P. Debye 1 (b. 1884) took, as an approximation to the actual body,an elastic solid, and considered the elastic waves in it. He showedthat for a fixed isotropic body of volume V, the number of naturalperiods or modes of vibration in the frequency-range v to v + dv is

where Ct is the velocity of transverse waves in the solid and Cl thevelocity of longitudinal waves; and he assumed that the energiesof these different sound waves vary in the same way as the energiesof the light waves in Planck's formula, so that each of these modeshas the energy

hv •eh·/kT - l'

thus the energy per unit volume of waves within the frequency-range v to v + dv is

1 Ann. d. Phys. xxxix (1912), p. 789

98


The total energy is to be obtained by integrating this with respectto v. But here a difficulty presents itself: for the number of naturalfrequencies of a continuous body is infinitely great: v extends fromzero to infinity. Debye (somewhat arbitrarily) dealt with thissituation by taking, in the integration with respect to v, an upperlimit Vm, such that the total number of frequencies less than Vm isequal to 3N, where N is the number of atoms in the body. Thus

. Vm is to be determined from the equation

3N = IVm41Tv( 23+ -;-)v2dv = 41TV( 23+ _~)vm3,o ct Ct 3 Ct CI

and the atomic heat of the body is

( 2 1) IV m x2e'" 241TkV 3+3 (", 1)2V dv whereCI CI 0 e -

or

hvx=kT

Ifwe write

where

hVm=0k '

hvmXm= kT.

we have Xm = °fT, and the atomic heat is a universal function of the ratioTf0, that is, the temperature T divided by a temperature °which is characteristicof the body.

Debye's theory is in good general accord with the experimental.results for many elements.1

When the temperature T is very great, Xm is very small, and theabove formula for the atomic heat becomes

9RIXtn 2d-3 x x orXm 0

3R,

which is Dulong and Petit's law, as would be expected.When on the other hand T is very small, we escape the difficulties

which arise from the fact that the body, as contrasted with thecontinuous elastic solid, has only a finite number of degrees offreedom. The above formula shows that the energy of the bodyis proportional at low temperatures to the integral

J 00 v3dv 100

or 0 v3e-h'WIdv,o eh./kT -1

that is, it is proportional to T'; so the atomic heat rif a body at low1 cf. the exhaustive report by E. SchrOdinger, Phys. ZS. xx (1919), pp. 420, 450, 474,

497,523

99 0~'U-2- ClYl2


temperatures is proportional to the cube if the absolute temperature. Thislaw has been carefully verified.1

In the same year in which Debye's theory of atomic heats waspublished, Max Born (b. 1882) and Theodor v. Karman (b. 1881) 2

attacked the problem from another angle; instead of replacing thebody by a continuous elastic medium, as Debye had done, theymade a dynamical study of crystals, regarded as Bravais space-lattices of atoms, in order to determine their natural periods. Theformulae obtained were of considerable complexity: Debye's methodis simpler, though Born and Karman's is more general and stringent.In the case of low temperatures, Born and Karman confirmedDeybe's result that the atomic heat is proportional to T3 whenTis small.3

The behaviour, as found by experiment, of the molecular heatof gases (i.e. the amount of heat required to raise by one degree thetemperature of one gramme-molecule of the gas, at constant volume)can be explained by quantum theory in much the same way asthat of solid bodies. According to classical theory,' a monatomicgas (such as helium, argon or mercury vapour) has three degreesof freedom (namely, the three required for translatory motion), andto each of them should correspond a mean kinetic energy tkT, soa gramme-molecule should have an energy ikNT when N isAvogadro's number, or iRT where R is the gas-constant per gramme-molecule: thus the molecular heat should be !R or approximatelythree calories, a result verified empirically.5

For the chief diatomic gases-H, N, 0 etC.-the molecular heatis 5 calories, which is explained classically by supposing that theyhave 3 translatory and 2 rotational degrees of freedom. The mole-cule may be pictured as a rigid dumbbell, having no oscillationsalong the line joining the atoms, and no rotations about this line asaxis. It was, however, found experimentally by A. Eucken 6 that themolecular heat of hydrogen at temperatures below 60° abs. falls to3 calories, the same value as for monatomic gases. This evidentlyimplies that the part of the molecular heat which is due to the tworotational degrees of freedom of the molecule falls to zero at lowtemperatures. The quantum theory supplies an obvious explanation

1 A. Eucken and F. Schwers, Verh. deutsch phys. Ges. xv (1913), p. 578; W. Nernstand F. Schwers, Berlin Ber. (1914), p. 355; W. H. Keesom and H. Kamerlingh Onnes,Amsterdam Proc. xvii (1915), p. 894; xviii (1915), p. 484

• Phys. -<:'S.xiii (1912), p. 297; xiv (1913), pp. 15,653 The diamond was studied specially by Born, Ann. d. Phys. xliv (1914), p. 605.

cf. Born, Dynamik der Kristallgitter (Leipzig, 1915), and many later papers. The detailedstudy of crystal-theory is beyond the scope of the present work.

• cf. Vol. I, p. 383• The quantity which is usually determined directly by experiment (from the velocity

of sound in the gas) is the ratio of the molecular heat at constant pressure to the molecularheat at constant volume, or (2+x)/x, where x is the molecular heat at constant volume.

• Berlin Sitz. (1912), p. 141. cf. also K. Scheel and W. Hf:llse, Ann. d. Phys. xl (1913),p. 473, who examined the specific heats of helium, and of nitrogen, oxygen and otherdiatomic gases, between +200 and -1800

; and F. Reiche, Ann. d. Phys. Iviii (1919),p.657.

100

THE BEGINNINGS OF Q.UANTUM THltORY

of this behaviour: it is, that no vibration can be excited except bythe absorption of quanta of energy that are whole multiples of hv,where v is the proper frequency of the vibration: and at lowtemperatures, the energy communicated by molecular impacts isinsufficient to do this, so far as the rotational degrees of freedomare concerned. For the translational degrees of freedom, on theother hand, v is effectively zero, so no limitation is imposed.

At very high temperatures the molecular heats of the permanentgases rise above 5 calories-to 6 or nearly 7-which evidently signifiesthat some additional degrees of freedom have come into action, e.g.vibrations along the line joining the two atoms in the molecule.For chlorine and bromine, this phenomenon is observed even atordinary temperatures, a fact which may be explained by reference tothe .looserconnection of the atoms in the molecules of these elements.

A new prospect opened in 1909, when Einstein 1 discussed thefluctuations in the energy of radiation in an enclosure which is ata given temperature T. From general thermodynamics it can beshown that at any place in the enclosure the mean square of thefluctuations of energy per unit volume in the frequency-range fromv to v + dv, which we may denote by~, is kT2 dE/dT, where k is Boltz-mann's constant and E is the mean energy per unit volume. Nowby Planck's law we have

E 87Thv3 dv= ~ e"-/l:'£ - 1

whence for E2 we obtain the value

or

If instead of Planck's law of radiation we had taken Wien's law, Iwe should have obtained

while if we had taken Rayleigh's law,3 we should have obtained- c3E2E2= ----

87Tv2dv'

Thus the mean-square of the fluctuations according to Planck's law isthe sum of the mean-squares of the fluctuations according to Wien's law andRayleigh's law, a result which, seen in the light of the principle thatfluctuations due to independent causes are additive, suggests that1 Phys. ZS. x (1909), p. 185 I cf. Vol. I, p. 381 • cf. Vol. I, p. 384

101


the causes operative in the case of high frequencies (for which Wien'slaw holds) are independent of those operative in the case of lowfrequencies (for which the law is Rayleigh's). Now Rayleigh's lawis based on the wave-theory oflight, and in fact the value c3E2/(81TV2dv)for the mean-square fluctuation was shown by Lorentz 1 to be a con-sequence of the interferences of the wave-trains which, according tothe classical picture, are crossing the enclosure in every direction :whereas the value hvE for the mean-square fluctuation is what wouldbe obtained if we were to take the formula for the fluctuation of thenumber of molecules in unit volume of an ideal gas, and supposethat each molecule has energy hv: that is, the expression is whatwould be obtained by a corpuscular-quantum theory. Moreover,the ratio of the particle-term to the wave-term in the completeexpression for the fluctuation is e'IV/kT - I : so when hv/kT is small, i.e.at low frequencies and high temperatures, the wave-term is pre-dominant, and when hv/kT is large, i.e. when the energy-density issmall, the particle-term is predominant. The formula thereforesuggests that light cannot be represented completely either by wavesor by particles, although for certain classes of phenomena the wave-representation is practically sufficient, and for other classes ofphenomena the particle-representation. The undulatory andcorpuscular theories are in some sense both true.

Some illuminating remarks on Einstein's formula were made byPrince Louis Victor de Broglie 2 (b. 1892). Planck's formula

E= 87Thv3 dvCS ehv/I<r - 1

may be writtenE = 87T~VS (e-hv/Itl' + e-2h,/1tl' +e-Shv/Itl' +. . .)dv

c

whereE 87Thvs -shv/fld.=--s- e v.c

00

= 2: shvE •.•-1

1 cr. Lorentz, us tMories statistiques en thermodynamique (Leipzig, 1916), p. 114• Comptes Rendus, c1xxv (1922), p. 811; J. de phys. iii (1922), p. 422. cr. W. Bothe,

ZS.f. P. xx (1923), p. 145; Nalurwiss. xi (1923), p. 965102

'tHE BEGINNINGS OF QUANTUM 'tHEORY

This resembles the first term hvE in Einstein's formula, but it isnow summed for all values of s. So it is precisely the result weshould expect if the energy E. were made up of light-quanta each ofenergy shv. Thus de Broglie suggested that the term E1 should beregarded as corresponding to energy existing in the form of quantaof amount hv, that the second term E2 should be regarded as corre-sponding to energy existing in the form of quanta of amount 2hv,and so on. So Einstein's formula for the fluctuations may be obtained onthe basis of a purely corpuscular theory of light, provided the total energyof the radiation is suitably allocated among corpuscles of different energieshv, 2hv, 3hv, .... 1

The theory of the fluctuations of the energy of radiation wasdeveloped further in many subsequent papers.2

In spite of the many triumphs of the quantum theory, its dis-coverer Planck was in 1911 still dissatisfied with it, chiefly becauseit could not be reconciled with Maxwell's electromagnetic theoryof light. In that year he proposed 3 a new hypothesis, namely, thatalthough emission of radiation always takes place discontinuouslyin quanta, absorption on the other hand is a continuous process,which takes place according to the laws of the classical theory.Radiation while in transit might therefore be represented by Max-well's theory, and the energy of an oscillator at any instant mighthave any value whatever. When an oscillator has absorbed anamount hv of energy, it has a chance of emitting this exact amount:but it does not necessarily take the opportunity, so that emissionis a matter of probability. In the new theory, therefore, there wereno discontinuities in space, although the act of emission involved adiscontinuity in time.

The system based on these principles is generally called Planck'sSecond Theory. He showed that it can lead to the same formulafor black-body radiation as the original theory of 1900; but thereis a notable difference, in that the mean-energy of a linear oscillatorof frequency v is now

1hv e_liv_/lT_+_I_

"2" ehv/Itr _ l'

which ill greater by thv than the value given by the earlier theory :so that at the absolute zero of temperature, the mean energy of the oscillatoris thv. This was the first appearance in theoretical physics of the

1 This corpuscular theory had been proposed in the previous year by M. Wolfke,Phys. ZS. xxii (1921), p. 375.

I M. von Laue, Verh. d. deu(sch ph/s. Ces. xvii (1915), p. 198; W. Bothe, ZS. j. P.xx (1923), p. 145; M. Planck, Berltn Silz. xxxiii (1923), p. 355; Ann. d. Phys. !xxiii(1924), p. 272; P. Ehrenfest, ZS.f. P. xxxiv (1925), p. 362; M. Born, W. Heisenberg, u.P. Jordan, ZS.j. P. xxxv (1926), p. 557; S. Jacobsohn, Phys. Rev. xxx (1927), pp. 936,944; J. Solomon, Ann. de phys. xvi (1931), p. 411; W. Heisenberg, Leipzig Ber. !xxxiii(1931), Math.-Phys. Klasse, p. 3; M. Born and K. Fuchs, Proc. R.S. clxxii (1939),p. 465

I Verh. d. deutsch. phys. Ces. xiii (1911), p. 138103

AETliER AND ELECTRIClTY

doctrine or zero-point energy, which later assumed great importance.In 1913 A. Einstein and O. Stern 1 made it the basis of a newproof of Planck's radiation-formula, and in 1916 W. Nernst 2

suggested that the aether everywhere might be occupied by zero-point energy.

Planck's Second Theory was criticised in 1912 by Poincare,3and in 1914 Planck 4 came to the conclusion that emission by quantacould scarcely be reconciled with classical doctrines, so he nowmade a new prorosal (known as his Third Theory), namely, that theemission as wel as the absorption of radiation by oscillators iscontinuous, and is ruled by classical electrodynamics, and thatquantum discontinuities take place only in exchanges of energy bycollisions between the oscillators and free particles (molecules, ionsand electrons). A year later, however, he 5 abandoned the ThirdTheory, having become convinced by a paper of A. D. Fokker 6

that the calculation of the stationary state of a system of rotatingrigid electric dipoles in a given field of radiation, when the calcula-tion was performed according to the rules of classical electrodynamics,led to results that were in direct contradiction with experiment.The Second Theory fell from favour with most physicists about thesame time, when the experiments of Franck and Hertz 7 showedthe strong analogy between optical absorption and the undoubtedlyquantistic phenomena which take place when slow electrons collidewith molecules.

Meanwhile, in a Report presented to the Physical Section of the83rd Congress of German men of science at Karlsruhe on 25 Sep-tember 1911, Sommerfeld 8 made a suggestion which was the firstgroping towards a new method. Referring to the name Quantumof Action which had been given to the quantity h by Planck, on accountof the fact that its dimensions were those of (Energy x Time) orAction, he remarked that there should be some connection betweenh and the integral which appears in Hamilton's Principle, namely,

J(T-V)dt

where T denotes the kinetic and V the potential energy of themechanical system considered. He proposed to achieve this bymaking the following hypothesis: In every purely molecular process,a certain d~finite amount of Action is absorbed or emitted, namely, the amount

1 Ann. d. Phys. xl (1913), p. 551• J. de phys.(5) ii (1912), p. 5, at p. 30• Berlin Siu:.., 8 July 1915, p. 512, Described below in Chapter IV

• Verh. d. deutsch. phys. Ges. xviii (1916), p. 83• Berlin Sit;;. 30 July 1914, p. 918• Ann. d. Phys. xliii (1914), p. 810• Verh. deutsch. phys. Ges. xiii (1911), p. 1074-

104


where L =T - V is the kinetic potential or Lagrangean function, and whereT is the duration of the process,. A discussion of the photo-electriceffect in the light of this principle was given in 1913 by Sommerfeldand Debye.1 The principle itself, however, was superseded in thelater development of the subject.

1 Ann. d. Phys. xli (1913), p. 873

105

Chapter IV,r

SPECTROSCOPY IN THE OLDER QUANTUM THEORY

In the nineteenth century, it was generally supposed that the luminousvibrations represented by line spectra were produced in the sameway as sounds are produced by the free vibrations of a material body.That is to say, the atom was regarded as an electrical system of somekind, which had a large number of natural periods of oscillation,corresponding to the aggregate of its spectral lines. The first physicistto break with this conception was Arthur William Conway (1875-1950), professor of mathematical physics in University College,Dublin, who in 1907, in a paper of only two and a half pages,!enunciated the principles on which the true explanation was to bebased: namely, that the spectrum of an atom does not representthe free vibrations of the atom as a whole, but that an atom producesspectral lines one at a time, so that the production of the completespectrum depends on the presence of a vast number of atoms. InConway's view, an atom, in order to be able to generate a spectralline, must be in an abnormal or disturbed state: and in this abnormalstate, a single electron, situated within the atom, is stimulated toproduce vibrations of the frequency corresponding to the spectralline in question. The abnormal state of the atom does not endurepermanently, but lasts for a time sufficient to enable the activeelectron to emit a fairly long train of vibrations.

Conway had not at his disposal in 1907 certain facts aboutspectra and atoms which were indispensable for the constructionof a satisfactory theory of atomic spectra: for until after the publica-tion of Ritz's paper of 1908 2 physicists did not realise that thefrequencies of the lines in the spectrum of an element are the differ-ences, taken in pairs, of certain numbers called ' terms'; and itwas not until 1911 that Rutherford 3 introduced his model atom,constituted of a central positively-charged nucleus with negativeelectrons circulating round it. But the revolutionary general prin-ciples that Conway introduced were perfectly sound, and showeda remarkable physical insight.

These principles were reaffirmed in 1910 by Penry VaughanBevan (1875-1913), in a paper4 recording experiments on theanomalous dispersion, by potassium vapour, of light in the regionof the red lines of the potassium spectrum. He first attemptedto explain his results theoretically in accordanc~ with Lorentz'smodernised version 5 of the Maxwell-Sellmeier theory, and found

1 Sci Proc. R. Dubl. Soc. xi (March 1907), p. 181• cr. p. 22 • Proc. R.S.<A), !xxxiv (1910), p. 209

106

t cr. Vol. I, p. 378• cr. Vol. I, p. 401


that it required him to postulate an impossibly great number ofelectrons per molecule: deriving from this contradiction the correctconclusion, that spectroscopic phenomena are to be explained bythe presence of a very great number of atoms, which at anyoneinstant "are in different states, and eachTOfwhich at that instant isconcerned not with the whole spectrum, but at most with only oneline of it.

The next advances were made by John William Nicholson I

at that time of Trinity College, Cambridge. He introducedinto spectroscopic theory the model atom which had beenproposed a few months before by Rutherford, namely a verysmall nucleus carrying practically all the mass of the atom, sur-rounded by negative electrons circling round it like planets roundthe sun. This was so much more precisely defined than earlier modelatoms that it might now be possible to calculate exact numericalvalues for the wave-lengths of lines in atomic spectra. Nicholson'ssecond advance was to recognise the fundamental fact, that theproduction of atomic spectra is essentially a quantum phenomenon., The fundamental physical laws,' he said, , must lie irr the quantumor unit theory of radiation, recently developed by Planck and others,according to which, interchanges of energy between systems of aperiodic kind can only take place in certain definite amounts deter-mined by the frequencies of the systems' 2; and he discovered theform which the quantum principle should take in its application tothe Rutherford atom: 'the angular momentum of an atom can only riseorfall by discrete amounts.' 3 Moreover, following Conway and Bevan,he asserted that the different lines of a spectrum are produced bydifferent atoms: 'the lines of a series may not emanate from thesame atom, but from atoms whose internal angular momenta have,by radiation or otherwise, run down by various discrete amountsfrom a standard value. For example, in this view there are variouskinds of hydrogen atom, identical in chemical properties and evenin weight, but differe'nt in their internal motions.' 4 In other words,an atom of a given chemical element may exist in many differentstates, resembling in many ways the energy-levels of Planck's oscil-lators. And' the incapacity' of an atom' for radiating in a continuousway will secure sharpness of the lines.' 5 Nicholson did not, however,fully appropriate Conway's idea that a single electron (among themany present in the atom) is alone concerned in the production ofa spectral line: on the contrary, he studied the vibrations of anumber of electrons circulating round a nucleus by methods recall-ing the work of Maxwell on Saturn's rings, retaining classical ideasfor the actual computation of the motion, and identifying the fre-quency of spectral lines with the frequency of vibration of a dynamicalsystem.

1 Mon. Not. R.A.S. lxxii, pp. 49 (November 1911), 139 (December 1911), 677, 693(June 1912), 729 (August 1912)

• loc. cit. p. 729 • loc. cit. p. 679 • lococit. p. 730 • ibid.107


The first successful application of quantum principles to spectro-scopy was in the domam not of atomic but of molecular spectra.In 1912 Niels Bjerrum 1 applied quantum ideas in order to explaincertain characteristics of the absorption spectra that had beenobserved with hydrochloric and hydrobromic acids in their gaseousforms. For these and similar compounds, two widely-separatedregions of absorption had been found in the infra-red, of which one,that of longest wave-length, was assigned by Bjerrum, followingDrude,2 to rotations of the molecules. For the absorption in theshort-wave infra-red he gave a new explanation. He assumed thatthe two atoms constituting a molecule are positively and negativelycharged respectively, and that they oscillate relatively to each otheralong the line joining them, say with frequency Vo; incident radia-tion of this frequency is absorbed. Moreover (following a suggestionof Lorentz), he assumed that the line joining the atoms rotates ina plane, and that the rotational energy must be a multiple of hv,where v is the number of revolutions per second.3 Denoting by Jthe moment of inertia of the rotating system, the rotational energyis U. (27TV)2, and we have

U. (27TV) 2 = nhv (n=O, 1,2,3, ... ):

or, denoting this value of v by Vn, we have

nhVn = 27T2J"

By comparing the linear and rotational motions,' oscillations areobtained of frequencies Vn, Vo + Vn, and Vo - Vn: so the absorption-spectrum should contain in the short-wave infra-red the equidistantfrequencies

hVo ± 27T2J'

3hVo ± 27T2J' etc.

Bjerrum's theory stimulated more careful experimental measure-ments by W. Burmeister,5 Eva von Bahr,6 J. B. Brinsmade andE. C. Kemble,' and E. S. Imes.s Burmeister, and afterwards Imes,found that the central frequency Vo was not observed, which wouldseem to indicate that rotation is always present.9

1 Nernst Ftstschrift, 1912, p. 90 • Ann. d. Phys.(4) xiv (1904), p. 677• The quantification of molecular rotations had been suggested by Nernst, ZS. f.

Elektrochtm. xvii (1911), p. 265.• The principle of this composition is due to Rayleigh, Phil. Mag.(5) xxiv (1892),

p.410.• Vtrh. d. dtutsch. phys. Ges. xv (1913), p. 389• Vtrh. d dtutsch. phys. Gts. xv (1913), pp. 710, 731, 1150, Proc. Nat. Acad. Sci. iii (1917), p. 420 • Astrophys. J. I (1919), p. 251• It was later found necessary to modify Bjerrum's theory by associating the lines

not with the actual rotations, but with transitions from one state of rotation to another.108

SPECTROSCOPY IN THE OLDER Q.UANTUM THEORY

In the year following ~jerrum's paper, P. Ehrenfest 1 improvedthe quantum theory of rotation by assuming that if v is the number ofrevolutions per second, then the rotational energy must be a multipleof thv (not hv, as Bjerrum had supposed); the factor t beinginserted because the rotational energy is purely kinetic, and not, asin the case of an oscillator, half potentia1.2 If J is the moment ofinertia, so that the rotational energy is tJ . (27TV)', we have

(n=O, 1,2,3, ... )or

nhv= 47T2J

The angular momentum 27TVJ therefore has the value nhj27T. Thequantity hj27T is now generally denoted 3 by n. Thus the law ofquantification of angular momentum is that it must be a whole multipleof n.

The culmination of the efforts to explain atomic spectra cameinJuly 1913, when Niels Bohr (b. 1885), a Danish research studentof Rutherford's at Manchester, found 4 the true solution of theproblem. With unerring instinct Bohr seized upon whatever wasright in the ideas of his predecessors, and rejected what was wrong,adjoining to them precisely what was needed in order to make themfruitful, and eventually producing a theory which has been thestarting-point of all subsequent work in spectroscopy.

Bohr accepted Conway's principles that (1) atoms producespectral lines one at a time, and (2) that a single electron is theagent in the process, together with Nicholson's principles that(3) the Rutherford atom provides a satisfactory basis for exactcalculations of wave-lengths of spectral lines, (4) the production ofatomic spectra is a quantum phenomenon, (5) an atom of a givenchemical element may exist in different states, characterised bycertain discrete values of its angular momentum and also discretevalues of its energy. He discovered independently 5 Ehrenfest'sprinciple (6) that in quantum-theory, angular momenta must bewhole multiples of n. He further adopted a principle which wassuggested by Ritz's law of spectral' terms,' and had been in somedegree adumbrated but perhaps not clearly grasped by Nicholson,namely (7) that two distinct states of the atom are concerned in theprod uction of a spectral line: and he recognised an exact corres-pondence between the' terms' into which the spectra were analysed

1 Verh. d. deutsch. phys. Ges. xv (1913), p. 451, Ehrenfest's assumption was confirmed by E. C. Kemble, Phys. Rev.(2) viii (1916),

p.689.3 Read' crossedh'• Phil. Mag. xxvi (1913), pp. I, 476, 875; xxvii (1914), p. 506; xxix (1915), p. 332;

xxx (1915),p. 394• Ehrenfest's paper was published on 15June 1913, and Bohr's in the July number

of the Phil. Mag.1°9


by Ritz and the states or ener~-levels of the atom described byNicholson. He also assumed (8) that the Planck·Einstein equationE = hv connecting energy and radiation-frequency holds for emissionas well as absorption: and finally he introduced (9) the principlethat we must renounce all attempts to visualise or to explain classically thebehaviour of the active electron during a transition of the atom from onestationary state to another. This last principle, which had not beendreamt of by any of his pr.::decessors,was the decisive new elementthat was required for the creation of a science of theoretical spectro-scopy.

Let us now see how the Balmer series of hydrogen is explainedby Bohr's theory. The atom of hydrogen consists of one proton withone negative electron circulating round it, the charges being e and- e respectively. We suppose that by some event such as a collision,the atom has been thrown into an 'excited' state in which theelectron describes an orbit more remote from the proton than itsnormal orbit, and that a spectral line is emitted when the electronfalls back into an orbit closer to the proton. Considering any par-ticular orbit, supposed circular for simplicity, let m denote the massand v the velocity of the electron, and r the radius of the orbit.Then the electrostatic force e2/r2 between the proton and electronmust be equal to the centripetal force mv2/r required to hold theelectron in its orbit, so

The quantum condition is that the angular momentum of themotion must be a whole multiple of Ii, say

mvr=n!i.

These equations give

The kinetic energy of the electron, !mv2, is me4/2n2li2. If the electronmakes a transition to an orbit nearer the nucleus for which theangular momentum is pli, there is a gain of kinetic energy

but a loss of potential energy

me4(11)or Iii ji2 - n2 ,IIO


SO altogether (remembering that h = 27T1i) the loss of energy of theatom is

~~~~,!el(12_ ~).

The equation E =hv shows that the frequency of the homogeneousradiation emitted is

v = 27T2e~'!l (1 _1).1z3 p2 n2

This can be identified with the formula for Balmer's series 1

provided we take p = 2 and

so we have obtained an expression for Rydberg's constant R mterms of e, m and h.

The valuee =4· 78 x 10-10

had been obtained by R. A. Millikan,2 the value

~=5·31 X 1017

m

by P. Gmelin 3 and A. H. Bucherer.4 On the basis of Planck's theory,Bohr obtained 5

h = 7·27 X 1018•

1 cr. Vol. I, pp. 376-8. In theoretical researches frequencies are often employedinstead of wave-numbers, so e.g. Balmer's formula would be written

v-R(~- ~)4 ml

where R stands for c times the R of the wave-number formula.It may be mentioned that three years previously A. E. Haas, Wien. Sitz. cxix (1910),

Abth. IIa, p. 119, had conceived the idea that Rydberg's constant should be expressiblein terms of the constants e, m, h, which were already known. By an argument whichwas highly speculative, he obtained the result that this constant has the value 16Tf'e'm/h·,which differs from the correct value only by a simple numerical factor. However, thisis not surprising in view of the fact that e'mh-I is the only product of powers e, m and h,which has the same dimensions as the Rydberg constant.

I B.A. Rep. 1912, p. 410 • Ann. d. Phys. xxviii (1909), p. 1086• Ann. d. Phys. xxxvii (1912), p. 597• Calculated from the experiments of E. War burg, G. Leithaiiser, E. Hapka and

C. Miiller, Ann. d. Phys. xl (1913), p. 611III


Using these values, he got

R = 3·26 X 1015 (sec)-1

in close agreement with observation.1 This successfulprediction hadan effect which may be compared with the effect of Maxwell'scalculation of the velocity of light from his electromagnetic theory.

Bohr pointed out that while by taking p = 2 we obtain Balmer'sseries, by taking p = 3, n = 4, 5, 6, ... , we obtain a series in theinfra-red whose first two members had been discovered by F. Paschen2

in 1908. At the time of Bohr's paper, the series obtained bytaking p = 1 was not known observationally, but it was discoveredin 1914 by Th. Lyman,3 and the series obtained by taking p=4,n=5, 6, 7, ... , was observed in 1922 by F. Brackett.'

Bohr l"emarked that the frequency of revolution of the electronin the nIh state of the hydrogen atom is v/21Tr, or

But the frequency of the radiation emitted m the transition fromthe (n+ 1)lh state to the nIh is

21T2me' { 1 I}~ nS-(n+l)1

which when n is great has approximately the value

Thus for great values of n, the frequency of the radiation emitted in the transi-tion from the nIh orbit to the next orbit is equal to the frequency of revolutionin the nIh orbit. So we have an asymptotic connection between theclassical and guantum theories.

Evidence m favour of Bohr's theory was obtained from certainfacts regarding absorption-spectra. It was known that the linesof the Balmer series do not appear in the absorption-spectrumof hydrogen under ordinary terrestial conditions, and this was atonce explained by the circumstance that the hydrogen atoms havenormally no electrons in the orbits for which p =2, and therefore noelectrons can be raised from these orbits to higher orbits, as wouldbe necessary for the production of a Balmer absorption line. In the

1 For a comparison with the observational values of 1950, cf. R. T. Birge, Phys. Rev.!xxix (1950), p. 193.

• Ann. d. Phys. xxvii (1908), p. 565. This is the Bergmann series for hydrogen; cf.Vol. I, p. 379.

• Phys. Rev. iii (1914), p. 504; Phil. Mag. xxix (1915), p. 284, Nature cix (1922), p. 209

112


atmospheres of the stars, on the other hand, there are excited hydrogenatoms having electrons in the orbits for which p = 2, and these arccapable of giving the Balmer lines in absorption. Considerationsof the same kind can be applied to explain why under ordinaryterrestial conditions the principal series of the alkali metals can beobtained in absorption-spectra, but the two subordinate series cannot.

In the early days of quantum theory no-one troubled about thefact that most of the equations used were not invariant with respectto th~ transformations of relativity theory. This defect was evidentin the case of Bohr's frequency-equation oE = hv, which connectsthe lossof energy in the transition with the frequency of the emittedradiation. It was, however, shown in 1924 by P. A. M. Dirac 1

that, provided the radiation is emitted in a definite direction, the frequencyequation can be expressed in a form which is independent of theframe of reference; it can in fact be written as a vector equationin four-dimensional space-time

oE'=hv'

where the direction of this vector in space-time is the same as thatof the radiation.2

Some other problems which had been raised by the observationalspectroscopists were solved in Bohr's first papers. In 1896 theAmerican astronomer Edward Charles Pickering (1846-1919) dis-covered 3 in the spectrum of the star , Puppis, together with theBalmer series, a series of lines which had the same convergence-number as the Balmer series. Now it is a property 4 of the' diffuse'and' sharp' subordinate series of the alkali metals, that the morerefrangible members of the doublets converge to the same limit inthe two series; and this fact suggested to Rydberg 5 that the Picker-ing and Balmer series were actually the 'sharp' and 'diffuse' sub-ordinate series respectively of hydrogen. If this were true, then thewave-lengths of the lines of the principal series would be immedi-ately calculable, the first of them being at ;\4687·88: and in facta line was observed in the spectrum of' Puppis at ;\4686, very nearthis position: the higher members of the series could not be expectedto be seen, since they were beyond the limit for which our atmosphereis transparent.

The line ;\4686 was observed by the English spectroscopist AlfredFowler 6 (1868-1940) at the Indian eclipse of 22 January 1898, ipthe spectrum of the sun's chromosphere. In 1912, in an ordinary

1 Proc. Camb. PhiL. Soc. xxii (1924), p. 432I On the relation of Bohr's theory to relativity theory, cr. K. Forsterling, ZS. f. P.

iii (1920), p. 404, and E. Schrodinger, Phys. ZS. xxiii (1922), p. 301.I As/roph . .1. iv (1896), p. 369; v (1897), p. 92• cf. Vol. 1, p. 377• As/roph. J. vi (1897), p. 233; cr. also H. Kayser, ibid. v (1897), pp. 95, 243• Phil. Trans. cxcvii (1901), p. 202


discharge tube containing a mixture of hydrogen and helium, hefound 1 other lines very near the positions calculated by Rydbergfor the supposed' principal series of hydrogen,' and, moreover, founda new series in the ultra-violet which had the same convergence-limit as this supposed principal series, and which he provisionallynamed the 'second principal series of hydrogen.' The small dis-crepancies in wave-length with Rydberg's calculations were, how-ever, unexplained, and also the fact that the presence of heliumappeared to be necessary.

Bohr accounted in the most natural way for Pickering's' Puppisseries, and the two series found by Fowler, by sug-gesting that theywere not due to hydrogen at all, but to ionised helIum. The heliumnucleus has a charge 2e, and in ionised helium this is accompaniedby one negative electron circulating round it. A calculation similarto that carried out for the hydrogen atom shows that in this casethe frequency of the radiation emitted is

27T2e4mwhere as before R =--.h3

If in this we take p = 3 and n = 4, 5, 6, ... , we obtain a series whichincludes the two series found by Fowler : while if we take p =4and n = 5, 6, 7, . . ., we obtain the series observed by Pickering inthe spectrum of 'Puppis. Every alternate line in the series thuscalculated would be identical with a line in the Balmer series ofhydrogen.

There still, however, remained the difficulty arising from theslight discrepancies in wave-length. This was accounted for byBohr, who remarked that the geometrical centre of the circularorbits of the electron is, strictly speaking, not the nucleus but thecentre of gravity of the nucleus and the electron. This makes itnecessary to multiply the value of the Rydberg constant by theratio of the mass of the nucleus to the combined mass of the nucleusand electron. Remembering that the nucleus of the helium atomhas a mass four times as great as the mass of the hydrogen nucleus,the slight difference of the Rydberg constants for hydrogen and forhelium can be calculated. As Bohr 2 pointed out, ionised heliummust be expected to emit a series of lines closely but not exactlycoinciding with lines of the ordinary hydrogen spe€trum : thealternate members of the' Puppis series cannot be superposed onthe Balmer hydrogen line~, but should be slightly displaced withrespect to them. Thus near the hydrogen lines Ha(,,6563), HIl(A4861),Hy(A4340·5), Ho(A4102), there are helium lines at ,,6560·37,10.4859·53,,,4338·86, ,,4100·22 respectively, and the observed discrepancies arecompletely explained.

1 Mon. Not. R.A.S. !xxiii (1912), p. 62; Phil. Trans. ccxiv (1914), p. 225I Nature xcii (1913), p. 231. The attribution of the line ,\4686 to helium was.con-

firmed experimentally by E. J. Evans, ibid. p. 5.114

SPECTROSCOPY IN THE OLDER QUANTUM: THEORY

A. Fowler 1 noticed that some lines which he had observed inthe spectrum of magnesium could be arranged in a series with aRydberg constant which, like the Rydberg constant for ionisedhelium, was approximately four times the normal Rydberg constant.The explanation obviously was that they were produced by ionisedatoms. The nucleus of magnesium has a positive charge 12e; andwhen the metal is ionised, the outermost electron, which describesthe orbits concerned, is under the influence of the nucleus togetherwith 10 negative electrons in the inner orbits, that is, the effectivecentral charge is 2e; and so, as in the case of ionised helium, theRydberg constant must be multiplied approximately by 4.

In 1923 this principle was carried further by F. Paschen andA. Fowler. Paschen 2 found a series in the spark spectrum ofaluminium capable of being represented by a series of the Rydbergtype in which the Rydberg constant had nine times its normalvalue. Since the nucleus of aluminium has a positive charge 13e,this series obviously belonged to doubly-ionised atoms, for whichthere would be 10 inner negative electrons, and the effective centralcharge on the outermost or active electron would be 3e.

Fowler 3 went further still and discovered in the case of siliconnot only series with a Rydberg constant 9 R, due to double ionisation,but also series with a Rydberg constant 16R, due to trebly-ionisedatoms of silicon.

Another type of experimental investigation which led to resultsconfirming Bohr's theory must now be mentioned. In the courseof an investigation on the ionisation of gases by collisions of electronswith the atoms, James Franck (b. 1882) and Gustav Hertz (b. 1887)(a nephew of Heinrich Hertz) found 4 that the collision of slowelectrons with the atoms of mercury vapour led in some cases toemission of the mercury line .\2536. So long as the kinetic energyof the electrons is smaller than hv, where v is the frequency of theline, they are reflected elastically by the mercury atoms; but whenthe kinetic energy is greater than this, light of frequency v is emitted.Evidently the collision brings about an excited state of the atom,and the radiation is emitted when the atom falls back into its normalstate. The subject was pursued in many papers by these and otherauthors,S with the following general results: Every transition if anelectronfrom one orbit to another, corresponding to a line if the atom's spectrum,can be brought about by the collision of a free electron with the atom, theelectron losing an amount hv of kinetic energy, where v is the frequency of theline. Which transitions occur, depends on the state of excitement of the atom.With a normal or unexcited atom, the transitions are those that correspond tothe lines of the absorption-spectrum if the unexcited atom.

1 Nature, xcii (1913), p. 232; Phil. Trans. ccxiv (1914), p. 225• Ann. d. Phys. lxxi (May 1923), p. 142 • Proc. R.S. ciii (June 1923), p. 413• Verh. deutsch. phys. Ges. xvi (May and June 1914), pp. 457, 517• An extensive bibliography of the experimental work is given at the end of a paper

by Franck and Hertz, Phys. -?-S.xx (1919), p. 132.115


Moreover, there is a direct connection between the ionisationof an atom and its spectrum. If eV denotes the energythat must be givento an electron in order that it may ionise the atom, so that V is the' ionisation-potential,' then

eV =hv

where v denotes thefrequency of the ultra-violet limit of a series in the spectrum:for the unexcited atom, the ionisation-potential is given by this equation,where v is the limiting frequency of the absorption series of the unexcited atom.Thus in the case of mercury vapour, the ionisation-potentiall is 10·27volts, correspond-ing to the ultra-violet limit at AII88 of the seriesof which ;>..2536is the first term.2

In one notable case, namely that of helium, the value derivedspectroscopically proved more reliable than that obtained in thefirst place by direct observation of ionisation brought about byelectronic bombardment. The latter value was believed until 1922to be 25·3 volts, whereas Lyman found in that year 3 that the limitof the spectral series concerned was at ;>"504,which would correspondto an ionisation-potential of24·5 volts. The discrepancy led Franck 4

to re-examine his experimental data, with the result that he founda source of error which led to reduction of the value 25·3 volts by0·8 volts, bringing it into coincidence with the spectroscopic value.

Bohr's theory proved adequate also to account for some typicalphenomena of fluorescence observed by R. J. Strutt.5 It was well-known that if sodium vapour is illumined by the D-light emitted bya sodium flame, it emits D-light as a resonance radiation. Nowthe D-lines constitute the first doublet in the principal series ofsodium, the second doublet being in the ultra-violet, at ;>..3303.Strutt asked the question, whether stimulation of sodium vapourby this second doublet would give rise to D-light? He found thatit would. Moreover, he noticed that there is a line in the spectrumof zinc which practically coincides with the less refrangible memberof the sodium doublet at ;>..3303,but that there is no zinc line coin-ciding with the more refrangible member. So by making use of azinc spark he was able to stimulate the sodium vapour with lightof the wave-length of the less refrangible member only of the sodiumdoublet. It was found that both the D-lines were emitted. Theexplanation depends on the fact that both lines of both doubletscorrespond to transitions down to a certain orbit which is the samein all four cases, and which may be called orbit I. By the absorptionof the ultra-violet light of the second doublet, an electron is raisedfrom orbit I to a higher orbit, and collisions with other atoms mayshake it into somewhat lower orbits, from which it falls into orbit I

1 F. N. Bishop, Phys. Rev. x (1917), p. 244I In making the calculation it is useful to note that according to the equation eV-h.,

one electron volt corresponds to '\12336. This would make '\1188 correspond to 10·4 volts.I Nature, ex (1922), p. 278 ; Astroph. J. Ix (1924), p. I• <Sf. P. xi (1922), p. ISS • Proc. R.S.(A), xci (1915), p. 511

II6


with emission of the D-lines. Bohr's theory is thus seen to be offundamental importance in regard to fluorescence.

In the first year of the century W. Voigt 1 had predicted theexistence of an electric analogue to the Zeeman effect-a splittingof spectral lines by an intense electric field. However, discussingthe matter by classical physics, he concluded that with a potential-fall of 300 volts per cm. in the field, the effect would be only aboutthe 20,000th part of the separation between the D-lines of sodium,and hence would be unobservable. Notwithstanding this unfavour-able opinion, in 1913Johannes Stark 2 (b. 1874), when investigatingthe light emitted by the particles which constitute the canal rays,examined the influence of an electric field on this light, and observeda measurable effect. By spectroscopic observation in a directionperpendicular to the field, it was found that the hydrogen lines Hp,Hy were each split into five components, the oscillations of the threeinner components (which were of feeble intensity) being parallel tothe electric field and the oscillations of the two outer components(which were stronger) being at right angles to the field. The distancebetween the components was proportional to the electric force. Forhelium, it was found that the effect of the electric field on the linesof the principal series and the sharp series was very small and hardlydistinguishable, but the effect on the lines of the diffuse series S wasof the same order of magnitude as for the hydrogen lines, thoughof a different type.

The Stark-e.Jfeet, as it has since been called, was extensivelystudied from the experimental side in the years immediately followingits discovery, chiefly by Stark and his disciples.4 Some curiousproperties were noticed, such as that in some cases, where no splittingcould be observed with the fields employed, terms were displacedtowards the ultra-violet; that terms belonging to the same serieswere not as a rule affected in the same way, but that there was oftena similarity in behaviour between terms which belonged to differentbut homologous series, and which had the same term-number:and that when the spectroscopic observation was in a directionparallel to the field, only those components of an electrically-splitline appeared which, when the observation was at right angles tothe field, oscillated linearly perpendicularly to the field; but thatthese components were now unpolarised.

1 Ann. d. Phys. iv (1901), p. 197J Berlin Sitz. 20 November 1913,p. 932; reprinted Ann. d. Phys. xliii (1914)". 965.

The effect was discovered independently at the same time by A. Lo Surdo, Ren • Linceixxii (1913), p. 664; xxiii (1914), p. 82.

J It may be noted that lines of the diffuseseriesare broadened when the gas-pressureis increased, while those of the other series are not much affected.

• J. Stark, Verh. deutsch. P'JYs. Cu. xvi (1914), p. 327; J.. Stark and G. Wendt, Ann.d. Phys. xliii (1914), p. 983; J. Stark and H. Kirschbaum, Ibid. pp. 991, 1017; J. Stark,Ann. d. Phys. xlviii (l91~), pp. 193,210; H. N~quist, Phys. Rev. ii (1917), p. ~2? ;J. Stark,O. Hardtke and C. Liebert, Ann. d. Phys. IVI (1918), p. 569; J. Stark, Ibid. p. 577 ;G. Liebert, ibid. pp. 589, 610 ; J. Stark and O. Hardtke, Ann. d. Phys.lviii (1919),p. 712 ;J. Stark, ibid. p. 723

117


A study of the Stark effect, from the standpoint of Bohr's theory,was undertaken by E. Warburg 1 and by Bohr himself.2 I twasassumed that the field influences the stationary states of the emittingsystem, and thereby the energy possessed by the system in thesestates, splitting a term into two or more components. Warburgshowed that the effect to be expected, according to quantum theory,of an electric field on the spectral lines of hydrogen, would be of thesame order of ma~nitude as that observed experimentally by Stark;but the investigatIOns were imperfect as compared with those pub-lished two years afterwards by Schwarzschild and Epstein, whichwill be described later.

The next ~reat advance in theoretical spectroscopy was theremoval of a lImitation characteristic of the original Bohr theory,namely that it took into consideration only a single set of circularorbits round each atom: it was obviously desirable to extend Bohr'sprinciples by taking into account more than one degree of freedom.This was achieved independently and almost simultaneously in 1915by William Wilson S (b. 1875) and Sommerfeld.4 Their idea recallsthat which had inspired Sommerfeld's paper of 1911.5

In the circular orbits of the steady states in Bohr's theory of thehydrogen atom, let q denote the angle which the line joining theelectron to the proton makes with a fixed line; then the momentumP corresponding to the co-ordinate q is the angular momentum of theelectron round the proton, and for steady states this must be a multipleof Ii. Thus, since h = 27T1i, we have

PJdq = a multiple of h

where the integration is taken once round the circle; or, remember-ing that fpdq is the definition of the Action, we haveIncrease of Action in going once round the orbit = a multiple of h.

Wilson and Sommerfeld generalised this into the statement thatunder certain circumstances, in a system with several degrees offreedom, if qh q2> . . . are the co-ordinates, and PI, P2' . . . are thecorresponding momenta, then the steady states of the system are suchthat fpldql' fp2dq2, ... , are multiples of h, when the integrations areextended over periods corresponding to the co-ordinates.6

1 Verh. d. deutsch. phys. Ges. xv (December 1913), p. 1259I Phil. Mag. xxvii (March 1914), p. 506; xxx (1915), p. 404I Phil. Mag. xxix (1915), p. 795; xxxi (1916), p. 156• Miinchen Sitz. 1915, pp. 425, 459; Ann. d. Phys. Ii (1916), p. 1. At almost the same

time Jun Ishiwara, Tokyo Sugaki-But. Kizj(2) viii, No.4, p. 106, Proc. Math. Phys. Soc. Tok;'o,viii (1915), p. 318 published proposals which in some respects resembled those of Wilsonand Sommerfeld; cf. also the work of Planck on systems with several degrees of freedom,Verh. d. deutsch. phys. Ges. xvii (1915), pp. 407, 438; Ann. d. Phys. I (1916), p. 385.

• cf. p. 104• The rule as thus broadly stated is evidently not independent of the choice of co-

ordinates, a point on which see Einstein, Verh. d. deutsch. phys. Ges. xix (1917), p. 82,and other papers discussed later in the present chapter.

118

whereVi = (r!!.) I + rl (t£O) I

dt dt'Zelv=- -r '

so the momenta corresponding to 0 and r are mrS dO/dt and m dr/dtrespectively. The quantum conditions specifying a steady state weretherefore taken by Sommerfeld to be

Jmr2 ~O dO=nh (1)dt

and Jm t!!. dr = n' h (2)dt

where nand n' are whole numbers, and the integrations are to betaken once round the orbit.

From (1) and (2) we have

(n+n')h= JmvSdt.

But from the known properties of Keplerian elliptic motion, wehave

denoting by a the major semi-axis of the orbit. Therefore

(n + n') h = 21Te(ZmaY~. (3)Also from (1)

h 2 dO JdO 2 S dOn = mr di = 1Tmr dt

son 1 2 dO--=---mr -

n+n' e(Zma)I dt

ba

119

(4)


from the known properties of elliptic motion, where b denotes theminor semi-axis. Equations (3) and (4) connect the quantumnumbers nand n' with the semi-axes of the orbit.

Ze2Also, total energy of electron tmv2 - r

27T2e4Z2mh2(n+n')1

RZ2h- (n+·n'F (5)

where R is Rydberg's number.The total number of possible steady orbits has been greatly

increased by Sommerfeld's introduction of ellipses in addition toBohr's circular orbits: but from equation (5) it seems as if thetotal number of possible values of the energy has not been increased,since the energy depends only on the single number (n + n'). Thusapparently in the case of the hydrogen atom (for which Z = I) weshould get exactly the same spectra as before. However, as Sommer-feld pointed out, the theory has so far supposed the orbit to be aKeplerian ellipse, the electron being regarded as a particle ofconstant mass; whereas, since the velocity in the orbIt is great,the relativistic increase of mass with velocity ou~ht to be taken intoaccount.1 Sommerfeld showed that the orbit IS an ellipse with amoving perihelion, the motion of the perihelion being great or smallaccording as (for the quantified ellipses) the quantity e2/cn is greator small. Actually e2/cn, which is generally denoted by a andcalled the fine-structure constant, has a value 2 which at that time wasbelieved to be about 7.10-3, and has since been found to be 1/137.It represents the ratio of the velocity of the electron in the first Bohrorbit to the velocity of light.

The formula for an energy-level or ' term' of the hydrogen atomnow becomes, in the first approximation,

T=To+T1

where To is the uncorrected value, so To= R/n2, n being the' principalquantum number,' and T1 is a correction term given by

T1 = Ra2(~ _~)n4 k 4

1 The necessity for a relativity correction had been pointed out already by Bohr,Phil. Mag. xxix (1915), p. 332.

• The fact that e'jcli is a pure number had been pointed out by Jeans, Brit. Ass. Rep.1913, p. 376, who suspected that it might have the value 1/(47T)·.

120


where k is a second quantum number. The frequency of a radiationcorresponding to the fall of an electron from one energy-level toanother will depend on the values of nand k for both levels. It istherefore evident that to each spectral line of the original theory(which depended only on the principal quantum numbers) therewill now correspond a group of lines very close together; thesepredicted lines were found to agree remarkably well with linesobserved 1 in what was called the fine-structure of the spectrum.

As we shall see later, Sommerfeld's theory is not the completeexplanation of the fine-structure, which depends also on a propertydiscovered later, that of the spin of the electron; but it was rightlyacclaimed at the time as a great achievement. The quantumnumber k now introduced in connection with hydrogen was foundlater to account for the distinction between the principal andsubordinate series of e.g. the alkali metals.

Sommerfeld's extension of Bohr's theory led Karl Schwarzschild 2

(1873-1916), director of the Astrophysical Observatory at Potsdam,and Paul Sophus Epstein 3 (b. 1883), a former student of Sommerfeld'sat Munich, to investigate theoretically the Stark effect. In a station-ary state of an atom the active electron has three degrees of freedom,and we may therefore expect that the state will be specified by threequantum numbers. Success in the mathematical treatment of theproblem depends on the possibility of finding three pairs of variablesqi, pi (where qi is a co-ordinate and pi the corresponding momentum),such that the Hamilton's partial differential equation belonging tothe classical problem 4 can be solved by separation of variables.The integrals fpidqi corresponding to these separate pairs of variablesare then equated to multiples of h. By carrying out this programme,Epstein found in the case of the Balmer series of hydrogen the follow-ing simple formula for the displacements of the Stark componentsfrom the original position of the spectral line :

where

Here E is the applied electric force (in the electrostatic system ofunits), (mI, m2, ms) are the quantum numbers of the outer orbitfrom which the electron falls into an inner orbit of quantum numbers(nI, n2, ns), I-' is the mass of the electron, and v is the reciprocal

1 Especially by the measurements, by F. Paschen, Ann. d. Phys. I (1916), p. 901, ofthe fine structure of Fowler's spectra of ionised helium; cf. E. J. Evans and C. Croxson,Nature, xcvii (1916), p. 56

• Berlin Sitz. April 1916,p. 548I Phys. ZS. xvii (1916), p. 148; Ann. d. Phys. I (1916), p. 489• cr. Whittaker, Anafytical Dynamics, § 142

121


of the wave-length. The sums (rnl+rn2+rnS) and (nl+n2+nS)correspond to the ordinal number of the term in the Balmer series,so nl + n2 + ns = 2, while rnl + rn2 + rna = 3 for the line Ha, 4 for Hp,5 for Hy, etc. In the case of the Ha-line, Z could take the valuesO~1, 2, 3, 4, 5; and the wave-numbers calculated by the formulaagreed well with Stark's observations, not only as regards themagnitude of the displacements but also as regards polarisation,which depends on whether (rns -ns) is odd or even. Similar satis-factory results were found for the other Balmer lines.

The theory of the Stark effect given by Schwarzschild andEpstein is valid so long as the electric field is strong enough to makethe Stark separation of the components much greater than theseparation due to the fine-structure. The case when the field isso weak that this condition is not satisfied was treated by H. A.Kramers,l who found that the fine-structure lines were split intoseveral components, whose displacements were pro:portional to thesquare of the electric field-strength. Kramers also mvesfigated theStark effect on series lines in the spectra of elements of higher atomicnumber.

The methods of quantification employed by Sommerfeld,Schwarzschild and Epstein may be justified by a theory known asthe theory of adiabatic invariants, which was originally created bythe founders of thermodynamics,2 who studied quantities that areinvariant during adiabatic changes. It was now developed byPaul Ehrenfest 3 (1880-1933) of Leiden, whose startin~-point wasa theorem proved by Wien in the establishment of his dIsplacementlaw,4 namely that if radiation is contained in a perfectly-reflectinghollow sphere which is slowly contracting, then the frequency Vp

and the energy Ep of each of the (infinitely-many) principal modesof vibration of the cavity increase together during the compressionin such a way that their ratio remains constant. Ehrenfest showedthat this property is really the basis of the law that the ratio ofenergy to frequency Elv can take only the values 0, h, 2h, ... :for a different assumption, such as that E/v2 is proportional to0, h, 2h, ... , could be shown to lead to a conflict with the secondlaw of thermodynamics. What impressed Ehrenfest was the circum-stance that although Wien's theorem was deduced by purely classicalmethods, it was valid in quantum theory, and actually ind1cated afundamental quantum law. He was thus led to suspect that if in

1 ZS.f. Phys. iii (1920), p. 199I cf. R. Clausius, Ann. d. Phys. cxlii (1871), p. 433; cxlvi (1872), p. 585; cI (1873),

p. 106; English translations, Phil. Mag. xlii (1871), p. 161; xliv (1872), p. 365; xlvi(1873), pp. 236,266; C. Szily, Ann. d. Phys. cxlv (1872), p. 295 -Phil. Mag. xliii (1872),p. 339; L. Boltzmann, Vorlesungen iiber die Principe der Mechanik, ii Teil (Leipzig, 1904),ch. iv

I Ann. d. Phys. xxxvi (1911), p. 91; Ii (1916), p. 327; Verh. d. deutsch. phys. Ges. xv(1913), p. 451; Proc. Arnst. Acad, xvi (1914), p. 591 j xix (1917), p.576; Phys. ZS. xv(1914), p. 657; Phil. Mag. xxxiii (1917), p. 500

• cf. Vol. I, p. 379122


a dynamical system some parameter is slowly changed (as was theradius of the sphere in Wien's theorem), and if some quantity J isfound to be invariant according to classical physics during thischange (as was Elv in Wien's case), then the system may be quantifiedby putting J equal to a multiple of h. An example of ' adiabaticchange' is furnished by the motion of a simple pendulum when thelength of the suspending cord is very slowly altered.

The general problem he formulated as follows. Let there be asystem of differential equations (for simplicity taking two inde-pendent variables)

where the letter m stands for one or more constant parameters.These equations will possess integrals

Then it is yossible that some functions of the constants of integrationCl, C2, and oj the parameters m, exist, which maintain their values unalteredwhen the parameters m are varied in an arbitrary manner, provided the variationis very slow. Such functions are called adiabatic invariants; andEhrenfest now showed that the quantities which had been put equal towhole-number multiples of h in earlier papers on quantum theory were alwaysadiabatic invariants; and, moreover, that the rule thus indicated wasvalid generally. In other words, stationary orbits are adiabaticallyinvariant.

Now it had been shown by J. Willard Gibbs I and Paul Hertz,2that in the case of a dynamical system with one degree of freedomwhose solutions are periodic and whose equations of motion are

so that the trajectory is represented in the (q, p) plane by a curve ofconstant energy

H(q, p) = W,

then the area ffdq dp enclosed by this curve is an adiabatic invariant;or, Jpdq taken round the curve is an adiabatic invariant: that is,the increment of the Action, in a complete period, is an adiabatic invariant.

This theorem is true even when the system has more than one,degree offreedom, provided the solution is periodic. As an example,s

I Principles qf statistical mechanics, 1902, p. 157 I Ann. d. Phys. xxxiii (1910), p. 537• Other examples of adiabatic invariants are given by W. B. Morton, Phil. Mag.

viii (1929), p. Hl6 and by P. L. Bhatnagar and D. S. Kothari, Indian J. Phys. xvi (1942),p.271.

123


consider the case of a planet describing an elliptic orbit under theNewtonian law of force to the focus; and suppose that the mass mof the planet is slowly increased by moving through a dusty atmos-phere, while gradual changes also take place in the strength J.t ofthe centre of force. It is well known that the velocity v, the radiusvector r and the mean distance a, are connected by the equation

v2= J.t (~-~).

Varying this, we have

or

When the planet picks up a small particle Om previously at rest,by the theorem of conservation of linear momentum we have

ov Omo(mv) = 0, or - = --v m'

so the previous equation becomes

oa = _ ~(20m + oJ.t).a2 J.t m J.t

,

Suppose the changes in m and J.t to be brought about so graduallythat the increments Om and oJ.t are spread over a large number oforbital revolutions; we may then, before integrating the lastequation, replace v2 by its time-average. Now since the averagevalue of l/r IS lla, we see from the equation

v2 = J.t(~-~)that the average value of v2 is J.t/a. When this is inserted, the varia-tional equation becomes

Integrating, we havem2 ~ = Constant:

and since the increment of the Action in a complete period is knownto be 21TmJ.t1ai, this equation verifies for Keplerian motion the theorem

124

But

is the sum of two integrals, namely fm (dr/ dt) dr and fmr9 (dO/dt) dO: andsince the alterations of I-' and m do not affect the angular momentumround the centre of force, we see that the second of these integrals,taken singly, must be an adiabatic invariant. Whence it follows thatthe first must be also. Thus a Justification is provided for Sommerfeld'schoice of integrals to be equated to integral multiples qf h.

We shall now establish a connection (not restricted to Keplerianmotion) between the adiabatic invariant representing the incrementof Action in a complete period (which we shall denote by J), the totalenergy of the motion (which we shall denote by W) and the frequencyv of the motion (i.e. the reciprocal of the periodic time).

The solution of the differential equations of motion will consistin representing the original Hamiltonian variables in terms ofQ=vt+ € (where € is an arbitrary constant) and a variable P whichis conjugate to Q and is actually constant: the transformationfrom (q, p) to (Q, P) being a contact-transformation, so that

pdq-PdQ=do.

where dO. is the differential of a function which resumes its valueafter describing the periodic orbit.

Integrating this equation once round the orbit, we have

Jpdq-PJdQ= Jdo.=O.

JdQ= J~9-dt=vJdt= 1.

Therefore P = fpdq integrated round the orbit: that is, P is the adi-abatic invariant which we have called J. The Hamiltonian equa-tions in terms of the variables P and Q are

dQ_oHdt - OP'

The former equation is

so we have

where W denotes the total energy.

dP oHdt =oQ=O.

dHv= dP

dWv= dJ

125


The methods employed by Schwarzschild and Epstein in themathematical treatment of the Stark effect led other workers inquantum theory 1 to study dynamical systems which can be solvedby separation of variables, a subject which had been studied exten-sively in 1891 and the following years by Paul Stackel of Kie1.2 Insuch systems the integral of Action,

J2Tdt

where T denotes the kinetic energy, can be separated into a sumof integrals each of which depends on one only of the co-ordinates,

In general, the motion of each co-ordinate is a libration, i.e. it oscillatesbetween two fixed limits, the values of which are determined by theintegrated equations of motion.3 For such systems it was provedby J. M. Burgers' that the single integrals

Jk = J,,,/ {Fk(qt) }dqk'

where the integration is taken over a raJ,gJ In which qk oscillates once upand down between its limits, are adiabatic invariants 5: and thereforethe rule that must be followed in order to quantify is to write

Jk=nkh

where nk is a whole number, and is called a quantum number.Denoting by T the average value of the kinetic energy, we have

2T=_I_JB 2TdtA+B -A

where A and B are large numbers. Hence

Now

2T=A-lB[A~ J{ Fk(qk)}dqk'

Jk =J,/ {Fk(qk) }dqk

1 P. Debye, GOlt. Nach. 1916, p. 142 ; Phys. <S. xvii (1916), pp. 507, 512 ; A. Sommer-feld, Phys. <S. xvii (1916), p. 491

• Habititatiansschrift, Halle, 1891; Camptes Rendus, cxvi (1893), p. 485; Clod (1895).p. 489; Math. Ann. xlii (1893), p. 545

• There is, however, often among the co-ordinates an azimuthal angle which canincrease indefinitely but with respect to which the configuration of the system is periodic:an increase of 2" in it takes the place of the libration of the other co-ordinates.

• Phil. Mag. xxxiii (1917), p. 514; Ann. d. P~ys. Iii (1917), p. 195• Leaving aside some special cases of degeneration

126

(k = 1, 2, • • • n)


taken over a range of integration described in the periodic time of theoscillation of q". So

1 "A+B2T=-- L--l"A+B" T"

or

where VI- is the frequency associated with the co-ordinate q".Now it was shown in 1887 by Otto Staude 1 of Dorpat (for the

case of two degrees of freedom) and by Stackel 2 in 1901 (for anynumber of degrees of freedom) that a dynamical system, for whichHamilton's partial differential equation can be integrated byseparation of variables, is multiply periodic,3 that is to say, the co-ordinates (qIl q2, •.• , qn) can be expressed by generalised Fourierseries of the type

qr=2 Qrcos {27T(T1V1+ TzV2+ ••• + TnVn)t+yr}

where Qr and Yr are functions of T1, T2, ••• Tn, where the V" havethe same meanings as before, and where summation is over integralvalues of the parameters T1) Tz, ... Tn.

I t can be shown, as in the case of a system with one degree offreedom, that

oWv,,=-oj"where W denotes the total energy, so that the frequency of theradiation emitted in the transition from a state Wr to a state W,is given by the equation

Now suppose that the quantum numbers belonging to the states Wrand W, are large, and that the quantum numbers of the state Wr differtrom those of the state W, by Tl' T2, ••• Tn, respectively, so that

t..j 1= T1h, Llj 2 = Tzh, • • ., LlJn = Tnh.

Since consecutive orbits with large quantum numbers do not differgreatly from each other, we can represent the increment Wr- W. ofthe energy approximately by

oW oW oWLlW = aj~ LlJ1 + aJ2 LlJz+ ••• + aJ~ LlJn

, Math. Ann. xxix (1887), p. 468; Sitz. d. Dorpater Naturfor., April 1887• Math. Ann. liv (1901), p. 86• The German term, introduced by Staude, loc. cit., is bedingt-periodisch.

127


so we haveoW

+ OJn Tnh

or

that is to say, the frequency of the spectroscopic line emitted in the transitionr-+-s is approximately V1T1 +V2T2+ . . . +VnTn, which is the frequencyof one term in the multiple Fourier series representing the classical solutionof the problem, namely that for which

where (' n1,rn2, . . . ) are the quantum numbers in the state r, and('n1, 'n2, . . . ) are the quantum numbers in the state s. This was calledby Ehrenfest the correspondence theorem for frequencies.

In the same year (1916) in which Schwarzschild and Epsteinpublished their explanations of the Stark effect, Sommerfeld I andDebye 2 showed that the Zeeman effect also could be brought withinthe compass of the quantum theory. If we consider the motion ofan electron under the influence of a fixed electric charge at theorigin, and a magnetic field H parallel to the axis of z, the classicalequations of motion may be written in the Hamiltonian form

dpr _ oKdt - - oqr (r= 1,2,3)

where q1 is the radius vector from the origin to the electron, q2 isthe angle between q1 and the intersection of the plane of xy withthe plane of instantaneous motion of the electron (which we maycall the plane of the orbit), q3 is the angle between the fixed axis Oxand the intersection of the plane of xy with the plane of the orbit,PI is the component of linear momentum along the radius vector,P2 is the angular momentum uf the electron round the origin, and P3is the angular momentum of the electron about the axis of z. Thesevariables are separable. During the motion, the plane of the orbitprecesses 3 uniformly round the axis of z with angular velocityeH/2mc, so the dynamical equations involving q3 and P3 must be

dq3_ eHdt - 2mc'

dP3 =0dt

1 Phys. ZS. xvii (1916), p. 491I Phys. ZS. xvii (1916), p. 507; cf. also A. W. Conway, Nature, cxvi (1925), p. 97

and T. van Lohuizen, Arnst. Proc. xxii (1919), p. 190I This is the Lamwr precession, described in Vol. I, pp. 415--Q

128


and therefore the term involving qs and ps in the Hamiltonianfunction K must be

Proceeding now to the quantification, there will be three quantumconditions of the kind usual in rroblems for which the variablescan be separated, and the third 0 them will be

ps=m/i

where mj is a whole number which will be called the magneticquantum number: so the existence of the magnetic quantum numberis an assertion that the component of angular momentum in thedirection of the magnetic field can take only values which are whole-number multiples of Ii, Since this component attains its greatestvalue when it is equal or opposite to the total angular momentum,we see that mj can take only the values - j, - j + 1, , , " j - 1, j,where j is the quantum number which specifies the total angularmomentum.

Thus the Hamiltonian function K, which represents the energyof the motion, is increased (as compared with the case when themagnetic field is absent) by

mjeliH2mc'

Supposing that we are dealing with the hydrogen atom, the part ofthe energy corresponding to the unperturbed motion is (as inBohr's original theory, neglecting the fine structure)

27T2me4

-fi2hI

and adding to this the part we have just found, due to the magneticfield, we have for the total energy in the stationary state specifiedby the quantum numbers n, mj,

27T2me4 mjehH- --r1JiI + 47Tmc J

and therefore the frequency of the spectral line emitted in thetransition from the state (n, mj) to the state (n', m/) is

v = 27T2me

4 (l- _1-) +~ (mj - m/).h3 n'2 n2 47Tcm

Thus when the magnetic field is applied, in place of the single spectralline specified by (n, n'), we have a number of lines depending on mj and m/.

129


These are the Zeeman components. Their number, intensity and stateof polarisation are furnished by a principle which was not discovereduntil 1918, and which will be described presently. It will appearthat so far as the number, position and state of polarisation of thecomponents are concerned, the quantum theory giVes(for lines suchas those we are now considering, namely single lines which are notmembers of doublets or triplets) exactly the same results as Lorentz'soriginal theory. It will be noted that this became possible becausePlanck's constant h cancelled out in the magnetic part of the aboveexpression for the frequency.

We may note that since in the classical problem the angle awhich the plane of the orbit makes with Oz is given by

cos a=Ps,PI

therefore in the quantified problem, when both the total angularmomentum and its component in the direction of Oz are whole-number multiples of Ii, this angle a can take only certain discrete values.This is an example of what is called space quantification or directionquantification; the plane of the orbit is permitted to be inclined atonly certain definite angles to the direction of the field.

The principle of space quantification was strikingly confirmedby an experiment performed in 1921 by O. Stern 1 and W. Gerlach,2working in the department of Max Born at Frankfort-on-Main. Leta ray of atoms of silver produced by boiling silver in a furnace andpassing the vapour through two fine slits, be travellin~ in the x-direc-tion, and sUfpose that the ray encounters a non-umform magneticfield paralle to the axis of z. In the non-uniform magnetic fieldHz, a particle, whose magnetic moment in the z-direction is M,experiences a mechanical force MoHz/oz in the z-direction. Space-quantification ensures, however, that atoms orient themselves in themagnetic field in certain ways, in fact that M can take only valueswhich correspond to the directions parallel and antiparallel to Hz,and so are equal and opposite. Hence the original ray of silver atomsis split into two rays in the plane of xz, corresponding to these twoopposite values of the z-component of the magnetic moment: inthe experiment, these rays strike a plane to which the atoms adhereand so produce an image which is observed.

It may be noted that the Stern-Gerlach effect concerns only asingle state of the atom, not (like the Zeeman effect) a transitionfrom one state to another.

The effect of crossed electric and magnetic fields on the radiationfrom a hydrogen atom was discussed by O. Klein,3 W. Lenz 4 andN. Bohr.s

1 ZS.f. P. vii (1921),p.249 • ZS.f.P.viii(1921),p.llO; ix (1922),pp. 349,352I ZS.f. P. xxii (1924), p. 109 • ZS. f. P. xxiv (1924), p. 197• Proc. Phys. Soc. xxxv (1923), p. 275

SPECTROSCOPY IN THE OLDER 'IDJANTUM THEORY

Before 1918 the Bohr theory had been applied to determine onlythe frequencies of lines in spectra, and had not yielded any resultsregarding their intensity. Some definite questions concerning mtensitywere, however, suggested by spectroscopic observations; in partic-ular, certain lines, whose existence mIght be expected accordingto the Bohr theory, were found to be absent in the spectrum asobserved: which suggested that transitions of the active electronfrom certain orbits to certain other orbits never took place, so thatthe corresponding lines could not appear.

The explanation of this phenomenon was given by AdalbertRubinowicz 1 (b. 1889), a Pole then working at Munich, who, con-sidering atoms of the hydrogen type, and defining a stationary orbitby the numbers 11, n', introduced by Sommerfeld's quantum conditions

J dBmr2 -dB=nhdt '

remarked that the angular momentum of the atom was nli, and thatin a transition between an orbit of quantum numbers (m, m') andan orbit of quantum numbers (n, n'), this angular momentum wouldchange by 1m - n Iii. But by the principle of conservation of angularmomentum, any change in the angular momentum of the atommust be balanced by the angular momentum carried off by theradiation associated with the transition. By an argument basedpartly on classical electrodynamics and partly on quantum theory(and therefore perhaps not very secure), Rubinowicz found thatthe angular momentum radiated, when the radiation is circularlypolarised, is Ii; and when the radiation is linearly polarised, theangular momentum radiated is'zero. Thus we have

I m - n I Ii = 0 or 1,

and we obtain a selection-principle, namely that the azimuthal quantumnumber n can only change by 1, 0 or - 1.

In a footnote appended to his paper, Rubinowicz explained thatwhen his paper was ready for press there had appeared the firstPart of a memoir by Bohr,2 in which the same problem was approachedfrom quite a different standpoint, depending on the close relationswhich exist between the quantum theory and the classical theoryfor very great quantum numbers. We have seen an example ofthese relations in the correspondence theorem for frequencies; Bohrnow extended this theorem by assuming that there is a relationbetween the intensity of the spectral line radiated and the amplitudeof the corresponding term in the classical multiple-Fourier expansion:in fact, that the transition-probability associated with the genesis of

I Phys. ZS. xix, p. 441 (15 Oct. 1918), and p. 465 (1 November 1918)• D. Kgl. Danske Vid. Setsk. Skr., Nat. og Math. A/d., 8 Raekke,iv, 1 (1918) j cf.N.Bohr,

Ann. d. Phys. !xxi (1923), p. 228


the spectral line contains a factor proportional to the square of thecorresponding coefficient in the Fourier series. Moreover, heextended this correspondence principle for intensities by assumingits validity not only in the region of high-quantum numbers butover the whole range of quantum numbers: so that if any term inthe classical multiple-Fourier expansion is absent, the spectral line, whichcorresponds to it according to the correspondence-theorem for frequencies, willalso be absent. This is a selection-principle of wide application.

He postulated also that the polarisation of the emitted spectralline may be inferred from the nature of the conjugated classicalvibration. Thus 1 considering in particular the Zeeman splittingof a spectral line of hydrogen, when the transition is such that themagnetic quantum number is unchanged, the Zeeman componentwill occupy the same position as the original line, and the radiationwill correspond to that emitted in classical electrodynamics by anelectron performing linear oscillations parallel to the magnetic field;while in the case when the magnetic quantum number changes by±l, (which is the only other possibility permitted by the correspond-ence-principle), we shall obtain Zeeman components symmetricallysituated with respect to the original line, and the radiation willcorrespond to that emitted by a classical electron describing a circularorbit in a plane at right angles to the magnetic field, in one or theother direction of circulation. The polarisation of the emitted linewill therefore in all three cases be the same as that predicted byLorentz 2 on the classical theory.

An extensive memoir by H. A. Kramers 8 supplied convincingevidence of the validity of Bohr's correspondence-principle for thecalculation of the intensities of spectral lines: while W. Kossel andA. Sommerfeld 4 showed that the deductions from the selectionprinciple were confirmed by experiment in the case of many differentkinds of atoms.

The correspondence principle was extended to absorption byJ. H. van Vleck.6

We must now consider developments in the theory of quantumnumbers. We have seen that Sommerfeld specified an energy-levelor ' term' of an atom by two quantum numbers (leaving aside forthe moment the magnetic quantum number). The first of theseis the principal quantum number n, which had been introduced by Bohrin 1913, and which increases by unity when we pass from a termof a spectral series to the next higher term: and the other is theazimuthal quantum number k, which had been introduced by Sommerfeldhimself in 1915, and which distinguishes the different series fromeach other: thus in the sodium spectrum 6 the terms of the' sharp'series have the frequencies R/(n+s)2, where R is Rydberg's constant

1 N. Bohr, Proc. Phys. Soc. xxxv (1923), p. 275 I cf. Vol. I, p. 412• D. Kgl. Dansk. Vid. Selsk. Skr., Nat. og Math. Afd., 8 Raekke, iii, 3 (1919)• Verh. d. deutschphys. Ges. xxi (1919), p. 240 • Phys. Rev. XXIV (1924), p. 330• cf. Vol. I, p. 378

132


and n is a positive whole number: those of the 'principal' serieshave the frequencies Rf(n+ Pl)2 and Rf(n+ P2)2; those of the, diffuse' series have the frequencies Rf (n + d) 2, and those of the, fundamental' series have the frequencies Rf(n +1)2: and theseseries correspond respectively to k = 1, 2, 3, 4. It has becomecustomary to use in place of k the letter I, where 1= k - 1: the reasonbeing that when there is one active electron, its orbital angularmomentum is lIi. The series of energy-levels of the atom for which1=0, 1, 2, 3, ... , were denoted by s, p, d, j, ... , these being theinitial letters of the words sharp, principal, diffuse, fundamental, etc.I has the selection rule that in a transition it can change only to1+1 or I-I.

Evidently, however, the two quantum numbers n and I did notsuffice for the description of the terms of the alkali spectra, for theprincipal series was a series of doublets. To meet this situation,Sommerfeld 1 in 1920 introduced a third number j, which he calledthe inner quantum number and which is different for the two termsof a doublet. This number must arise from the quantification of amotion in some third degree offreedom, and it was natural to supposethat besides the orbital angular momentum of the atom, which wasaccounted for by l, there was yet another independent angularmomentum. This was at first conjectured to be the angular momen-tum of the atom's core, was denoted by sli and was supposed to have(for the alkalis) the value !Ii; so that when it was compounded withthe angular momentum iii, with which space-quantification compelsit to be either parallel or anti-parallel, the resultant total angularmomentum of the atom, jli, could have either of the two values 2

(l- !)Ii and (l + !)Ii. There is a selection-rule that j may passonly to (j+ 1), j, or (j- 1), and moreover transitions in which jremains zero are forbidden.

In the spectra of the alkaline earths there are series of tripletswhich were accounted for in the same way by supposing that theangular momentum sli can take the values 0 and Ii, giving for thetotal angular momentum the three possibilities (l- 1)Ii, Iii, (l + 1)Ii,and so for j the three possibilities j = 1- 1, j = l, j = l + 1. In otheratoms, every value 3 of l was supposed to yield a set of energy-levelsor terms corresponding to the values

j=l+s, I+s-l, II - s I+ 1, II - s I.We have said that at first the independent angular momentum

sli, which is compounded with the orbital angular momentum iiiin order to produce the resultant total angular momentum jli, wassupposed to be the angular momentum of the ' core' of the atom,i.e. possibly the nucleus together with the innermost closed shells

1 Ann. d. Phys. !xiii (1920), p. 221 ; !xx (1923), p. 32• Except when l- 0, in which case there is only one value ofj, namely!.I l=O yields only a single term

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of electrons. This hypothesis was overthrown by Wolfgang Pauli(b. 1900), a Viennese who, after studying with Sommerfeld at Munichand with Bohr at Copenhagen, had become a privat-dozent in theUniversity of Hamburg. Pauli showed 1 that if the angular momen-tum sli belonged to the atomic core, there would follow a certaindependence of the Zeeman effect on the atomic number, and thiseffect was not observed: he inferred that the angular momentumsn must be due to a new quantum-theoretic property of the electron,which he called 'a two-valuedness not describable classically.'This remark suggested later in the same year to two pupils ofEhrenfest, G. E. Uhlenbeck and S. Goudsmit of Leiden 2 the adop-tion of a proposal which had been made in 1921 by Arthur H.Compton,3 an American who was at that time working with Ruther-ford at Cambridge, namely that the electron itself possesses an angularmomentum or spin, and a magnetic moment. Uhlenbeck and Goudsmitproposed as the amount of angular momentum tli; and theysuggested that the values (l + t)1i and (l- t)1i which are possiblefor the total angular momentum jli in e.g. the alkali spectra, areobtained by compounding the angular momentum iii with theelectron-spin, which (since it exists in the magnetic field createdby the orbital revolution of the electron) is compelled by space-quantification to take orientations either parallel or anti-parallelto iii. Associated with the spin there is a magnetic moment whosevalue they asserted (for reasons to be discussed presently) to beeli/2mc.

The discovery of electron-spin raised a question as to the validityof Sommerfeld's explanation of the fine-structure of the hydrogenlines. For if the electron has a spin with a magnetic moment, thentwo different orientations of the spin must be allowed, and to thesemust correspond two different energy-levels for the atom, causing afurther resolution of each fine-structure component into a doublet.The measurements of Paschen had shown, however, that Sommer-feld's formula expressed the experimental data for the hydrogenspectrum satisfactorily: and it was found that the two correctionswhich in a more complete theory should be made to Sommerfeld'sanalysis, namely (1) replacing: the particle-dynamics of Sommerfeldby quantum-mechanics, and (2) taking account of the spin magneticmoment, more or less neutralised each other, producing only areplacement of the quantum number k by U+t).

The theory of spectra was much advanced by investigationsarising out of the new experimental work on the Zeeman effect.

In the first decade of the twentieth century no great progresswas made in the observational field, though in 1907 C. Runge 4

1 ZS.f. P. xxxi (1925), p. 373I Naturwiss, xiii (1925), p. 953; Nature, cxvii (1926), p. 264. It is said that R. de L.

Kronig had the same idea somewhat earlier, but finding it received unsympatheticallyby a colleague, did not publish it.

I Phil. Mag. xli (1921), p. 279 j J. Frankl. [TlSt.cxcii (1921), p. 144• Phys. ZS. viii (1907), p. 232

134


studied a number of spectral lines which show complex types ofresolution, and found that the distances (measured as differencesof frequency) of the components from the centre of the undisturbedline were connected by simple numerical relations with the frequency-difference which was given by Lorentz's theory of the Zeemantriplet, namely eH/47Tcm, where H is the external magnetic forcein electromagnetic units. In 1912, however, F. Paschen and E. Back 1

studied the Zeeman effect in lines which are members of doubletsor triplets in series spectra (e.g. of the alkali atoms), and found thatso long as the magnetic field is not strong the Zeeman splitting isvery complicated. The different lines of a doublet or triplet behavedifferently, thoug-h the separations between the components alwaysincrease proportIOnally to the field-strength. When, however, thefield has increased to such a strength that the separations betweenthe Zeeman components are of the same order of magnitude asthe separations between the components of the original doublet ortriplet, the individual Zeeman components become diffuse and tendto amalgamate: and ultimately, at very great field-strengths, thewhole system reduces to three components constituting a normalZeeman triplet,2 and having its centre at the centre of the originaldoublet or triplet. The difference in character between the Zeemaneffect in weak and strong external magnetic fields was seen at onceto be connected with the fact that the space-quantification of theelectron spin is governed by the magnetic field of the orbital motionswhen the external magnetic field is weak, but is governed by theexternal field when that is sufficiently strong.

The experimental knowledge regarding the splitting that is foundwith comEaratively weak fields in doublets and triplets, or theanomalous .{,eeman effect as it was called, was reduced to a mathematicalformula by Alfred Lande 3 of Tiibingen in 1923. He showed thatthe frequency of anyone of the Zeeman components can, like thefrequencies of the lines of the original spectrum, be represented bythe difference of two' terms' or energy-levels. In a magnetic field,each term W of the original spectrum is split into several termsW +Z. If (supposing for simplicity that there is only one activeor valence electron) the term W has the quantum numbers n, l, j, s,then for its Zeeman components we have

Z = mjegntI ergs2mc '

where e, Ii, m, c have their usual meanings; g, which is called theLandi splitting factor, is given by the equation

= 1+j(j+ I) +s(s+ 1) -l(l+ I)g 2j(j+I)

1 Ann. d. Phys. xxxix (1912), p. 897 ; xl (1913), p. 960• For a discussionof the phenomena taking place during the/assage from weak to

strong fields,cr. W. Pauli, ::;S.f. P. xx (1923), p. 371. • ::;S.. P. xv (1923), p. 189135


and mj is the magnetic quantum number, 1 which, as we have alreadyseen, can take the values

mj=j,j-I, ••. ,-j

so one spectral term splits into (2j + I) equidistant terms. As wehave seen, in transitions mj can change only by 0 or ±l, and thelines so arising are polarised in the same way as the lines of a LorentzZeeman triplet.

If the original term is a singlet level, we have s = 0 and l =j, sog = 1; and lines resulting from combinations of levels in singletseries have the appearance of Lorentz Zeeman triplets.

Observation of the Zeeman effect is of great assistance in deter-mining the character of an y particular spectral line, since theobservation furnishes the value of the Lande splitting factor, andthis knowledge generally leads to the determination of the quantumnumber i.

One naturally inquires why the quantum theory of the Zeemaneffect given earlier (which in fact is valid only for lines which belongto series of singlets) does not apply in general. The reason mustobviously be, that the Larmor procession, whose value was assumedto be eH/2mc in the earlier proof, has not always this value: andthis again can only mean that the ratio of magnetic moment tomechanical angular momentum is somehow different in the caseof the anomalous Zeeman effect from what is asserted in Larmor'stheory, where the ratio is that corresponding to the revolution of anelectron in an orbit large compared with its own size. We concludetherefore that the existence of a g-factor different from unity indicatesthat the ratio of magnetic moment to angular momentum is not thesame for the intrinsic spin of the electron as it is for the orbitalmotion. Now a state of the atom in which the angular momentumis due solely to electron-spin is specified by l = 0, s = t, j = t: andthe g-factor then has the value 2: so we are led to suspect thatfor the electron-spin, the ratio of magnetic moment to angularmomentum is twice as great as it is in the case of an electron circulat-ing in an orbit: that is, it is e/mc instead of e/2mc. The magneticmoment of the spinning electron is therefore conjectured to be eli/2mc; andit is found that with this assumption the Lande g-factor can besatisfactorily explained in all cases. The cause of the anomalousZeeman effect was therefore now revealed.

Some striking confirmations of the validity of the correspondence-principle were obtained in a series of papers which followed a dis-covery made in 1922 by Miguel A. Catalan,2 a Spanish researchstudent of Alfred Fowler's at the Imperial College in London.When investigating the spectrum of manganese, Catalan noticedthat there was a marked tendency for lines of similar character to

1 cf. A. Sommerfeld and W. Heisenberg, ;;:S.f. P. xxxi (1922), p.131• Phil. Trans. ccxxiii (1922), p. 127 (at p. 146)

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appear in groups, and that these groups included some of the mostintense lines in the spectrum: for instance, a group of nine linesbetween 1\4455 and 1\4462. For this kind of regularity he suggestedthe name multiplet. The lines arise from the combination of multipletenergy-levels. Spectral lines, particularly in multiplets, their relativeintensities and their Zeeman components, were studied by manywriters 1 in 1924-5, with results satisfactorily in accord with thepredictions of the correspondence principle.

The rapid development of spectroscopy from 1913 onwards ledto a much fuller understanding of the system of electrons whichsurrounds the nucleus of an atom. Investigators of this subjectnaturally based their work on the Newlands-Mendeleev periodictable,2 and were stimulated by the attempts of A. M. Mayer andJ. J. Thomson 3 to explain it in terms of stable configurations ofelectrons. It was obvious from chemical evidence that the twoelectrons possessed by the helium atom form a very stable con-figuration, which may be regarded as constituting a complete, shell' of electrons surrounding the nucleus. The next atom inorder of atomic number, lithium, must have this shell together withone loosely-attached electron outside it, and for the succeedingelements further electrons are added to this second shell until itcontains eight electrons, when the tenth element neon is formed,thereby completing the shell and arriving again at a very stableconfiguration. The eleventh element sodium has these two completeshells together with one loosely-attached electron outside them,and so on.

The chemical evidence on atomic structure was marshalled in1916-19 by two Americans, G. N. Lewis 4 and Irving Langmuir 5

(b. 1881). Lewis began by considering the different kinds of bondsthat unite atoms into molecules, interpreting one kind of chemicalbond as a couple of electrons held in common by two atoms; manyfacts, such as the tetrahedral carbon atom which is necessary for

1 F. M. Walters, J. Opt. Soc. Amer. viii (1924), p. 245; O. Laporte, ZS. j. P. xxiii(1924), p. 135; xxvi (1924), p. I. These two writers analysed the iron spectrum.H. C. Burger and H. B. Dorgels, ZS. j. P. xxiii (1924), p. 258; L. S. Ornstein andH. C. Burger, ZS.f. P. xxiv (1924), p. 41; xxviii (1924), p. 135; xxix (1924), p. 241 ;xxxi (1925), p. 355; W. Heisenberg, ZS.j. P. xxxi (1925), p. 617; xxxii (1925), p. 841;S. qoudsmit and R. de L. Kronig, Proc ..Amst. Ac. xxviii (1925), p. 418; H. Hon.I!.ZS.j. P.XXX1 (1925), p. 340; R. de L. Krorug, ZS. j. P. XXX1 (1925), p. 885; XXXlll (1925),p. 261; A. Sommerfeld and H. Honl, Berlin Sit;;. (1925), p. 141; H. N. Russell, Nature,cxv (1925), p. 835; Proc. N.A.S. xi (1925), pp. 314, 322; H. N. Russell and F. A.Saunders, Astroph. J. !xi (1925), .p. 38. This paper, arising out of an investigation ofgroups of lines in the arc spectrum of calcium, was of great importance for the studyofcomplexspectra. It took into account the simultaneous action of twodisplaced electrom;F. Hund, ZS.j. P. xxxiii (1925), p. 345 ; xxxiv (1925), p. 296.

• cr. p. 11• cr. p. 21• Joum. Amer. Chern. Soc. xxxviii (1916), p. 762; Lewis, Valence and the Structure of

Atoms and Molecules, New York, 1923• Journ. Amer. Chern. Soc. xli (1919), p. 868. Somewhat similar ideas were published

by W. Kossel, Ann. d. Phys. (1916), p. 229, who studied the transfer of electrons fromelectropositive to electronegauve atoms, resulting in the formation of iom.

137


the understanding of chemical processes in organic substances,indicate that the atom must have a structure in three-dimensionalspace (in contradistinction to a flat ring system) 1; and Lewisfavoured a cubical form. Langmuir, continuing Lewis's work,concluded that the electrons in any atom are arranged in a seriesof nearly spherical shells. The outermost occupied shell consistsof those electrons that do not belong to a closed configuration.These play the principal part in spectroscopic phenomena, and areknown as the active electrons: they also play the principal part inchemical phenomena, in which connection they are known asvalence electrons. The properties of the atoms depend much on theease with which they are able to revert to more stable forms bygiving or taking up electrons. The shells that are completed inhelium and neon respectively have already been mentioned: argon(atomic number 18) in addition to the innermost shell of 2 andthe next shell of 8, has a third shell of 8: while krypton has fourshells of 2, 8, 8 and 18 electrons: and so on. In the light of thismodel atom, Langmuir explained the chemical properties of theelements, and also their physical properties such as boiling-points,electric conductivity and magnetic behaviour.

It was early realised, however, that the formation of the shellscannot be quite regular: it was suggested in 1920 by R. Ladenburg 2

that in the case of the elements of atomic numbers 21 to 28 inclusive(scandium to nickel) the electrons newly added are not placed inthe outermost shell but are used in building up a shell interior tothis. This implies that a shell begins to be formed with potassium(atomic number 19) and calcium (atomic number 20) before thethird shell is really complete.

It was, however, from the study of X-ray spectra (i.e. thecharacteristic X-rays which are emitted by solid chemical elements,or compounds of them, when bombarded by a beam of high-energyelectrQIls) 3 that the greatest help was obtained in relating the shell-structure to the chemical elements. It was known that the character-istic X-rays constituted an additive atomic property and thereforethe X-ray spectra must belong to the atoms of the anti-cathode.Moreover, it was known that they consisted of lines which could bearranged in series, like those of optical spectra: and Moseley'slaw connecting the progression of X-ray spectra from element toelement with the amount of nuclear charge suggested that theirorigin should be sought in the innermost layers of the atom.

Almost irr..mediately after the publication of Bohr's theory,W. Kossel' explained them as being due, like the radiations ofoptical spectra, to transition-processes: they arise when the atom

1 On this see also. Born, Verh. d. deutsch. phys. Ges. xx (191~), p. 230; I.;allde, Verh.d. deutsch. phys. Ges. XXI (1919), p. 2; E. Madelung, Phys. ZS. XIX (1918), p. 524

• Naturwiss, viii (1920), p. 5; ZS.f. Elektrocltem, xxvi (1920), p. 262• cr. p. 16• Verh. d. deutsch.phys. Ges. xvi (November 1914), p. 953; xviii (1916), p. 339

138


is restored to its original state after a disturbance which consistsprimarily in dislodging an electron from one of the innermost shells :the place thus vacated is filled by an electron which falls from ashell at a greater distance from the nucleus, and the place of this isagain filled by an electron from a shell still more remote, and so on.The lowest level may be called, in harmony with Barkla's nomen-clature of three years earlier,! the K-shell, and the fall of an electroninto a vacant place in this shell yields an X-ray of the K-series.The next lowest levels are three known as Ll) Lu and LIm and sofor the others.2

Now if Z denotes the atomic number of the atom considered, bytaking account of Z in the calculation on page Ill, we see thataccording to Bohr's original theory, the frequency of the radi,ationemitted when an electron passes from a circular orbit of angularmomentum nli to one of angular momentum pli is

27T2me4Z2 (1.._1..)h3 p2 n2·

But if we put p = 1, this formula represents precisely the frequenciesof the K-lmes. If p = 2, it represents the L-lines; and so on: fromwhich fact we infer that the electrons in the K, L, M, . . . shellsmove in orbits which have respectively the principal quantumnumbers 1, 2, 3, ... 3 The principal quantum number increases byunity from each shell to the next. That the characteristic X-rays are ofhigh frequency as compared with the radiations in optical spectrais explained by the presence of the factor Z2 in the formula: afactor whose presence accounts at once for Moseley's law that thesquare root of the frequency of any particular line, such as theKa-line, is proportional to the atomic number.

In the early days of X-ray spectroscopy, lines of the K-seriescould not be observed for the lighter elements (atomic numbersbelow 11), because their wave-lengths were longer than those ofX-rays and yet shorter than that of ultra-violet light. The gapbetween X-rays and the ultra-violet was filled about 1928 by JeanThibaud 4 of Paris, Erik Backlin 5 of Upsala, and A. P. R. Wadlund 6

of Chicago, and it was then found possible to trace the K-linescontinuously down to the lightest elements. As might be eXl?ected,since the K-lines represent transitions down to the level of pnncipalnumber 1, they pass into the Lyman series of hydrogen, while theL-lines, which represent transitions down to the principal number 2,

1 cf. p. 16• The y-rays emitted by radio-active bodies come from the nucleus of the atom,

and depend on nuclear levels, so they constitute a phenomenon altogether differentfrom the characteristic X-rays of the K. L, M, ... series.

• Sommerfeld, Ann. d. Phys.(4) Ii (1916), pp. I and 125• Phys. ZS. xxix (1928), p. 241 ; Journ. 0pl. Soc. Amer. xvii (1928), p. 145• Inaug. Diss. Uppsala Universilets Arsskrifl, 1928• Proc. Nal. Ac. Sc. xiv (1928), p. 588

139


pass into the Balmer series, and the M-lines, which represent transi-tions down to the principal number 3, pass into the Paschen seriesof hydrogen.

The Bohr theory, as applied by Kossel, yields at once the lawsof absorption of X-rays. Remembering that the first line of theK-series is emitted when an electron falls from the L-shell to theK-shell, it is obvious that this line could not be expected to appearin the absorption-spectrum: for if in absorption an electron weredislodged from the K-shell, there would normally be no vacant placein the L-shell to receive it: indeed the K-absorption only setsin suddenly, when the incident energy is sufficient to separate aK-electron completely from the atom: that is, the absorption-edgecoincides with the series-limit. This explains why in X-ray spectrathere are no absorption-lines,! but only absorption-edges: and thefrequency of every line in the X-ray emission-spectrum is the difference of thefrequencies of two absorption-edges.

What was known in 1921 regarding the structure of the atomin relation to the physical and chemical properties of the elementswas set forth in an extensive survey by Bohr,2 whose principles werevindicated in the following year in a somewhat dramatic way. Oneof the missing elements in the Newlands-Mendeleev table was thatof atomic number 72. Now the elements immediately preceding (ofatomic numbers 57 to 71 inclusive) belong to the group of the' rareearths,' and it was expected by many chemists that number 72would also belong to this group. Indeed in 1911 G. Urbain,S byfractionation of the earths of gadolinite, believed that he had dis-covered a new rare earth to which he gave the name of Celtium ;and this was later identified by Urbain and Dauvillier with themissing element 72. Bohr, however, gave a rational interpretationof the occurrence of the rare earths in the periodic system, assertingthat they represent a gradual completion of the shell of electronsfor which the principal quantum number is 4, while the number ofelectrons in the shells of principal quantum numbers 5 and 6 remainsunchanged. With the rare-earth lutecium (71) the shell n =4 attainsits full complement of 32 electrons, and it follows that the element 72cannot be a rare earth, but must have an additional electron in theshells n = 5 or n = 6: it must in fact be a homologue of zirconium.In 1922 D. Coster and G. Hevesy of Copenhagen verified thisprediction 4: examining the X-ray spectrum of a Norwegian zir-conium mineral, they found lines which, by Moseley's rule, mustcertainly belong to an element of atomic number 72: and for thisthey proposed the name of Hafnium (Hafniae = Copenhagen). Itschemical properties showed that it was undoubtedly analogous totitanium (22) and zirconium (40).

1 For a refinement of this general statement cf. Kossel, ;::S.j. P. i (1920), p. 119.• Fysisk Tideskrift, xix (1921), p. 153- Theory qf Spectra and Atomic Constitution, Cam-

bridge, 1922 • Comptes Rendus, clii (1911), p. 141; Chern. News, ciii (1911), p. 73• Nature, cxi (1923), p. 79


In 1922 and the following years great additions were made tothe accurate knowledg-e of X-ray spectra and their relation toatomic number.l GUIded by this work, in 1923 N. Bohr andD. Coster 2 introduced new symbols for the layers of electrons inthe atoms, based on their spectroscopic behaviour with respect toX-rays. The innermost group was now denoted by I (1, 1)K, thenext groups by 2(1, I)L1, 2(2,I)Ln, 2(2, 2)Lm, and so on outward.Here the letters with the Roman subscript numbers indicate thepreviously-recognised shells with their subsidiary levels, while thesymbols of the form n(k, j) define the subsidiary levels more closely,n being Bohr's principal quantum number, while k and j are wholenumbers which were later to be identified with Sommerfeld's azi-muthal quantum number and a function of Sommerfeld's innerquantum number respectively. Then in January 1924A. Dauvillier S

showed experimentally (by examining the absorption relative to thelevel) that the 8 electrons in the L-level must be partitioned intosub-groups of 2, 2 and 4. These results quickly led to the under-standing of the orbits and energies of the various groups of electronswhich was finally accepted, and which was proposed in 1924-5 byEdmund C. Stoner of Cambridge 4 (whose arguments were basedon physical reasoning) andJ. D. Main Smith of Birmingham 5 (whoapproached the matter from the chemical side). According to thissystem, to each complete shell corresponds a definite value of theprincipal quantum number n. Within this shell the subsidiaryquantum number l can take the values 0, I, ... (n -I) : and theinner quantum number j of an electron can then take the values (l + t)or (/-t), (unless l = 0, in which casej can take only the value t). Thenumber of electrons in the sub-group (n, l, j) is simply (2j+ I), andtherefore the total number of electrons with the quantum numbersnand lis 2(2l + 1), and the total number of electrons in the n-shell is

2{1+3+5+7+ ... +(2n-I)}or

2n'.Helium has altogether 2 electrons, forming a complete K-shell. Forneon there are 10 electrons, namely 2 forming a complete K-shelland 8 forming a complete L-shell. For argon there are 18 electrons,of which 2 form a complete K-shell, 8 form a complete L-shelland 8 form an incomplete M-shell : the M-shell is, however, complete

1 Dirk Coster, Phil. Mag. xliii (1922), p. 1070",xliv (1922), p. 546; A. Lande, ZS.f. P.xvi (1923), p. 391; Manne Siegbahn and A. Lli~ek, Ann. d. Phys. lxxi (1923), p. 187 ;M. Siegbahn and B. B. Ray, Ark.f. Mat, Ast.och Fys. xviii (1924), No. 19; M. Slegbahnand R. Thoraeus, Phil. Mag. xlix (1925), p. 513; cr. L. de Broglie and A. Dauvillier,Phil. Mag. xlix (1925), p. 752

• ZS.f. P. Xli (1923), p. 342 • Comptes Rendus, clxxviii (1924), p. 476• Phil. Mag. xlviii (1924), p. 719• J. Chern. Ind. xliii (1924), p. 323 ; xliv (1925), p. 944; Chemistry and Atomic Structure,

London, 1924; cf. A. Sommerfeld, Ann. d. Phys. lxxvi (1925), p. 284; Phys. ZS. xxvi(1925), p. 70


in the next noble gas, krypton, which has 2 K-, 8 L-, 18 M-, and8 N-electrons. For any complete shell the orbital, spin and totalangular momentum are all zero.

We have seen that the number of electrons in the sub-group(n, l,j) is (2j+ I) : but for a given value ofj, the magnetic quantumnumber mj can take precisely the (2j + I) values j, j - I, . . ., - j.Thus there is one and only one electron corresponding to eachdistinct state or energy-level, i.e. each distinct set of the four quan-tum numbers (n, I, j, mj). In 1924 Pauli 1 based on this fact ageneral principle, that two electrons in a central field can never be in statesof binding which have the same four quantum numbers. This assertion canbe extended to systems in which there is not a single central field:e.g. it applies to electrons which are in the field of two nuclei at thesame time: the states of these electrons can be described by quantumnumbers, and it is still true that no two electrons can have the samestate, i.e. be described by the same set of quantum numbers. Thestatement in this general form is called Pauli's exclusion principle.I t is valid for protons as well as for electrons, and indeed for allelementary particles whose spin is tiL

Another discovery of Pauli's, made in the same year,2 relatedto a structure much finer than the ordinary multiplet structure,which is observed in some spectra, and which is called the hyperfinestructure of spectral lines. Pauli showed that this is to be ascribednot to the electron-shells, but to the influence of the atomic nucleus,which may itself have an angular momentum and a ma~neticmoment: and in the case when the spectrum is that of a mIxtureof isotopes, the differences in nuclear mass of the isotopes will alsocause small differences of position of lines in their spectra, and socontribute to the hyperfine structure.

In 1923 the domain of quantum theory was enlarged, when thediffraction of a parallel beam of radiation by a gratin~ was explainedon quantum principles by William Duane.3 ConsIder an infinitegrating with the spacing d between its rulings. If the grating moveswith constant velocity in a direction in its own plane perpendicularto the rulings, it will return to its original aspect when it has movedthrough a distance d: so we can regard it as a periodic system towhich the Wilson-Sommerfeld quantum rule

Jpdq=nh

can be applied, where p denotes momentum in the direction of thesurface of the grating perpendicular to its rulings, the spacing dbeing the domain over which the integration must be extended:and therefore

pd=nh,1 ZS.f. P. xxxi (1925), p. 765 • Natunviss, xii (1924), p. 741• Proc. N. A. S. ix (1923), p. 158. Duane's treatment was considerably improved

as regards its justification by A. H. Compton, ibid. p. 359.142


SO the grating can pick up momentum! on{y in multiples of hid: themomentum of radiation is transferre to and from matter in quanta.If a photon of energy hy, and therefore of momentum hylc, falls onthe grating in a direction making an angle i with the normal, and isdiffracted in a direction making an angle r with the normal, thentaking the components of momentum in a direction of the surfaceof the grating at right angles to its rulings, we have the equation ofconservation of momentum

(hYlc) (sin i-sin r) = nhld

which in the language of the wave-theory would be

nA= d (sin i-sin r).

This is the ordinary equation giving the directions of the diffracted radiation,now obtained from the corpuscular (photon) theory of light.

It may be noted that Duane's equation pd = nh can be appliedto a photon, if d be interpreted as wave-length, so we obtain pA=h:since A= ely, this shows that the momentum of the photon is hyle.

Duane's principle is closely related to a principle introducedlater in the same year (1923) by L. de Broglie, which will be con-sidered in Chapter VI.

The Duane method was extended to finite gratings, includingeven the case of only two reflecting points, by P. S. Epstein andP. Ehrenfest in 1924-.1

• Proc.N. A. s. x (1924), p. 133; xiii (1927), p. 400

14-3

Chapter VGRAVITATION

WE have seen 1 that for many years after its first publication, theNewtonian doctrine of gravitation was not well received. Even inNewton's own University of Cambridge, the textbook of physicsin general use during the first quarter of the eighteenth century wasstill Cartesian: while all the great mathematicians of the Continent-Huygens in Holland, Leibnitz in Germany, Johann Bernoulli inSwitzerland, Cassini in France-rejected the Newtonian theoryaltogether.

This must not be set down entirely to prejudice: many well-informed astronomers believed, apparently with good reason, that theNewtonian law was not reconcilable with the observed motions ofthe heavenly bodies. They admitted that it explained satisfactorilythe first approximation to the planetary orbits, namely that they areellipses with the sun in one focus: but by the end of the seventeenthcentury much was known observationally about the departures fromelliptic motion, or inequalities as they were called, which were pre-sumably due to mutual gravitational interaction: and some of theseseemed to resist every attempt to explain them as consequences ofthe Newtonian law.

The inequalities were of two kinds: first, there were disturbanceswhich righted themselves after a time, so as to have no cumulativeeffect: these were called periodic inequalities. Much more seriouswere those derangements which proceeded continually in the samesense, always increasing the departure from the original type ofmotion: these were called secular inequalities. The best knownof them was what was called the great inequality rif Jupiter and Saturn,of which an account must now be given.

A comparison of the ancient observations cited by Ptolemy inthe Almagest with those of the earlier astronomers of Western Europeand their more recent successors, showed that for centuries past themean motion, or average angular velocity round the sun, of Jupiter,had been continually increasing, while the mean motion of Saturnhad been continually decreasing. This indicated some strikingconsequences in the remote future. Since by Kepler's third law thesquare of the mean motion is proportional to the inverse cube of themean distance, the decrease in the mean motion of Saturn impliedthat the radius of his orbit must be increasing, so that this planet,the most distant of those then known, would be always becoming-more remote, and would ultimately, with his attendant ring and

1 cr. Vol. I, pp. 29-31144

GRAVITATION

satellites, be altogether lost to the solar system. The orbit of jupiter,on the other hand, must be constantly shrinking, so that he mustat some time or other either collide with one of the interior planets,or must be precipitated on the incandescent surface of the sun.

No explanation of the secular inequality of jupiter and Saturncould be obtained by any simple and straightforward application ofNewton's gravitational law, and the French Academy of Sciencesoffered a prize in 1748, and again in 1752, for a memoir relatingto these two planets. On each occasion Euler made considerableadvances 1 in the general treatment of planetary perturbations, andreceived the award: but the result of his investigations was to makethe observed secular accelerations of jupiter and Saturn moremysterious than ever, for they appeared to be quite inconsistentwith the tolerably complete theory which he created. Lagrange,who wrote on the problem 2 in 1763, and gave a still more completediscussion, likewise failed to obtain a satisfactory agreement withthe observations.

In 1773 the matter was taken up by Laplace. S He began bycarrying the approximation to a higher order than his predecessors,and was surprised to find that in the final expression for the effectof Jupiter's disturbing action upon the mean motion of Saturn, theterms cancelled each other out. The same result, as he showed,held for the effect of any planet upon the mean motion of anyother: thus the mean motions of the planets cannot have any secularaccelerations whatever as a result of their mutual attractions. Laplaceaccordingly concluded that the accelerations observed in the caseof Jupiter and Saturn could not be genuinely secular: they mustreally be periodic, though the period might be immensely long.

With this key to the mystery, he completely solved it, in a greatmemoir of 1784.4 He realised that an inequality of long periodcould be produced only by a term of long period in the perturbingfunction: denoting this term by p sin qt, then in order that it maybe oflong period, q must be extremely small. By a double integrationwith respect to the time, such as happens in the course of solvingthe differential equations, this term would become (P/q2) sin qt, and(P/q2) might be quite large even though p were very small. Thusa great inequality of long period might be produced by a termin the perturbing function which was so small that it had beenneglected altogether by preceding investigators. This explainedwhy Euler and Lagrange had failed to solve the problem, andnothing remained to be done except to inquire more closely into theidentity of the term in the perturbing function. Now five times themean motion of Saturn is very nearly equal to twice the meanmotion of jupiter: so if n, n' are the mean motions, then 5n - 2n'

1 Recueil des pieces qui ani remportl les prix de ['Acad., tome vii (1769)I Mllanges de phil. el de math. de la Soc. Roy. de Turin pour ['annie 1763, p. 179 (1766)• Mlm. des Savans IIrang. vii (1776). Read 10 Feb. 1773• Mlm. de I'Acad., 1784, p. 1

145


is very small: and the term in the perturbing function whose argu-ment is (5n - 2n')t, which would have an extremely small coefficient,would satisfy all the conditions required. Thus Laplace was ableto assert that the great inequality of Jupiter and Saturn is not a secularinequality, but is an inequality of long period, in fact 929 years: and it isdue to thefact that the mean motions of the two planets are nearly commensurable.When the results of his calculations were compared with the observa-tions, the agreement was found to be perfect.l

The story of the great inequality of Jupiter and Saturn illustratesa distinctive feature of the situation, namely that the truth of theNewtonian or any other law of gravitation cannot be tested by meansof controlled experiments in a laboratory, and its verification mustdepend on the comparison of astronomical observations, extendedover centuries, with mathematical theories of extreme complexity.

After the triumphant conclusion of Laplace's researches on thegreat inequality of Jupiter and Saturn, there was still outstandingone unsolved problem which formed a serious challenge to theNewtonian theory, namely the secular acceleration of the meanmotion of the moon. From a study of ancient eclipses recorded byPtolemy and the Arab astronomers, Halley 2 had concluded in 1693that the mean motion of the moon has been becoming continuallymore rapid ever since the epoch of the earliest recorded observations.The mean distance of our satellite must therefore have been con-tinually decreasing, and it seemed that at some time in the remotefuture the moon must be precipitated on the earth. The Academyof Sciences of Paris proposed the subject for the prize in 1770, andagain in 1772 and 1774, and prizes were awarded to Euler sandLagrange,' who made valuable contributions to general dynamicalastronomy: on the question proposed, however, they found onlythe negative results that no secular inequality could be producedby the action of Newtonian gravitation when the heavenly bodieswere regarded as spherical, and, moreover, that the observedphenomena could not be explained by taking into account thedepartures of the figures of the earth and moon from sphericity.Laplace now took up the matter, and showed, first, that the effectwas not due to any retardation of the earth's diurnal rotation dueto the resistance of the aether: he then investigated the consequencesof another supposition, namely that gravitational effects are prop-agated with a velocity which is finite 5: but this also led to nosatisfactory issue, and at last he found the true solution,6 which

1 A!JLaplace's discovery showed, the existence of I small divisors' makes it a matterof great difficulty to investigate the convergence of the series that occur in CelestialMechanics; cf E. T. Whittaker, Proc. R.S. Ed. xxxvii (1917), p. 95.

a Phil. Trans. xvii (1693), p. 913• Recueil des pieces qui ont remportl les prix de I'Acad. ix (1777)• Recueil des pieces qui ont remportl les prix de I' Acad. ix (1777), for the competition of

1772; Mem. des Savans Etrang. vii, for that of 1774• cf. Vol. I, p. 207• It was presented to the Academy on 19 March 1787; Mem. de l' Acad. 1786, p. 235

(published 1788).

GRAVITATION

may be described as follows. The mean motion of the moon roundthe earth depends mainly on the moon's gravity to the earth, butis slightly diminished by the action of the sun upon the moon.This solar action, however, depends to a certain extent on theeccentricity of the terrestial orbit, which is slowly diminishing, asa result of the action of the planets on the earth. Consequentlythe sun's mean action on the moon's mean motion must also bediminishing, and hence the moon's mean motion must be con-tinually increasing, which is precisely the phenomenon that isobserved. The acceleration of the moon's mean motion will con-tinue as long as the earth's orbit is approaching a circular form:but as soon as this process ceases, and the orbit again becomes moreelliptic, the sun's mean action will increase and the accelerationof the moon's motion will be converted into a retardation. Theinequality is therefore not truly secular, but periodic, though theperiod is immensely long, in fact millions of years. This strikingvindication of the Newtonian theory came exactly a century afterthe p,ublication of the Principia.

The moon, in the present day,' wrote Robert Grant,! ' is abouttwo hours later in coming to the meridian than she would havebeen if she had retained the same mean motion as in the time of theearliest Chaldean observations. It is a wonderful fact in the historyof science that those rude notes of the priests of Babylon shouldescape the ruin of successive empires, and, finally, after the lapseof nearly 3,000 years, should become subservient in establishmga phenomenon of so refined and complicated a character as theinequality we have just been considering.'

In the nineteenth and twentieth centuries, however, manyastronomers formed the opinion that Laplace's great memoir hadnot completely cleared up the situation. The fact to be explainedis that the moon has relative to the sun an apparent acceleration ofits mean motion of about 22" per century per century.2 Laplace'stheoretical value was of about this amount, but J. C. Adams 3 •

found, by including terms of higher order in the calculation, thatLaplace's value was much too great, the amount explicable bypurely gravitational causes being only 12·2" per century per cen-tury: and his conclusion was substantiated by later workers in lunartheory. In 1905 P. H. Cowell 4 redetermined (from ancient eclipses)the observed secular acceleration, and found that it was almosttwice the theoretical value, and that there was also deducible fromobservation a secular acceleration of the sun (Le. of the earth'sorbital motion). His own tentative explanation 5 depended on the

1 History oj Physical Astrononry (London, 1852), p. 63, There is some confusion of language on this subject. The coefficient of t' in the

expression for the longitude is about II", and the true secular acceleration is thereforeabout 22"; but many writers speak of the acceleration as ' II" in a century.'

, Phil. Trans. cxliii (1853), p. 397; Mon. Not. R.A.S. xl (1880), p. 472• Mon. Not. R.A.S. !xv (1905), p. 861• Mon. Not. R.A.S. Ixvi (1906), p. 352

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notion of tidal friction, while J. H. Jeans 1 suggested a modificationof the Newtonian law, and E. A. Milne 2 proposed a dependenceof the Newtonian constant of gravitation on the a~e of the universe.These matters were discussed by J. K. Fothenngham 3 in 1920and in 1939 by H. Spencer Jones,' who showed that the secularaccelerations of the sun, Mercury, and Venus are proportional totheir mean motions, and can be accounted for by the retardationof the earth's axial rotation by tidal friction.5 This retardationmust also produce, in addition to its direct apparent effect, a realsecular acceleration of the moon's mean motion, in order that thetotal angular momentum of the earth-moon system may be con-served: but the amount of this acceleration cannot be predictedtheoretically.

We may now refer to other phenomena in the solar system whichwere not explained with complete satisfaction by Newton's formula.The anomalous motion of the perihelion of the planet Mercury hasalready been referred to 6: it might possibly be accounted for if theinverse square law is modified by adding a term involving thevelocities of the bodies 7: or it might, as H. Seeliger 8 showed, beexplained by the attraction of the masses forming the zodiacal light.The node of the orbit of Venus was also found by Newcomb 9 to havea secular acceleration which was five times the probable error, andfor which no explanation could be offered: and a secular increasein the mean motion of the inner satellite of Mars, discovered in 1945by B. P. Sharpless,1°is so far·not accounted for.

Certain comets also present problems. Among them the bestknown is an object which was discovered by Jean-Louis Pons in1818, but is generally called Encke's comet from a long series ofmemoirs 11devoted to it by J. F. Encke, who showed in 1819 that itwas periodic, with a period of 1,207 days, and later that its motionshowed an acceleration which was not explicable by the Newtoniantheory. Encke himself proposed to explain this by postulating aresisting medium whose density was inversely proportional to the

1 Mon. Not. R.A.S. lxxxiv (1923), p. 60 (at p. 75)• Proc. R.S.(A), clvi (1936), p. 62 (at p. 81)I Mon. Not. R.A.S. !xxx (1920), p. 578 • Mon. Not. R.A.S. xcix (1939), p. 541• The retardation of the earth's rotation due to tidal friction increases the length

of the day by about I{IOOOofa second per century, so each century is 36i seconds lonserthan the one preceding. There is, moreover, a variability in the rate of rotation, whIchis slower in February than in August. This fluctuation, which was discovered by meansof clocks formed of vibrating quartz crystals, is probably of meteorological origin.

• cr. Vol. I, p. 2087 For work more recent than that referred to in Vol. I, cr. Paul Gerber, ;::'S.f, Math.

U. Phys. xliii (1898), p. 93; Ann. d. Phys. Iii (1917), p. 415, and the comments on thelatter paper by H. Seeliger, Ann. d. Phys. liii (1917), p. 31; liv (1917), p. 38, andS. Oppenheim, ibid. liii (1917), p. 163

I Miinchen Ber. 1906, p. 595I S. Newcomb, The elements qf the four inner planets; Supplement to the American

Ephemeris and Nautical Almanac for 1897; Washington, 1895II Ast. J. Ii (1945), p. 18511 Mostly in the Berlin Abhandlungen and the Astronomische Nachrichten; cr. specially

Comptes Rendus, xlviii (1858), p. 763

GRAVITATION

square of the distance from the sun: but O. Backlund, who devotedmany years to the study of the comet, showed that Encke's assumptionis impossible.l An alternative hypothesis is that the comet hasencounters with a swarm of meteors.

In 1910 P. H. Cowell and A. C. D. Crommelin 2 computed withgreat care the motion of another periodic comet, that of Halley,between 1759 and 1910, and predicted the time of perihelion for itsreturn in 1910: the time deduced later from actual observationswas about 2·7 days later than this. The discrepancy could not beaccounted for by any defect in the calculations, and it would seemtherefore that there is some small disturbing cause or causes at work,other than the gravitational attraction expressed by Newton's law.3

At the meeting of the Amsterdam Academy of Sciences on31 March 1900, Lorentz communicated a paper entitled Considera-tions on Gravitation,4 in which he reviewed the problem as it appearedat that time-a problem which, as the above recital shows, wasstill far from a completely satisfactory solution. So many phenomenahad been successfully accounted for by applications of electro-magnetic theory that it seemed natural to seek in the first place anexplanation in terms of electric and magnetic actions. As we haveseen,6 the assumptions of Laplace's investigation, which led him toconclude that the velocity of propagation of gravitation must bevastly greater than that of light, do not to twentieth-century mindsseem very plausible 6: and Lorentz felt free to put forward a theorydepending on electromagnetic actions propagated with the speed oflight. The first possibility he considered was suggested by Le Sage'sconcept of ultra-mundane corpuscles.7 Since it had been found thata pressure against a body could be produced as well by trains ofelectric waves as by moving projectiles, and that the X-rays withtheir remarkable penetrating power were essentially electric waves,it was natural to replace Le Sage's corpuscles by vibratory motions.Why should there not exist radiations far more penetrating than eventhe X-rays, which mis-ht account for a force whIch, so far as is known,is independent of all mtervening matter?

Lorentz therefore calculated the interaction between two ionson the assumption that space is traversed in all directions by trainsof electric waves of very high frequency. If an ion P is alone in a

1 Backlund's conclusions are summarised in Bull. astronomique, xi (1894), p. 473;cr. also A. Wilkens, .1str. Nach. cxcvi (1914), p. 57. On the perturbations of Encke's cometcf. D. Brouwer, Ast. J. Iii (1947), p. 190.

• Investigation rif the motion of Halley's Comet from 1759 to 1.910; Appendix to the 1909volume of Greenwich Observations (1910). Essay on the return of Halley's comet, Publ. Astr.Ges., Lpz., No. 23 (1910)

• On the effect of loss of mass by evaporation when a comet is near the sun; cr.F. Whipple, Astroph. J. cxi (1950), p. 375

• Proc. Arnst. Acad. ii (1900), p. 559; French translation in Arch. Neerl. vii (1902),p.325

• Vol. I, pp. 207-8• The same remark applies to the ideas of R. Lehmann-Filhes, Miinchen. Ber. xxv

(1895), p. 71., cf. Vol. I, p. 31

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field in which the propagation of waves takes place equally in alldirections, the mean force on it will vanish. But the situation willbe different as soon as a second ion Q has been placed in the neigh-bourhood of P : for then, in consequence of the vibrations emittedby Q after it has been exposed to the rays, there might be a forceon P, of course in the direction of the line QP. It was found, how-ever, that this force could exist only if in some way or other electro-magnetic energy were continually disappearing : and after fullconsideration, Lorentz concluded that the assumptions he had madecould not provide a satisfactory explanation of gravitation.

He then considered a second hypothesis, which may be regardedas having been foreshadowed in the one-fluid electrical theory ofWatson, Franklin and Aepinus.1 According to this theory, asdeveloped in 1836 by O. F. Mossotti 2 (1791-1863), electricity isconceived as a continuous fluid, whose atoms repel each other.Material molecules are also supposed to repel each other, but tohave with the aether-atoms a mutual attraction, which is somewhatgreater than the mutual repulsion of the particles which repel.The composition of these forces accounts for gravitation, exceptat very small distances, where the same mechanism accounts forcohesion.

Wilhelm Weber (1804-91) of Gottingen and Friedrich Zollners(1834-82) of Leipzig developed this conception into the idea that allponderable molecules are associations of positively and negativelycharged electrical corpuscles, with the condition that the force ofattraction between corpuscles of unlike sign is somewhat greaterthan the force of repulsion between corpuscles of like sign. If theforce between two electric units of like charge at a certain distanceis a dynes, and the force between a positive and a negative unitcharge at the same distance is y dynes, then, taking account of thefact that a neutral atom contains as much positive as negativeelectric charge, it was found that (y-a)/a need only be a quantityof the order 10-35 in order to account for gravitation as due to thedifference between a and y.

At the time of Lorentz's paper, no strong physical reason for anassumption of this kind could be given. Many years afterwardsEddington 4 suggested one. He had taken to heart a warning utteredby Mach.5 'Even in the simplest case, in which apparently we dealwith the mutual action between only two particles, it is impossibleto disregard the rest of the universe: Nature does not begin withelements, as we are forced to do. Certainly it is fortunate for us

1 cf. Vol. I, p. 50• Sur lesforces qui r!gissent la constitution intlrieure des corps, apperfu pour servir II Ladetermina-

tion tU la cause et tUs lois tU l'tu:tion moUculaire (Turin, 1836).• Erkliirung der universellen Gravitation, Leipzig, 1882; pp. 67-82 deal with Weber's

contributions; cf. alsoJ. J. Thomson, Proc. Camb. Phil. Soc. xv (1910), p. 65• Fundammtal Theory (Cambridge, 1946), p. 102• E. Mach, Die Mechanik in ihrer Entwickelung (5th edn., 1904), p. 249; Englisb

translation (London, 1893), p. 235150

GRAVITATION

that we can sometimes turn away from the overwhelming All, andallow ourselves to study isolated facts. But we must not forgetultimately to amend and complete our views by taking into accountwhat had been omitted.' Eddington applied Mach's generalprinciple to the interaction between two electric charges. If theyare of opposite sign, all their lines of force run from one to the other,and the two together may be regarded as a self-contained systemwhich is independent of the rest of the universe: but if the twocharges are of the same sign, then the lines of force from each of themmust terminate on other bodies in the universe, and it is naturalto expect that these other bodies will have some influence on thenature of the interaction between the charges. Following up thisidea by a calculation, Eddington arrived at the conclusion thatwhen two protons are at a distance r apart, which is of the sameorder of magnitude as the radius of an atomic nucleus, their mutua!energy contains, in addition to the ordinary electrostatic energycorresponding to the inverse-square law, a term of the form

where A and k are constants: if it could be supposed that this iscorrect, and is an asymptotic approximation, valid for values ofr of nuclear dimensions, to a function whose asymptotic approxima-tion, valid for values of r large compared with nuclear dimensions,is inversely proportional to r, then there would obviously be apossibility of accounting on these lines for gravitation.

From 1904 onwards the Newtonian law of gravitation wasexamined in the light of the relativity theory of Poincare andLorentz. This was done first by Poincare,l who pointed out thatif relativity theory were true, gravity must be propagated withthe speed of light, and who showed that this supposition was notcontradicted by the results of observation, as Laplace had sup-posed it to be. He suggested modifications of the Newtonianformula, which were afterwards discussed and further developed byH. Minkowski 2 and by W. de Sitter. 3 It was found that relativitytheory would require secular motions of the perihelia of the planets,which however would be of appreciable amount only in the case ofMercury, and even in that case not great enough to account for theobserved anomalous motion.

In 1907 Planck 4 broke new ground. It had been establishedby the careful experiments of R. v. Eotvos 5 that inertial mass (whichdetermines the acceleration of a body under the action of a given

I Comptes Rendus, cxl (1905), p. 1504; Palermo Rend. xxi (1906), p. 129I Cott. Nach. 1908, p. 53I Mon. Not. R.A.S. Ixxi (1911), p. 388; cr. also F. Wacker, Inaug. Diss., Tiibingen,

1909• Berl. Sitl;. 13June 1907, p. 542, specially at p. 544• Math. u. nat. Ber. aus Ungarn, viii (1891), p. 65

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force) and gravitational mass (which determines the gravitationalforces between the body and other bodies) are always exactly equal:which indicates that the gravitational properties of a body are essentiallyoj the same nature as its inertial properties. Now, said Planck, all energyhas inertial properties, and therefore all energy must gravitate. Sixmonths later Einstein 1 published a memoir in which he introduced 2

what he later called the Principle of Equivalence, which may be thusdescribed:

Consider an observer who is enclosed in a chamber withoutwindows, so that he is unable to find out by direct observationwhether the chamber is in motion relative to an outside world ornot. Suppose the observer finds that any object in the chamber,whatever be its chemical or physical nature, when left unsupported,falls towards one particular side of the chamber with an accelerationf which is constant relative to the chamber. The observer wouldbe justified in putting forward either of two alternative explanationsto account for this phenomenon:

(i) he might su~pose that the chamber is ' at rest,' and that thereis a field of joree, hke the earth's gravitational field, acting on allbodies in the chamber, and causing them if free to fall with accelera-tionj: or

(ii) he might explain the observed effects by supposing that thechamber is in motion: if he postulates that in the outside world thereare co-ordinate axes (C) relative to which there is no field of force,and if he moreover supposes the chamber to be in motion relativeto these axes (C) with an acceleration equal in magnitude butopposite in direction to 1, then it is obvious that free bodies insidethe chamber would have an accelerationj relative to the chamber.

The observer has no criterion enabling him to tell which of thesetwo explanations is the true one. If we could say definitely that thechamber is at rest, then explanation (i) would be true, while if wecould say definitely that the axes (C) are at rest, then explana-tion (ii) would be true. But by the Principle of Relativity, we cannotgive a preference to one of these sets of axes over the other: wecannot say that one of them is moving and the other at rest: andwe must therefore regard the two explanations as,equally valid, or,in other words, must assert that a homogeneous field of force isequivalent to an afparent field which is due to the acceleratedmotion of one set 0 axes relative to another: a uniform gravitational

field is physically equivalent to a field which is due to a change in theco-ordinate system.

In this paper Einstein also showed 3 by combining Doppler'sprinciple WIth the principle of equivalence, that a spectral linegenerated by an atom situated at a place of very high gravitationalpotential, e.g. at the sun's surface, has, when observed at a place

1 Jahrb. d. Radioakt. iv (4 Dec. 1907), p. 411 ; cr. Einstein, Ann. d. Phys. xxxv (1911),p.898

I At p. 454 • At pp. 458-9152

GRAVITATION

of lower potential, e.g. on the earth, a greater wave-length thanthe corresponding line generated by an identical atom on the earth.This may also be shown very simply as follows. Denoting by 0the gravitational potential at the sun's surface, the energy lost bya photon of frequency v in escaping from the sun's gravitationalfield is 0 x the mass of the photon, or Ohv/c2• Remembering thatthe energy hv is he/A, we see that the wave-length of the solar radia-tion as measured by the terrestial observer is 1+ (0/e2) times thewave-length of the same radiation when produced on earth.l

In 1911 Einstein followed up this work by an important memoir,2in which he argued that since light is a form of (electromagnetic)energy, therefore light must gravitate, that is, a ray of light passingnear a powerfully gravitating body such as the sun, must be curved: andthe velocity of light must depend on the gravitational field.

Einstein's paper was the starting-point of a theory publishedshortly afterwards by Max Abraham.3 Accepting the principlesthat the velocity of light e depends on the gravitational potential,and that the law of gravitation might be expressed by a dIfferentialequation satisfied by c, he postulated that the negative gradient ofc indicates the direction of the ~ravitational force, and that theenergy-density in a statical graVItational field is proportional toc-l(grad e)2. Einstein himself at almost the same time published 4

a somewhat different theory, in which the equations of motion ofa particle in a statical gravitational field, when gravity only isacting, are

d(l dx) 1 ocdt C2 de = - c ox'

and similar equations in y and z.To the same period belong the theories of G. Nordstrom 5 and

Gustav Mie.6 Though Mie's theory has not survived as the per-manent basis of mathematical physics, it had a marked influenceon thought, and some of its ideas appeared later in the researchesof other workers. It aimed at being a complete theory of physics,

I cr. also J. M. Whittaker, Proc. Camb. Phil. Soc. xxiv (1928), p. 414. The red-displacement due to a gravitational field with arbitrary motion of the source and of theobserver was calculated by H. Weyl in the fifth edition (1923) of his Raum Zeit Materie,Anhang III.

• Ann. d. Phys.(4) xxxv (21June 1911), p. 898• Lincei Alii, xx (Dec. 1911), p. 678; Phys. ZS. xiii (1912), pp. 1,4, 176, 310, 311,

793; N. Cimento(4)iv (Dec. 1912), p. 459, Ann. d. Phys. xxxviii (Feb. 1912), pp. 355, 443. A controversy followed, for which

see Abraham, Ann. d. Phys. xxxviii (1912), p. 1056; xxxix (1912), p. 444; and Einstein,Ann. d. Phys. xxxviii (1912), p. 1059; xxxix (1912), p. 704.

• Phys. ZS. xiii (Nov. 1912), p. 1126; Ann. d. Phys.(4) xl (April 1913), p. 856;ibid. xlii (Oct. 191.3),p. 533; Phys.ZS. xv (1914), p. 375; cf.A. Einstein.andA. D. Fokker,Ann. d. phys.(4) xhv (1914), p. 321; M. v. Laue, Jahrb. d. Rad. u. El. XIV (1917), p. 263

, Ann. d. Phys. xxxvii (1912), p. 511; xxxix (1913), p. I; xl (1913), p. I; Phys. ZS.xv (1914), pp. 115, 169,263; Festschrift fur J. Elster u. H. Geitel (Braunschweig, 1915),pp. 251-68; cf. A. Einstein, Phys. ZS. xv (1914), p. 176. There is a good short account ofMie's theory in H. Weyl, Raum Zeit Materie, 4th Aufl., (Berlin, 1921), § 26.

153

GRAVITATION

that we can sometimes turn away from the overwhelming All, andallow ourselves to study isolated facts. But we must not forgetultimately to amend and complete our views by taking into accountwhat had been omitted.' Eddington applied Mach's generalprinciple to the interaction between two electric charges. If theyare of opposite sign, all their lines offorce run from one to the other,and the two together may be regarded as a self-contained systemwhich is independent of the rest of the universe: but if the twocharges are of the same sign, then the lines of force from each of themmust terminate on other bodies in the universe, and it is naturalto expect that these other bodies will have some influence on thenature of the interaction between the charges. Following up thisidea by a calculation, Eddington arrived at the conclusion thatwhen two protons are at a distance r apart, which is of the sameorder of magnitude as the radius of an atomic nucleus, their mutualenergy contains, in addition to the ordinary electrostatic energycorresponding to the inverse-square law, a term of the form

where A and k are constants: if it could be supposed that this iscorrect, and is an asymptotic approximation, valid for values ofr of nuclear dimensions, to a function whose asymptotic approxima-tion, valid for values of r large compared with nuclear dimensions,is inversely proportional to r, then there would obviously be apossibility of accounting on these lines for gravitation.

From 1904 onwards the Newtonian law of gravitation wasexamined in the light of the relativity theory of Poincare andLorentz. This was done first by Poincare,! who pointed out thatif relativity theory were true, gravity must be propagated withthe speed of light, and who showed that this supposition was notcontradicted by the results of observation, as Laplace had sup-posed it to be. He suggested modifications of the Newtonianformula, which were afterwards discussed and further developed byH. Minkowski 2 and by W. de Sitter.3 It was found that relativitytheory would require secular motions of the perihelia of the planets,which however would be of appreciable amount only in the case ofMercury, and even in that case not great enough to account for theobserved anomalous motion.

In 1907 Planck 4 broke new ground. It had been establishedby the careful experiments of R. v. Eotvos 5 that inertial mass (whichdetermines the acceleration of a body under the action of a given

1 Comptes Rendus, cxl (1905), p. 1504; Palermo Rend. xxi (1906), p. 129I Coli. Nach. 1908, p. 53I Mon. Not. R.A.S. 1xxi (1911), p. 388; cr. also F. Wacker, Inaug. Diss., Tiibingen,

1909• Bert. Sit;:.. 13June 1907,p. 542, specially at p. 544• Math. u. nat. Ber. aus Ungarn, viii (1891), p. 65

151


force) and gravitational mass (which determines the gravitationalforces between the body and other bodies) are always exactly equal:which indicates that the gravitational properties of a body are essentiallyof the same nature as its inertial properties. Now, said Planck, all energyhas inertial properties, and therefore all energy must gravitate. Sixmonths later Einstein 1 published a memoir in which he introduced 2

what he later called the Principle of Equivalence, which may be thusdescribed:

Consider an observer who is enclosed in a chamber withoutwindows, so that he is unable to find out by direct observationwhether the chamber is in motion relative to an outside world ornot. Suppose the observer finds that any object in the chamber,whatever be its chemical or physical nature, when left unsupported,falls towards one particular sIde of the chamber with an accelerationf which is constant relative to the chamber. The observer wouldbe justified in putting forward either of two alternative explanationsto account for this phenomenon:

(i) he might sUJ?posethat the chamber is ' at rest,' and that thereis a field of jorce, hke the earth's gravitational field, acting on allbodies in the chamber, and causing them if free to fall with accelera-tionf: or

(ii) he might explain the observed effects by supposing that thechamber is in motion: if he postulates that in the outside world thereare co-ordinate axes (C) relative to which there is no field of force,and if he moreover supposes the chamber to be in motion relativeto these axes (C) with an acceleration equal in magnitude butopposite in direction to f, then it is obvious that free bodies insidethe chamber would have an accelerationf relative to the chamber.

The observer has no criterion enabling him to tell which of thesetwo explanations is the true one. If we could say definitely that thechamber is at rest, then explanation (i) would be true, while if wecould say definitely that the axes (C) are at rest, then explana-tion (ii) would be true. But by the Principle of Relativity, we cannotgive a preference to one of these sets of axes over the other: wecannot say that one of them is moving and the other at rest: andwe must therefore regard the two explanations as,equally valid, or,in other words, must assert that a homogeneous field of force isequivalent to an apparent field which is due to the acceleratedmotion of one set of axes relative to another: a uniform gravitational

field is physically equivalent to a field which is due to a change in theco-ordinate system.

In this paper Einstein also showed 3 by combining Doppler'sprinciple WIth the principle of equivalence, that a spectral linegenerated by an atom situated at a place of very high gravitationalpotential, e.g. at the sun's surface, has, when observed at a place

, Jahrb. d. Radioakt. iv (4 Dec. 1907), p. 411; cf. Einstein, Ann. d. Phys. xxxv (1911),p. 898

• At p. 454 • At pp. 458-9152

GRAVITATION

of lower potential, e.g. on the earth, a greater wave-length thanthe corresponding line generated by an identical atom on the earth.This may also be shown very simply as follows. Denoting by 0the gravitational potential at the sun's surface, the energy lost bya photon of frequency v in escaping from the sun's gravitationalfield is 0 x the mass of the photon, or Ohv/c2• Remembering thatthe energy hv is he/A, we see that the wave-length of the solar radia-tion as measured by the terrestial observer is 1+ (0/c2) times thewave-length of the same radiation when produced on earth.l

In 1911 Einstein followed up this work by an important memoir,2in which he argued that since light is a form of lelectromagnetic)energy, therefore light must gravitate, that is, a ray of light passingnear a powerfully gravitating body such as the sun, must be curved: andthe velocity of light must depend on the gravitational field.

Einstein's paper was the starting-point of a theory publishedshortly afterwards by Max Abraham.3 Accepting the principlesthat the velocity of light c depends on the gravitational potential,and that the law of gravitation might be expressed by a dIfferentialequation satisfied by c, he postulated that the negative gradient ofc indicates the direction of the ~ravitational force, and that theenergy-density in a statical graVltational field is proportional toc-l(grad C)2. Einstein himself at almost the same time published'a somewhat different theory, in which the equations of motion ofa particle in a statical gravitational field, when gravity only isacting, are

d(l dx) I ocdt C2 de = -c ax'

and similar equations iny and z.To the same period belong the theories of G. Nordstrom 6 and

Gustav Mie.6 Though Mie's theory has not survived as the per-manent basis of mathematical physics, it had a marked influenceon thought, and some of its ideas appeared later in the researchesof other workers. It aimed at being a complete theory of physics,

I ef. also J. M. Whittaker, Proc. Camb. Phil. Soc. xxiv (1928), p. 414. The red-displacement due to a gravitational field with arbitrary motion of the source and of theobserver was calculated by H. Weyl in the fifth edition (1923) of his Raum <:eit Malerie,Anhang III.

• Ann. d. Phys.(4) xxxv (21June 1911), p. 898• Lincei Alii, xx (Dec. 1911), p. 678; Phys. <:S. xiii (1912), pp. 1,4, 176, 310, 311,

793; N. Cimenlo(4) iv (Dec. 1912), p. 459• Ann. d. Phys. xxxviii (Feb. 1912), pp. 355, 443. A controversy followed, for which

see Abraham, Ann. d. Phys. xxxviii (1912), p. 1056; xxxix (1912), p. 444; and Einstein,Ann. d. Phys. xxxviii (1912), p. 1059; xxxix (1912), p. 704.

• Phys. <:S. xiii (Nov. 1912), p. 1126; Ann. d. Phys.(4) xl (April 1913), p. 856;ibid. xlii (Oct. 191.3),p. 533; Phys. <:S. xv (1914), p. 375; cf.A. Einstein.and A. D. Fokker,Ann. d. phys.(4) xlIV(1914), p. 321; M. v. Laue, Jahrb. d. Rad. u. El. XIV(1917), p. 263

• Ann. d. Phys. xxxvii (1912), p. 511 ; xxxix (1913), p. I ; xl (1913), p. I; Phys. ZS.xv (1914), pp. 115, 169,263; Festschrift fur J. Elster u. H. Geite1 (Braunschweig, 1915),pp. 251--68; cf. A. Einstein, Phys. <:S. xv (1914), p. 176. There is a good short account ofMie's theory in H. Weyl, Raum <:eit Malerie, 4th Aufl., (Berlin, 1921), § 26.

153


based on the principle that electric and magnetic fields andelectric charges and currents suffice completely to describe all thathappens in the material world, so that matter can be constructedfrom these elements. Moreover, he originated the notion of asingle world-function from which, by the aid of the Calculus of Vari-ations, all the laws of physical processes could be derived: as weshall see presently, this conception was developed afterwards byHilbert.

In Mie's theory all happenin~s, both in the field and in matter,are described by twenty functlOns, constituting two six-vectorswhich describe the field and two four-vectors which describe matter:namely

(i) a six-vector formed of the components of the electric displace-ment D and the magnetic force H

(ii) a six-vector formed of the components of the magnetic induc-tion B and the electric force E

(iii) a four-vector formed of the electric charge and current(iv) a four-vector formed of the electric scalar and vector

potentials.

In the Maxwell-Lorentz theory this last four-vector plays merelya mathematical part: but in Mie's theory its components are physicalrealities. In Chapter IV of his series of memoirs, Mie discussedquantum theory, and in Chapter V, gravitation.

The next advance owed much to a paper that had been writtenin 1909 by Harry Bateman 1 (1882-1946). At any place in theearth's gravitational field, take moving rectangular axes (xl, Xl, x3)

and a measure of time (XO), such that these axes constitute an inertialsystem (A), so that the path of a free particle relative to them is(at any rate near the origin) a straight line, and the vanishing of thedifferential form

is the condition that a luminous disturbance originating at thepoint (xl, Xl, XS) at the instant xO, should arrive at the point (x' + dx',xl+dxl, x3+dxS), at the instant (XO+ dxO).

In free aether, where there is no field of force, two differentinertial systems either can be derived from each other by simpletranslation and rotation in ordinary three-dimensional space, orelse they have a uniform motion of translation relative to each other(or, of course, a combination of these methods of derivation). Butwhen we move to a distant place in a field of force, e.g. if we moveto the antipodes in the earth's gravitational field, although we canhere a~ain find axes (say (B)) which are inertial (that is, free particlesin theIr vicinity move relatively to them with uniform velocity instraight lines), a framework (B) does not move with uniform velocity

1 Proc. L.M.S.(2) viii (1910), p. 223; cr. also Amer. J. Math. xxxiv (1912), p. 325154

GRAVITATION

relative to a framework (A) (in fact the two frameworks are inaccelerated motion relative to each other), so the relation betweentwo inertial frameworks which holds in the relativity theory ofPoinacre and Lorentz does not hold when a gravitational fieldis present. We cannot therefore find co-ordinates (XO, Xl, x2, x3)

describing position and time over the whole field such that the intervalds at any place in the field is given by the equation

(ds) 2 = (dxO) 2 - Jz { (dXl)2 + (dx2) 2 + (dx3) 2}.

Instead of this, we can now find at every place in the field a localframework of inertial axes (XO, Xl, X2, X3), such that the intervalwill be given approximately for points in the neighbourhood of theorigin by

Let (XO, xl, x2, x3) now be any co-ordinates specifying position andtime over the whole field. Then at each place, the differentials dXPwill be expressible in terms of the xl' and the dxP by equations of theform

3

dXP= Lap, dx'.,=0

Substituting this in the expression for (tis)2, we have3

(ds) 2 = L gpg dxpdxq

I',FO

where

(I)

The vanishing of this form (I) is now the condition that a luminousdisturbance originating at the space-time point (XO, xl, x2, x3) shouldarrive at the space-time point (XO+ dxO, Xl + dxl, x2 + dx2, x3 + dx3).

The form (I) must be invariant for all transformations of theco-ordinates (XO, xl, x2, x3) ; and its coefficientsgpg, which are functionsof (XO, xl, x2, x3), are characteristic of the field.

Bateman realised the connection of his work with the tensor-calculus of Ricci and Levi-Civita 1: in fact, since (dxO, dxl, dx2, dxl)is a contravariant vector, it follows from the invariance of thequadratic differential form that the set of the gpg is a symmetriccovariant tensor of rank 2.2

I d. p. 58• These ideas were applied by Bateman in order to investigate a scheme of funda-

mental electromagnetic equations which are not altered by very general transformations.155


Bateman's ideas were carried over into a more profound treatmentof the problem of gravitation in the second half of the year 1913 byEinstein,! who in the years 1912-14 worked in partnership (asregarded the mathematics) with a Zurich geometer, Marcel Gross-mann. In these papers the theory was put forward, that just as therectilinear motion of a particle in free aether when there is nofield is determined by the equation

where 3 is the symbol of the Calculus of Variations, and

so now (making a step analogous to that in Bateman's paper), themotion of a free material particle in a gravitational field is determinedby the equation

3 (Jdr)=Owhere

S

(dr)1= L gpq dxP dxq,p,q=O

the coefficients gpq being characteristic of the state at thepoint (XO, xl, Xl, Xl) in space-time, and dr being invariant withrespect to arbitrary transformations of (XO, xl, Xl, Xl). As withBateman, dr= 0 is the condition that a luminous disturbanceoriginally at the world-point (XO, xl, Xl, Xl) should arrive at theworld-point (XO + dxo, x' + dxI, Xl + dxI, Xl + dx3). In geometricallanguage, the path of a free material particle in a gravitational field is ageodesic in the jour-dimensional curved space whose metric is specified by theequation a

s(dr)· = L gpq dxP dxq•

p,q-O

This was a tremendous innovation, because it implied the abandon-ment of the time-honoured belief that a gravitational field can bespecified by a single scalar potential-function: instead, it proposed

1 A. Einstein and M. Grossmann, ZS. f. M. u. P. lxii (1913), p. 225; lxiii (1914),p. 215; A. Einstein, Vierteljahr. d. Nat. Ges. Zurich, lviii (1913), p. 284; Archives des sc.phys. et nat.(ol) xxxvii (1914), p. 5; Phys. ZS. xiv (15 Dec. 1913), p. 1249; Berlin Sit;;.1914,p. 1030

• A theory that matter consists in ' crinkles' of space had been published by W. K.Clifford in 1870; cf. Proc. Camb. Phil. Soc. ii (1876), p. 157-Clifford's Math. Papers,p.21.

156

GRAVITATION

to specify the gravitational field by the ten functions gpq which couldnow be spoken of as the gravitational potentials.

Einstein justified 1 this new departure by showing that the theoryofa single scalar gravitational potential led to inacceptable inferences.He compared, for instance, two systems, in the first of which amoveable hollow box with perfectly-reflecting walls is filled withpure-temperature radiation, while in the second the same radiationIS contained inside a fixed vertical pit which is closed at the topand bottom by moveable pistons connected by a rod so as to bealways at a fixed distance apart, the pit walls and pistons all beingperfectly-reflecting: and he showed that on the single-scalar-potentialtheory the work necessary to raise the radiation upwards againstthe force of gravity would in the second system be only one-thirdof the work required in the first system: a conclusion which wasobviously wrong. He admitted, however, that in his own mindthe strongest reason for rejecting the single-scalar-potential theorywas his conviction that relativity in physics exists not only withrespect to the Lorentz group of linear orthogonal transformationsbut with respect to a much wider group.

The ten coefficients gpq not only specify the force of gravitation,but they determine also the scale of distance in every direction,and the rate of clocks. The metric defined by

3

(ds)2= 2: gpqdx'Pdxq

",q=O

is not, in general, Euclidean: and since its non-Euclidean qualitiesdetermine the gravitational field, we may say that gravitational theoryis reduced to geometry, in accordance with an idea expressed by Fitz-Gerald 2 in 1894 in the words' Gravity is probably due to a changeof structure of the aether, produced by the presence of matter.'The ' aether' of FitzGerald was called by Einstein simply , space'or' space-time': and FitzGerald's somewhat vague term' structure'became with Einstein the more precise 'curvature.' Thus weobtain the central proposition of the Einsteinian theory: 'Gravityis due to a change in the curvature of space-time, produced bythe presence of matter.'

In comparing FitzGerald's statement with Einstein's, it may beremarked that if we consider a gravitational field which is statical,i.e. such as would be produced by gravitating masses that are per-manently at rest relative to each other, then feeble 3 electromagneticphenomena taking place in it can be shown to happen exactly inaccordance with the ordinary Maxwellian theory of electromagneticphenomena taking place in a medium whose specific inductivecapacity and magnetic permeability are aelotropic and vary from

1 § 7 of the paper in the ZS.f. M. u. P. I FitzGerald's Works, p. 313• i.e. so feeble that they do not appreciably change the curvature of the field.

157


point to point. In particular,! if we consider an electric point-chargeat rest in the field of a single gravitating point-mass, the electric fieldis the same as would be obtained, in ordinary electrostatics, bysupposing that the specific inductive capacity and magnetic per-meability of the medium vary with the distance from the gravitatingmass according to the law (r+ 1)3/r2(r-l).

It is possible that when FitzGerald said' Gravity is probably dueto a change of structure of the aether,' he was actually thinking ofa change which would show itself in alterations of the dielectricconstant and magnetic permeability, and that he had in mind anelectrical constitution of matter, on account of which matter wouldbe subject to forces depending on the values of the dielectric constantand magnetic permeability: by analogy with the fact that in aliquid whose dielectric constant varies from point to point, anelectrified body moves from places of lower to places of higherdielectric constant. 2

What differentiates the Einsteinian theory from all previousconceptions is that the older physicists had regarded gravity asmerely one among many types of natural force-electric, magnetic,etc.-each of which influenced in its own way the motion of materialparticles. Space, whose properties were set forth in Euclideangeometry, was, so to speak, the stage on which the forces playedtheir parts. But in the new theory gravity was no longer one of theplayers, but part of the structure of the stage. A gravitational fieldconsisted essentially in a replacement of the Euclidean properties bya much more complicated kind of geometry: space was no longerhomogeneous or isotropic. An analogy may be drawn from thegame of bowls. Bowling-greens, in the north of England, are notflat, but rise to a slight elevation in the centre. An observer whofailed to notice the central elevation would find that a bowl(supposed without bias) always described a path convex toward thecentre of the green, and he might account for this by postulat-ing a centre of repellent force there. A better-informed observerwould attribute the phenomenon to a geometrical feature-theslope. The two explanations correspond respectively to the New-tonian and the Einsteinian conceptions of gravity: for Newtonit is a force, for Einstein it is a modification of the geometry ofspace.

When the metric of space-time is specified by an equation3

(ds) 2 = 2: gpqdxpdxq,7' q=O

an observer moving in any manner will have a world-line consistingof the points of space-time which he successively occupies : and at

1 E. T. Copson, Proc. R.S.(A), cxviii (1928), p. 184I This idea was later developed by E. Wiechert, Ann. d. Phys. !xiii (1920), p. 30!.

158

GRAVITATION

any point of his world-line he will have in his immediate neighbour-hood an instantaneous three-dimensional space, formed by the aggregateof all the elements of length which are orthogonal to his world-lineat the point: orthogonality being defined, as already explained,lby the statement that two vectors (X) and (Y) are saId to beorthogonal if

3

L Xpyp=o,p=0

where (Xp) is the covariant form of one vector and (YP) is the con-travariant form of the other.

Einstein laid down the principle that the equations which describeany physical process must satisfy the condition that their covariancewith respect to arbitrary substitutions can be deduced from theinvariance of ds. In other words, the laws of nature must be repre-sented by equations which are covariantive for the form L gpgdxpdxq

p,gwith respect to all point-transformations of co-ordinates. Laws ofnature are assertions of coincidences in space-time, and therefore mustbe expressible by covariant equations.

I t might be thought that by following up the consequences ofthis principle we should obtain important positive results. However,Ricci and Levi-Civita 2 had shown long before that from practicallyany assumed law we can derive another law which does not differfrom it in any way that can be tested by observation, but which iscovariant. The fact that a formula has the covariant property doesnot, therefore, tell us anything as to whether it is correct or not.We are, however, perhaps justified in believing that a conjecturallaw which can be expressed readily and simply in covariant form ismore worthy of attention (as being more likely to be true) than onewhose covariant form is awkward and complicated.

Not only must the general laws of physics be covariant, itis also necessary that every single assertion which has a physicalmeaning must be covariant with respect to arbitrary transforma-tions of the co-ordinate system. Thus the assertion that an electronis at rest for an interval of time of duration unity cannot have aphysical meaning, since this assertion is not covariant.s

In Einstein's general theory, the velocity of light at any placehas always the value c with respect to any inertial frame of reference forthis neighbourhood, and the velocity of any material body is lessthan c. Thus there is no difficulty in the fact that the fixed starshave velocities greater than c with respect to axes fixed in the rotatingearth: for such axes are not inertial.

1 See p. 64• Math. Ann.liv (1901), p. 125; cr. E. Kretschmann, Ann. d. Phys.liii (1917), p. 575,

and A. Einstein, Ann. d. Phys. Iv (1918), p. 241, D. Hilbert, Math. Ann. xcii (1924), p. I

159


Some physicists called attention to the fact that when light ispropagated in a medium where there is anomalous dispersion, themdex of refraction may be less than unity, whence it seemed as ifthe velocity of light in the dispersive medium might be greater thanthe velocity of light in vacuo. The difficulty was removed when itwas pointed out by L. Brillouin 1 and A. Sommerfeld I that theVelOCItyof light with which the index of refraction is concerned isthe phase velocity, whereas the velocity of a signal is the group velocity,which is never greater than c.

It has sometimes been supposed, by a misunderstanding, thatthe general Einsteinian theory requires us to regard the Copernicanconception of the universe as no more true than the Ptolemaic, andthat it is indifferent whether we regard the earth as rotating onher axis or regard the stellar universe as performing a completerevolution about the earth every twenty-four hours. The root ofthe matter, by which everything is explained, is that the Copernicanaxes are inertial, while the Ptolemaic are not. The earth rotateswith respect to the local inertial axes. 3

In his first paper in the Zeitschrift fur Math. u. Phys.,' Einsteingave the form which Maxwell's equations of the electromagneticfield must take when the metric of space-time is given by a quadraticdifferential form

5

(ds)2= 2: gpq dx'P dxq•'P,q-O

To obtain these, it will be necessary to introduce some other conceptsof Ricci's absolute differential calculus, or tensor-calculus 6 as Einsteinhenceforth called it. Suppose that we are given a quadratic dif-ferential form in any number of variables

n

(ds)2= 2: g'Pqdx'P dxq'P,q=l

then, following Elwin Bruno Christoffel 8 (l~29-l900), we introducewhat are called Christoffel symbols of the first kmd, defined as

[PlJ=t(~::+~~q~-Offx'i) (p, g, l=l, 2, ... n),

I Comptes Rendus, clvii (1913), p. 914I Ann. d. Phys. xliv (1914), p. 177• cr. G. Giorgi and A. Cabras, Rend. Lineei, ix (1929), p. 513• At page 241• The word tensor had been used by W. Voigt in 1898, in connection with the elasticity

of crystals.• J.fur Math. !xx (1869), pp. 46, 241

160

GRAVITATION

and Christo.ffel symbols of the second kind, defined as

(p, q, 1= 1, 2, •. n).

As Christoffel showed, the Christoffel symbols enable us to formnew tensors from known tensors by a process of generalised differen-tiation. If (Xl' Xi' ... Xn) is any covariant vector, then thequantities (Xp)q defined by

ax n { }(Xp)q=-f- L P/ XrOx ,-1

constitute a covariant tensor of rank 2. This process was called covariantdifferentiation by Ricci. Similarly if (Xpq) is any covariant tensor ofrank 2, then the quantities (Xpq). defined by the equations

constitute a covariant tensor of rank 3 which is called the covariant ilerivativeof Xpq• The covariant differentiation of sums and products iseffected by rules similar to those that apply to ordinary differentia-tion. Moreover if (Xl) is any contravariant vector, then thequantities (Xl). defined by

(Xl).= a;l +i {qt} xqq=1

define a mixed tensor of rank 2 which is called the covariant derivativeof (Xl).

We know that Maxwell's equations in Euclidean space consistof the Ampere-Maxwell tetrad

ad",+ adll + ad. = 47Tox ay a.{'. P


and the Faraday tetrad

ohz + oh, + oh. = 0ox oy 0<;

ad. ad, 10hzay- - 0<; = - c atadz ad. 1 oh,0<; - ox = - c at

Now write xO=ct, x1=x, x'=y, x3=<;. We have seen in Chapter IIthat the electric and magnetic vectors together constitute a six-vector

Thus the Ampere-Maxwell tetrad becomes

OXOI OX02 OX03--+--+--=41Tpox1 ox· ox·

oX'O aXil ox18-- + -- + -- = 41TPVIIOXO ox1 ox.

oX'O aXil OXSBOXO+ oxl + ox' = 41TpV••

Now if (T") is a contravariant vector, the quantity L (T")" (where

the suffix outside the bracket denotes covariant differentiation) isa scalar which is called the divergence of (T"), and denoted by div (t") :we can easily show that

div (T") = Jg ~ o~,,(yg T").

If (T"g) is a contravariant tensor of rank 2, then the quantity

GRAVITATION

which is a vector, is called the vectorial divergence of (Tpq), and isdenoted by ~iv (Tpq). We can show that if (Tpq) is skew, i.e. ISa six-vector, then

Remembering that when the space is Euclidean, '\,Ig is a constant,we see that for a six-vector (Tpq) in Euclidean space we have

Comparing this with the above form of the Ampere-Maxwell tetrad,we see that the tensorial (i.e. covariant) form of this tetrad must be

(A)

where JP denotes the four-vector which represents the electric chargeand current, namely po dxP/ds, where po is the proper density of thecharge, i.e. the chaq~e divided by the volume it occupies, as measuredby an observer movmg with it.

Next consider the Faraday tetrad of equations, which may clearlybe written

Now it can be shown that if X" is a six-vector, and if (P, q, T, s) isan even permutation of the numbers (0, 1, 2, 3), then a six-vectorYpq can be defined by the equations

Ypq= v( -g) X"

and that we then have

Xpq= - '\,I( _g) yr>.163


The two six-vectors X1"1 and Y1"1 are said to be dual to each other.If Ypq is the six-vector dual to the electromagnetic six-vector X1"1'

so that in Euclidean space

YIO = hx, yso =hy, Y30 = hz, Y23 = dx, Y3l = d'l, Y12 =dz,

then the Faraday tetrad may evidently be written

oYOI oYos oY03oxl + oxS + ox3 = 0

or(B).

Since the equations (A) and (B) are tensor equations, and aretherefore covariant with respect to all transformations of the co-ordinates (XO, xl, Xi, x3), we may assume with Einstein that theyrepresent the equations of the electromagnetic field in space-time of any metricwhatever.

The six-vector of the electromagnetic field may be expressed interms of potentials, in the same way as in Euclidean space it isexpressed by the equations

dx= _ 04> _ oaxox cot

etc., hx= oaz _ oa'loy 0<;

etc.

For if we write (4)0' 4>1> 4>s, 4>3) for (4), - ax, - a'l, - az), these equa-tions become, as we have seen on page 76,

(p, q=O, 1,2,3).

Writing this

where the suffixes outside the brackets repres~nt covariant differen-tiation, we see that the potential (4)0, 4>1> 4>2> 4>3) is a covariant vector.

Electromagnetic theory leads naturally to physical optics, and164

GRAVITATION

this again to geometrical optics. Now in the relativity theory ofPoincare and Lorentz, for which the line-element dT in the worldof space-time is given by

(dT) 2 = (dt) 2 - ~ { (dX)2 + (dy)2 + (dz) 2 },

the geodesics of the world are straight lines, and the null geodesics(i.e. the geodesics for which dT vanishes) are the straight lines forwhich

(dx) 2 + (dy)2+ (dz) 2 _ ,

(dt)' -c,

so the null geodesics are the tracks of rays of light. When Einsteincreated his new general theory of relativity, in which gravitationwas taken into account, he carried over this principle by analogy,and asserted its truth for gravitational fields. The principle was,however,not proved at that time: and indeed there was the obviousdifficulty in proving it, that strictly speaking there are no 'rays'of light-that is to say, electromagnetic disturbances which arefiliform, or drawn out like a thread-except in the limit when thefrequency of the light is infinitely great: in all other cases diffractioncauses the' ray' to spread out.

The matter was investigated in 1920 by M. von Laue,l who,starting from the partial differential equations of electromagneticphenomena in a gravitational field, obtained a particular solutionwhich corresponded to light of infinitely high frequency, and showedthat the path of this disturbance satisfied the differential equationsof the null geodesics: thus for the first time proving the truth ofEinstein's assertion. It was afterwards shown by E. T. Whittaker I

that the law is really an immediate deduction from the theory ofthe characteristics of partial differential equations, and that it isnot necessary to introduce the notion of frequency at all: in fact,that in a gravitational field, any electromagnetic disturbance which is filiformmust necessarily have the form of a null geodesic of space-time.

At any point of space-time, the directions which issue from itmay be classified into those that have a spatial character and thosethat have a temporal character: the two classes are separated fromeach other by a cone whose generators are the paths of rays of light:this is called the null-cone.

It can be shown mathematically that if we know the geodesicsof space-time (i.e. the paths of free material particles) and alsoknow which of them are null geodesics (Le. paths of rays of light),then the coefficients gpq of the equation defining the metric

3

(ds) 2 = L gpqdxpdxqp,q=O

1 Phys. ZS. xxi (1920), p. 659 • Proc. Camb. Phil. Soc. xxiv (1928), p. 32165


are completely determinate. In fact, when the null geodesics aregiven, we can infer the metric save for a factor, say

(dr)2 = A(XO, xl, Xl, x3) X (a determinate quadratic form in the dxl')

where A is unknown: and when we are also given the non-nullgeodesics, the factor A(XO, xl, Xl, x3) can be determined.

In the papers we have referred to, which are of date earlier thanNovember 1915, Einstein gave, as we have seen, a satisfactoryaccount of the behaviour of mechanical and electrical systems ina field of gravitation which is supposed given: his formulae werederived fundamentally from the principle of equivalence, i.e. theprinciple that the systems behave just as if there were no gravitationalfield, but they were referred to a co-ordinate-system with an accelera-tion equal and opposite to the acceleration of gravity. But he hadnot as yet succeeded in obtaining an entirely satisfactory set offundamental equations for the gravitational field itself, i.e. equa-tions which would play the same part in his theory that Poisson'sequation 1

played in the Newtonian theory. This defect was repaired, and thetheory (now known as General Relativiry) substantially completed, ina series of short papers published in November-December 1915, inthe Berlin Sitzungsberichte.2

Let us first inquire what covariants of the form L gpqdxl'dxq can7'. q

be formed from the gpq's and their derivatives alone. It canbe shown that these can all be derived from a certain tensorof rank 4, known as the Riemann tensor, which must now beintroduced.

Let D. denote the operation which when applied to the Christoffelsymbol of the first kind is

then we define the Riemann tensor 3 by the equation

1 cf. Vol. I, p. 61• Berlin Sitz;. 1915, pp. 778, 799, 831, 844. An eight-line abstract, dated 25 March

1915, is given at p. 315.• It was discovered by Riemann, in a memoir Commentata mathematica qua respondere

tentatur •.• , which was sent to the Paris Academy in 1861and published posthumously.Werke (1892), p. 401 ; it was afterwards used by Christoffel, loc. cit.

166

GRAVITATION

From the Riemann tensor we can obtain a tensor of rank 2which is defined by the equation

" ,it is called the Ricci-tensor or contracted curvature-tensor: and from theRicci tensor we obtain what is called the scalar curvature of the space,defined by the equation

For a two-dimensional space, e.g. a surface in Euclidean three-dimensional space, - tK is the ordinary Gaussian measure ofcurvature, 1/Plpt.

In applications to the theory of gravitation, we are dealing withthe four-dimensional world of space-time. For such a world, K maybe defined geometrically in the following way. At any world-point,take any four directions that are mutually orthogonal with respectto the metric of the world. These four directions, taken in pairs,determine six surface-elements or orientations, in each of whichthere is an aggregate of geodesics issuing from the point, forminga 'geodesic surface': then K is minus twice the sum of theGaussian measures of curvature of these six surfaces at the givenworld-point. It is independent of the choice of the four orthogonaldirections.

Now Mach had introduced long before a principle, that inertiamust be reducible to the interaction of bodies: and Einsteingeneralised this into what he called Mach's principle, namely thatthe field represented by the ten potentials gpq is determined solelyby the masses of bodies. The word ' mass' is here to be understoodin the sense given to it by the theory of relativity, that is, as equi-valent to energy: and as energy is expressed covariantively byMinkowski's energy-tensor T pq, it follows that in the fundamentalequations of gravitation, corresponding to the equation "'ltV = - 47TPof the Newtonian theory, we may expect the tensor T pq or somelinear function of it to take the place of Poisson's p. We expectto find on the other side of the equation, corresponding to V2V, atensor of the same rank as T pq, that is, the second rank, containingsecond derivatives of the potentials but no higher derivatives.The only covariant tensors of this character are the Ricci-tensorKpq, with Kgpq and gpq. Einstein first supposed that Kpq might bea simple constant multiple of Tpq: but this is not satisfactory forreasons that will be more evident later 1: and he finally proposedthe equations

(p, q=O, 1,2,3)1 The divergence of T",g is zero, and of K!g is not in general zero.

167


where T = 2: gpqT pq, and I< is a constant depending on the Newtonianp,q

constant of gravitation. These are the general field-equations ofgravitation.

Multiplying them by gpq, summing with respect to p and q, andremembering that L gpqgpq = 4, we have K = I<T, so the equations may

l',qbe written

(p, q= 0, 1, 2, 3).

These are ten equations for the ten unknowns gpq: there are fouridentities between them, as might be expected: for four of thegpq's can be assigned arbitrarily as functions of the xl', correspondingto the fact that the equations are invariant under the most generaltransformation of co-ordinates.

According to Mach's principle as adopted by Einstein, thecurvature of space is governed by physical phenomena, and we haveto ask whether the metric of space-time may not be determinedwholly by the masses and energy present in the universe, so thatspace-time cannot exist at all except in so far as it is due to theexistence of matter. The point at issue may be illustrated by thefollowing concrete problem: if all matter were annihilated exceptone particle which is to be used as a test-body, would this particlehave inertia or not? The view of Mach and Einstein is that itwould not: and in support of this view it may be urged that,according to the deductions of general relativity, the inertia of abody is increased when it is in the neighbourhood of other largemasses: it seems needless, therefore, to postulate other sources ofinertia, and simplest to suppose that all inertia is due to the presenceof other masses. When we confront this hypothesis with the factsof observation, however, it seems that the masses of whose existencewe know-the solar system, stars and nebulae-are insufficient toconfer on terrestial bodies the inertia that they actually possess:and therefore if Mach's principle were adopted, it would be necessaryto postulate the existence of enormous quantities of matter in theuniverse which have not been detected by astronomical observation,and which are called into being simply in order to account forinertia in other bodies. This is, after all, no better than regardingsome part of inertia as intrinsic.

The relation of Einstein's to Newton's laws of motion in thegeneral case was discussed by L. Silberstein, 1 who showed thatthe differential equations of a geodesic in General Relativity arerigorously identical with the Newtonian equations of motion ofa particle,

dzgp oildti = agp

1 Nature, cxii (1923), p. 788168

(p= 1,2,3)

GRAVITATION

SO long as the frame of reference for the differential equations ofthe ~eodesic is a system which is momentarily at rest relative to thepartIcle. For simplicity consider a ' statical' world, specified by ametric

3

(ds)1 = VI(dt)1 - ~ L apq dxPdxqC p,q=l

where V and the apq are functions of (Xl, Xl, x3) only; and consideran observer who is stationary, i.e. whose co-ordinates (xl, Xl, x3) donot vary. Then the components of the gravitational force on theparticle can be shown to be

gO=O, (p=1,2,3),

and it can be shown from Einstein's fundamental equation

Kpq-!gpqK= -KTpqthat

3

6. IV =!K (Tolo + L apqT"q),V V ",q=l

where 6.IV denotes the Second Differential Parameter of V in thethree-dimensional space.

Now suppose that the field is of the kind considered in Poisson'sequation, namely that the space is approximately Euclidean andthat there is a volume-density p of matter at rest, and no radiation;then the only sensible element of the energy-tensor is the energy-density due to the equivalence of mass and energy, which has thevalue Clp. Moreover we can take V = 1+y, where y is small; sothe above equation becomes

6.aY = !KC·p.

But writing x, y, Z for the co-ordinates, since

approximately, we have

_ I (oly oly Oly)6..y-c oxl + 8y1 + oz•.Thus


Now in the light of the above equations for the gravitational force,we have

- c2y = the gravitational potential 0,so

020 020 020ax! + Oy! + OZI = - !KC

2p.

Comparing this with Poisson's equation

020 020 020ox! + iry2 + OZ2 = - 41TfJp

where fJ is the Newtonian constant of gravitation, we have

81TfJK=-

cl

an equation connecting the Einsteinian and Newtonian constants of gravitation.Since

fJ = 6·67 X 10-8 gr-1 cm3 secl

this givesK = 1·87 X 10-27 cm.gr-l•

Almost simultaneously with Einstein's discovery of GeneralRelativity, David Hilbert! (1862-1943) gave a derivation of thewhole theory from a unified principle. Defining a point in space-time by generalised co-ordinates (XO, xl, xl, x3), he adopted Einstein'sten gravitational potentials gpq and a four-vector (epo, epl, eps, ep3)representing the electrodynamic potential: and assumed the follow-ing axioms:Axiom I (Mie's axiom of the world-function). All physical happenings(gravitational, electrical, etc.) in the universe are determined by a scalar

function H (called the world-function) which involves the arguments gpqand their first and second derivatives with respect to the x's, and involves alsothe ep's and their first derivatives with respect to the x's: and the laws ofphysical processes are obtained by annulling the variation of the integral

(where g denotes the determinant of the gpq) for each of the fourteenpotentials gpq, ep.. The reason for the occurrence of the factor yg

1 GOtt. N(Jf;h., 1915,p. 395; read 20 Nov. 1915. The investigation was carried furtherby: H. A. Lorentz, Proc. Arnst. Ac. xix (1916), p. 751 ; J. Tres1ing, Proc. Arnst. Ac. xix(1916), E' 892; A. Ein~tein, Berlin Sitz, 1916,p. 1111 ; F. Klein, Gott. N,ach. 1917,p. 469 ;H. Wey , Ann. d. Phys. bv (1917), p. 117; A. D. Fokker, Proc. Arnst. Ac. XlX (1917), p. 968 ;A. PaIatini, Palerrrw Rend. xliii (1919), p. 203; D. Hilbert, Math. Ann. xcii (1924), p. 1 ;E. T. Whittaker, Proc. R.S.(A), cxiii (1927), p. 496.

qo

GRAVITATION

is that yg dxodx1dx2dx3, which is called the invariant hypervolume, ISinvariant under all transformations of co-ordinates.Axiom II (axiom of general invariance). The world-function H lS

invariant with respect to arbitrary transformations of the x'.These axioms represented a distinct advance on Einstein's

methods, in which Hamilton's Principle had played only a verysubordinate part.

We suppose that the world-function H is a sum of two terms,of which the first represents gravitation, i.e. whatever is inherentin the intrinsic structure of space-time, while the second termrepresents all other physical effects. The first term we take to beproportional to K, the scalar curvature of space-time at the point(XO, Xl, x2, x3); this amounts to supposing that the mutual energyofall the gravitating masses in the world can be expressed analyticallyas an integral taken over the whole of space, namely a numericalmultiple of ffffKyg dxodxldx2dx3• With regard to the second term,we shall suppose for simplicity that the only other physical effectto be considered is an electromagnetic field in free aether. Now inEuclidean space there is, in the field, electric energy (dx2 + d,l + dz2)/8rrper unit volume, and magnetic energy (hx2 + hy2 + hz2) /8rr per unitvolume; and since these are of opposite type, in the sense in whichkinetic and potential energy are of opposite type, we should expecttheir difference to occur in the world-function. In general co-ordinates this difference is represented by

L= __ l_L x"qxpq16rr b"

where XT'q is the electromagnetic six-vector. We shall take thesecond part of the world-function to be a numerical multiple of L.We must therefore have

where K is a constant: or

A calculation shows that

where Kpq is the Ricci tensor: and

Dyg= -tyg L gpqDg"q.",q

171


Let us now find the value of

SJHJLyg dxodx1dx2dx8•

The part that depends on the variations of the gpq is

HH~q, o~~;:,g) Sgpq dx°dx1dx2dx8

where 'is 1 or! according as p is equal or unequal to q: or

!2 HHTpq Sgpq • yg dxOdx1dx2dx8

p,q

whereT =~, o(Lyg) -n _oL+~,o( yg)L.pq yg ogpq ogpq yg ogpq

Now it may easily be shown that

,~= -g gpqogpqso

Moreover, when we regard L as a function of the gpq and the 4>.,remembering that Xpq is a function of the rp. only, not involvingthe gpq, we must write

L 1 ~ ple ql X X= -- L. g g pq leI16'7T le,/, 1', q

so we have

and therefore

T pq= - 41

'7T2Xq. Xp' + 196pq2: X" X"• 7T r. 6

or

which is the expression previously found for Minkowski's electromagneticenergy-tensor.

GRAVITATION

Thus the variational integral, in so far as the variations of thegpg are concerned, is

pio JHJ(Kpg - igpg K + KT pg) ogpg. yg tJx°dx1dx!dx' = 0

and therefore the variational equations derived .from the tenns in gpg are

(p, q = 0, 1, 2, 3)

which are identical with Einstein's gravitational equations, previously given.Hilbert showed moreover that the Ricci tensor satisfies the four

identical relations expressed by the equation

Lliv (K~-i o~K) =0.

We must now find the part of the variational integral thatinvol'les the variations of the electromagnetic potentials. It is(disregarding a constant multiplier)

HHL Xpg yg oXpq tJx°tJx1dx·dx·.P. g

But

so the part of the variational integral with which we are concerned is

which after integration by parts yields

Interchanging p and q in the summation of the second term, thisbecomes

and therefore the variational equations obtained from the variationsof the potentials are

L o(Xpgyg) -0g oxq

173

(p=0, 1,2,3)


or

and these are precisely the Ampere-Maxwell tetrad of electromagneticequations in free aether. The Faraday tetrad of course follows fromthe expression of the Xpq in terms of the potentials. Thus bothEinstein's gravitational equations and the electromagnetic equations can beobtained by the variation of the integral of Hilbert's world-Junction.

It was proposed by Cornel Lanczos 1 that the world-function Hoccurring in the integral ffffH vig dxodx1dx2dx3 should be formedin a way different from that adopted by Hilbert, namely that itshould be a quadratic function of the Ricci tensor Kpq, and in factshould be of the form

H=L KpqKpq+CK2p,q

where C is a numerical constant. The advantage of this form isthat in the course of the analysis, a four-vector is found to occurnaturally, and this vector can be identified with the electromagneticpotential-vector: thus electromagnetic theory can be unified withthe theory of General Relativity.

In 1920 a criticism of General Relativity was published in theTimes Educational Supplement 2 by the mathematician and philosopherAlfred North Whitehead (1861-1947). 'I doubt,' he said, 'thepossibility of measurement in space which is heterogeneous as toIts properties in different parts. I do not understand how the fixedconditions for measurement are to be obtained.' He followed upthis idea by devising an alternative theory which he set forth ina book, The Principle of Relativity,3 in 1922. 'I maintain,' he said,'the old-fashioned belief in the fundamental character of simul-taneity. But I adapt it to the novel outlook by the qualificationthat the meaning of simultaneity may be different in different in-dividual experiences.' This statement of course admits the relativitytheory of Poincare and Lorentz, but it is not compatible with theGeneral Theory of Relativity, in which an observer's domain ofsimultaneity is usually confined to a small region of his immediateneighbourhood. Whitehead therefore postulated two fields ofnatural relations, one of them (namely space and time relations)being isotropic, universally uniform and not conditioned by physicalcircumstances: the other comprising the physical relations ex-pressed by laws of nature, which are contingent.

A profound study of Whitehead's theory was published in 1952by J. L. Synge.' As he remarked, the theory of Whitehead offerssomething between the two extremes of Newtonian theory on the

1 ZS.f. P. Ixxiii (1931), p. 147; !xxv (1932), p. 63; Phys. Rev. xxxix (1932), p. 716I Times Educ. Suppl., 12 Feb. 1920, p. 83I The Principle of Relativity, with applications to physical scieru:e, Camb. Univ. Press, 1922, Proc. R.S.(A), ccxi (1952), p. 303

174

GRAVITATION

one hand and the General Theory of Relativity on the other. Itconforms to the requirement of Lorentz invariance (thus overcomingthe major criticism against the Newtonian theory), but it does notreinstate the concept offorce, with the equality of action and reaction,so that its range of applicability remains much lower than that ofNewtonian mechanics. However, it does free the theorist from thenearly impossible task of solving a set of non-linear partial differentialequations whenever he seeks a gravitational field. It is not a fieldtheory, in the sense commonly understood, but a theory involvingaction at a distance, propagated with the fundamental velocity c.

Whitehead's doctrine, though completely different from Einsteins'in its formulation, may be described very loosely as fitting theEinsteinian laws into a flat space-time; and no practicable observa-tional test has hitherto been suggested for discriminating betweenthe two theories.1

The idea of mapping the curved space of General Relativity ona flat space, and making the latter fundamental, was revived manyyears after Whitehead by N. Rosen.2 He and others a who developedit claimed that in this way it was possible to explain more directlythe conservation of energy, momentum, and angular momentum,and also possibly to account for certain unexplained residuals inrepetitions of the Michelson-Morley experiment. 4

In 1916 K. Schwarz schild made an important advance in theEinsteinian theory of gravitation, by discovering the analyticalsolution of Einstein's equations for space-time when it is occupiedby a single massive partlcle.6

The field being a statical one, we can take the quadratic formwhich specifies the metric of space-time to be

(ds)1= VI(dt)l_l (dl) Ic·

where dl is the line-element in the three-dimensional space, S, sos

(dl) I = L apqdxP dxqp,~l

and the functions V, apq, (p, q= 1, 2, 3), do not involve t. Therewill be a one-to-one correspondence between the points of thespace S, which contains the massive particle, and the points of a

1 cr. G. Temple, Proc. Phys. Soc. xxxvi (1924), p. 176; W. Band, Phys. Rev. !xi (1942),p. 698

I Phys. Rev. 1vii (1940), pp. 147, 150, 154I M. Schoenberg, Phys. Rev. lix (1941), p. 616; G. D. Birkhoff, Proc. Nat. Ac. Sci.

xxix (1943), p. 231 ; A. Papapetrou, Proc. R.I.A., Iii (A) (1948), p. 11, b. C. Miller, Proc. Nat. Ac. Sci. xi (1925), p. 306, K. Schwarzschi1d,Bert. Sit;;. 1916, p. H19; cr. also: A. Einstein, Bert. Sit;;. 1915,

p. 831; .0: Hi1be~t,Gott. !Vach. 1917,p. 53; J. Droste, Proc. Arnst. Ac. xix (1917), p. 197;A. Pa1atlm, N. Cmlento, XlV (1917), p. 12; C. W. Oseen, Ark./. Mat. Ast. och. Fys. xv(1921), No.9; J. L. Synge, Proc. R.I.A. liii (1950), p. 83.

175


space S' which is completely empty. Now the latter space can bespecified by co-ordinates (r, 8, c/» in such a way that its line-elementis given by

the origin of S' corresponding to the location of the massive particlein S. The effect of the presence of the particle will be to distortS as compared with S', but the distortion will be symmetrical withrespect to the particle, so the line-element in S will be expressiblein the form

and if we take a new variable R such that .yg(r) . r= R, this maybe written

where A is sc,me function of R. By symmetry, V is also a functionof R, and the problem is to determine A and V.

Thus, now writing r for R, the quadratic form which defines themetric in the space around the partIcle is

(ds)2= L gpqdxpdxqp,q

whereXO = t, x1= T,

and gpq is zero when p is different from q.Since the energy-tensor is zero everywhere except at the origin,

and since K = 0 when Kpq = 0, Einstein's gravitational equationsreduce to

Kpq=O.

Calculating the Ricci tensor, and denoting differentiations by dashes,we find

V" A'V' 2A'Kll = V - AV - Ar

(1 rV' rA' )KII=C

2 A2+AsV-As-1.176

GRAVITATION

Forming the combination (A'lc'VI) Koo + Kll, and equating it to zero,we have

so AV=a constant; and since when r-+ ex:> the space tends to thatdefined by the quadratic form

(ds)' = (dt) , _! {(dr)l + r'(d8)' + rl sin'8(dq,) I},cl

that is, A...•1 and V -+ 1, we must have

The equation K2I =0 now becomes

1 2rA'--- -1=0AI A' .

so denoting (1IA') - 1 by u, we have

u + r du = 0, or ru = Constant,dr

so

Thus the metric in the space around the particle is specified by

(ds)' = (1-~)(dt)l- ~ {id~~ + r'(d8), + r' sin'8(dq,), }.

r

This is Schwarzschild's solution.If at any instant, in the plane 8= !7T, we consider a circle on which

r is constant, we see that the length of an element of its arc is givenby the equation

(dl) 1= rl(dq,)'

so dl = rdr/>,and the circumference of the whole circle is 27Tr. Thisdetermines the physical meaning of the co-ordinate r. But if weconsider a radius vector from the origin, the element of length alongthis radius is given by

(dl)'= (dr)',l-~

r177


SO when r, in decreasing, tends to a, then dl-+oo: that is to say,the region inside the sphere r = a is impenetrable.

The differential equations of motion of a particle (supposed tobe so small that it does not disturb the field), under the influenceof a single gravitating centre, are the differential equations of thegeodesics in space-time with Schwarzschild's matric. These equa-tions can be written down and integrated in terms of ellipticfunctions.1

A particular case is that of rectilinear orbits along a radius vectorfrom the central mass. We then have dep= 0, and can readily derivethe integral

(I-~)~=Constant=l say,r ds I-'

and we have

Eliminating ds between these equations, we have

__ 1__1 (r!!)1 = _1__ 1-'1.

cl ( 1- i) dt 1- ~

Let dl denote the element of length along the radius, and let d-rdenote the element of time, so

(dl) 1= (dr) 1 ,

I-~r

then the preceding equation becomes

1- (!!!)8 = I_I-'I + al-'Icl d-r r

which obviously corresponds to the Newtonian equation of theconservation of energy. Differentiating it, we have

d8[ = _ ap.ac8 (I _~)!

d-rl 2r8 r·1 cr. W. de Sitter, Proe. Arnst. Ae. xix (1916), p. 367; T. Levi-Civita, Rena. Lincei,

xxvi (i) (1917), pp. 381, 458; xxvi (ii) (1917), p. 307; xxvii (i) (1918), p. 3; xxvii(ii) (1918), pp. 183,220,240, 283, 344; xxviii (i) (1919), pp. 3, !OI ; A. R. Forsyth,Proe. R.S.(A), xcvii (1920), p. 145 (Integration by Jacobian elliptic functions); F.Morley, Am. J. Math. xliii (1921), p. 29 (by Weierstrassian elliptic functions); C. deJans, MIm. de l'Ae. de Belgiq~, vii (1923) (by Weierstrassian functions); K. Ogura,Jap. J. qf Phys. iii (1924), pp. 75, 85; a very complete study of the trajectories of a smallparticle in the Schwarzschild field was made by Y. Hagihara, Jap. J. qf Ast. ana Geoph.viii (1931), p. 67.

178

GRAVITATION

When r is large compared with a, this becomes approximately

d2r aci

(jj2= - 2r2

which is the Newtonian law of attraction.Comparing it with the Newtonian law of attraction to the

sun,

where y is the Newtonian constant of attraction and M is the sun'smass,we have

2yMa=---

c2

which givesa=2'95 km.

This is the value of the constant a for the sun.By approximating from the elliptic-function solution for the

general case of motion round a gravitating centre, we can study theorbits when the distance of the small particle from the centre of forceis large compared with a, in which case we can have orbits differinglittle from the ellipses described by the planets under the Newtonianattraction of the sun. It is found that the line of apsides of such anorbit is not fixed, but slowly rotates: if I is the semi-latus-rectum,then the advance of the perihelion in one complete revolutionis 37Talt, which for the case of the planet Mercury revolvinground the sun amounts to about 0"·1: since Mercury makesabout 420 revolutions in a century, the secular advance ofthe perihelion is 42", which agrees well with the observed valueof 43".1

The paths of rays of light in the field of a single gravitatingcentre are, of course, the null geodesics of the Schwarzschild field.Many of them have remarkable forms. I t is found that a ray oflight can be propagated perpetually in a circle of radius h aboutthe gravitating centre: and there are trajectories spirally asymptoticto this circle both externally and internally, the other terminus ofthe trajectory being on the circle r = a in the former case and atinfinity in the latter. There are also trajectories of which one

1 Einstein first showed that the new gravitational theory would explain the anomalousmotion of the perihelion of Mercury in the third of his papers in the Berlin Sit;;. of 1915,p. 831. On the whole subject cf. G. M. Clemence, Proc. Amer. Phil. Soc. xciii (1949),p. 532. The secular motion of the earth's perihelion, as revealed by observation, wasfound by H. R. Morgan, Ast. }.Ii (1945), p. 127, to agree with that calculated by Einstein'stheory, which is 3'·84.

179


terminus is at infinity and the other on the circle r= a, trajectorieswhich at both termini meet the circle r= a, and quasi-hyperbolictrajectories both of whose extremities are at infinity. For theselast-named trajectories it is found that if a is the apsidal distance,the angle between the asymptotes is approximately equal to 2a/a.A ray coming from a star and passing close to the sun's gravitationalfield, when observed by a terrestrial observer, will therefore havebeen deflected through an angle 2a/a.1 If we take the radius ofthe solar corona to be 7 x 105 km., so that a= 7 x 105 km. anda = 3 km., then Einstein found that the displacement of the star is 2.3/7.105in circular measure, or 1"·75.

The notion that light possessesgravitating mass, and that there-fore a ray of light from a star will be deflected when it passes nearthe sun, was far from being a new one, for it had been put forwardin 1801 by J. Soldner,2 who calculated that a star viewed near thesun would be displaced by 0"·85. Einstein's prediction was tested bytwo British expeditions to the solar eclipse of May 1919,who found forthe deflection the values 1"·98±0"·12 and 1"·61±0"·30 respectively,so the prediction was regarded as confirmed observationally: andthis opinion was strengthened by the first reports regarding theAustralian eclipse of 1922 September 2I. Three different expedi-tions found for the shift at the sun's limb the values 1"'72±O"'ll,1"·90 ±0"·2 and 1"·77±0"'3, all three results differing from Einstein'spredicted value by less than their estimated probable errors. How-ever, a re-examination of the 1922 measures gave about 2"'2: atthe Sumatra eclipse of 1929 the deflection was found 3 to be 2"·0to 2"·24: and at the Brazilian eclipse of May 1947 a value of2"·01 ±0"·27 was obtained': while it must not be regarded asimpossible that the consequences of Einstein's theory may ultimatelybe reconciled with the results of observation, it must be said that atthe present time (1952) there is a discordance.

A second observational test proposed was the anomalous motionof the perihelion of Mercury. This is quantitatively in agreementwith Einstein's theory,5 but as we have seen,6 there are alternativeexplanations of it.

A third proposed test was the displacement to the red of spectrallines emitted in a strong gravitational field. This, however, was, aswe have seen,? explained before General Relativity was discovered,and does not, properly speaking, constitute a test of it in contra-

1 cf. F. D. Murnaghan, Phil. Mag. xliii (1922), p. 580; T. Shimizu, Jap. J. Phys.iii (1924), p. 187; R. J. Trumpler, .l.R.A.S. Canada,xxiii (1929), p. 208

I BerlinerAstr. Jahrb. 1804, p. 16f; reprinted Ann. d. Phys. Ixv (1921~, p. 593I E. Freundlich, H. v. Kltiber and A. v. Brunn, ZS.f. Astroph. iii 1931), p. 171;

vi (1933), p. 218; xiv (1937), p. 242; d.J.Jackson, Observatory,liv (19 I), p. 292• G. van Biesbroek, Ast. J. Iv (1950), p. 49; cr. E. Finlay-Freundlich and W. Leder-

mann, Mon. Not. R.A.S. civ (1944), p. 40• d. G. M. Clemence, Ast. J. I (1944)• d. p. 148, p. 152-3; cr. G. Y. Rainich, Phys. Rev. xxxi (1928), p. 448

180

(dx) 2 (dy) 2 (dz) 2.where v2= {Ii + dt + di

GRAVITATION

distinction to other theories.l In any case, the observed effect s6far as solar lines 2 are concerned is complicated by other factors.3

With regard to stellar lines, the greatest effect might be expectedfromstars of great density. Now Eddington showed that in a certainclass of stars, the 'white dwarfs,' the atoms have lost all theirelectrons, so that only the nuclei remain: and under the influenceof the gravitational field, the nuclei are packed together so tightlythat the density is enormous. The companion of Sirius is a starof this class: its radius is not accurately known, but the star mightconceivably have a density about 53,000 times that of water, inwhich case the red-displacement of its lines would be about 30 timesthat predicted for the sun. The comparison of theory with observa-tion of the red-displacement can, however, hardly be said to furnisha quantitative test of the theory.4

We may remark that it is easy to construct models which behaveso as to account for the observed facts of astronomy. For example,it has been shown by A. G. Walker 5 that if a particle is supposedto be moving in ordinary Euclidean space in which there is aNewtonian gravitational potential ep (so that the contribution toep from a single attracting mass M is -yM/r, where y is the New-tonian constant of attraction) and if the kinetic energy of theparticle is

T ,-2"'/e' 2=te v

(which reduces to T = iv2 when c-..oo), and if the potential energyof the particle is

I The derivation of the effect from Einstein's equations may readily be constructedfrom the following indications. For an atom at rest in a gravitational field (e.g. on thesun) the proper-time is given by

dT'=V'dt',

where, as we have seen, V = I - (I/e') 0 if 0 is the gravitational potential; while for anatom at rest at a great distance from the gravitational field (e.g. on the earth) we have

V = I and dT' -dt'.

The period of the radiation, as measured at its place of emission, is to its period as measuredby the terrestrial observer, in the ratio that dT{dt at the place of emission bears to dT{dtat the place of the observer, that is, the ratio V to unity; so, as before, we find that thewave-length of the radiation produced in the strong gravitational field and observed by theterrestrial observer is (I +O{e') times the wave-length of the same radiation produced onearth.

, C. E. 51. John, Astroph. J. Ixvii (1928), p. 195, cf. Miss M. G. Adam of Oxford, Mon. Not. R.A.S. cviii (1948), p. 446, For a discussion as to whether the perihelion motion of Mercury, the deflection

of light rays that pass near a gravitating body, and the displacement of spectral linesin the gravitational field, are to be regarded as establishing General Relativity, cr.E. Wiechert, Phys. ZS. xvii (1916), p. 442, and Ann. d. Pltys. !xiii (1920), p. 301.

• Nature, c!xviii (1951), p. 961181


(which reduces to t/> when c..•.oo), then the Lagrangean equationsof motion of the particle are

d{ 1 dx} l+v!jclot/>dt 1- vlje! dt = - 1-v2/eS ax

and two similar equations iny and z (which reduce to d2x/dt2 = - oep/oxand two similar equations when c....•.oo), and that the trajectorieshave the following properties:

(i) the velocity of the particle can never exceed c;(ii) a particle moving initially with velocity c continues to move

with this velocity;(iii) the perihelion of an orbit advances according to the same

formula as is derived from General Relativity ;(iv) if the particles moving with speed c are identified with

photons, they are deflected towards a massive gravitating bodyaccording to the same formulae as is derived from General Relativity.

It is unwise to accept a theory hastily on the ground of agree-ment between its predictions and the results of observation ina limited number of instances: a remark which perhaps is speciallyappropriate to the investigations of the present chapter.

An astronomical consequence of General Relativity which hasnot yet been mentioned was discovered in 1921 by A. D. Fokker/namely that, as a result of the curvature of space produced by thesun's gravitation, the earth's axis has a precession, additional tothat deducible from the Newtonian theory, of amount 0"·019 perannum; unlike the ordinary precession, it would be present evenif the earth were a perfect sphere. Its existence cannot howeverbe tested observationally, since the shape and internal constitutionof the earth are not known with sufficient accuracy to give a reliabletheoretical value for the ordinary precession, and the relativisticprecession is only about -doo of this.

Besides the Schwarzschild solution, a number of other particularsolutions of the equations of General Relativity were obtained inthe years following 1916, notably those corresponding to a particlewhich has both mass and electric charge,S and to fields possessingaxial symmetry,S especially an infinite rod,4 and two parncles.6 In1938 A. Einstein, L. Infeld and B. Hoffman published a method 6

for finding, by successive approximation, the field due to n bodies.1 Proc. Arnst. Ac. xxiii (1921), p. 729; cr. H. A. Kramers, ibid. p. 1052, and

G. Thomsen, Rend. Lincei, vi (1927), p. 37. The existence of an effect of this kind had beenpredicted in 1919 by j. A. Schouten, Proc. Arnst. Ac. xxi (1919), p. 533.

• H. Reissner, Ann. d. Phys.l (1916), p. 106; H. Weyl, Ann d. Phys.liv (1917),p.1l7;G. Nordstrom, Proc. Arnst. Ac. xx (1918), p. 1236; C. Longo, N. Cimento, xv (1918), p. 191 ;G. B. jeffery, Proc. R.S.(A), xcix (1921), p. 123.

• T. Levi-Civita, Rend. Lincei(5) >Lxviii(i) (1919), pp. 4, 101• W. Wilson, Phil. Mag.(6) ,d (1920), p.. 703• H. E.j. Curzon, Proc. L.M.S.(2) xxiii (1924-5), pp. xxix and 477• Ann. of Math. xxxix (1938), p. 65; cr. H. P. Robertson, ibid., p. 101 and T. Levi-

Civita, Mem. des Sc. Math., fasc. cxvi (1950)182

GRAVITATION

Static isotropic solutions of the field equations, and symmetricdistributions of matter, have been discussed by M. Wyman.1

In 1916 and the following years attention was given 2 to thepropagation of disturbances in a gravitational field. If the distri-butIOn of matter in space is changed, e.g. by the circular motionof a plate in its own plane, gravitational waves are generated,which are propagated outwards with the speed of light.

If such waves impinge on an electron which is at rest, the prin-ciple of equivalence shows that the physical situation is the sameas if the electron were moving with a certain acceleration, andtherefore an electron exposed to gravitational waves must radiate. 3

In 1917 Einstein 4 pointed out that the field-equations of gravita-tion, as he had given them in 1915, do not satisfy Mach's Principle,according to which, no space-time could exist except in so far as itis due to the existence of matter (or energy). Einstein's equationsof 1915, however, admit the particular solution

gpq =Constant, (p, q=O, 1,2,3)

so that a field is thinkable without any energy to generate it. Hetherefore proposed now 6 to modify the equations by writing them

(p, q=O, 1,2,3).

The effect of the A-term is to add to the ordinary gravitationalattraction between particles a small repulsion from the origin vary-ing directly as the distance: at very great distances this repulsionwill no longer be small, but will be sufficient to balance the attraction:and in fact, as Einstein showed, it is possible to have a staticaluniverse, spherical in the spatial co-ordinates, with a uniformdistribution of matter in exact equilibrium. 6 This is generally calledthe Einstein universe. The departure from Euclidean metric ismeasured by the radius of curvature Ro of the spherical space, andthis is connected with the total mass M of the particles constitutingthe universe by the equation

I Phys. Rev. Ixvi (1944), p. 267; !xxv (1949), p. 1930• A. Einstein, Berlin Sitz. 1916, p. 688; 1918, p. 154; H. Weyl, Raum Zeit Mate,w,

4th edn. (Berlin, 1921) p. 228 (English edn., p. 252); A. S. Eddington, Proc. R.S.(A),cii (1922), p. 268; H. Mineur, Bull. S.M. Fr.lvi (1928), p. 50; A. Einstein and N. Rosen,J. Frankl. Inst. ccxxiii (1937), p. 43; M. Brdicka, Proc. R.I.A. liv (1951), p. 137.

• This problem was investIgated by W. Alexandrow, Ann. d. Phys. Ixv (1921), p. 675.• Berlin Sitz., 1917, p. 142; Ann. d. Phys. Iv (1918), p. 241• The new equations can be derived variationally by adding a constant to Hilbert's

World-Function.• The suggestion that our universe might be an Einsteinian space-time of constant

spatial curvature seems to have been made first by Ehrenfest in a conversation with deSitter about the end of 1916.

183


where y denotes the Newtonian constant of gravitation. The totalvolume of this universe is 27T2Ro3.

It was shown by J. Chazy 1 and by E. Trefftz 2 that when theA-term is included in the gravitational equations, Schwarzschild'smetric for the space-time about a single gravitating centre mustbe modified to

If in this we put a =0, which amounts to supposing that thereis no mass at the origin, so the world is completely empty,3 weobtain

(dsr = (1 - ~ r2) (dt) 2 _ ~ f (dr)-=-+ r2(d8) 2 + r2sin28(dep) 2}.3 c (1 _ A r2

3

Now this metric had been discovered by W. de Sitter 4 in1917. It is the metric of a jour-dimensional space-time oj constantcurvature.

The de Sitter world, as it was called, was the subject of manypapers, partly on account of its intrinsic geometrical interest to thepure mathematician, and partly because of the possibility that someor all of its features might be similar to those of our actual universeas revealed by astronomical observation.5

Let us consider the universe as it would be if all minor irregu-larities were smoothed out: just as when we say that the earth isa spheroid, we mean that the earth would be a spheroid if all mountainswere levelled and all valleys filled up. In the case of the universethe levelling is a more formidable operation, since we have to smoothout the earth, the sun and all the heavenly bodies, and to reducethe world to a complete uniformity. But after all, only a very smallfraction of the cosmos is occupied by material bodies: and it is

1 Comptes Rendus, clxxiv (1922), p. 1157• Math. Ann.lxxxvi (1922), p. 317; cr. M. von Laue, Berlin Sitz;., 1923, p. 27I This, of course, shows the invalidity of the reason Einstein had originally given for

introducing the -'-term.• Proe. Arnst. Acad. xix (31 March 1917), p. 1217; xx (1917), pp. 229, 1309; Mon.

Not. R.A.S. !xxviii (Nov. 1917), p. 3• cr. F. Klein, Gott. Nach. 6 Dec. 1918, = Ges. Math. Abh. i, p. 604; K. Lanczos,Phys. ZS. xxiii (1922), p. 539; H. Weyl, Phy,. ZS. xxiv (1923), p. 230; Phil. Mag.(6)xlviii (1924), p. 348; Phil. Mag.(7) ix (1930), p. 936; P. du Val, Phil. Mag.(6) xlvii(1924), p. 930; M. von Laue "and N. Sen, Ann. d. Phys. Ixxiv (1924), p. 252; L. Silber-stein, Phil. Mag. (6) xlvii (1924), p. 907; H. P. Robertson, Phil. Mag.(7) v (1928),p. 835; R. C. Tolman, Astroph. J. lxix (1929), p. 245; G. Castelnuovo, Lineei Rend.xii (1930), p. 263; M. von Laue, Berlin Sitz;. 1931, p. 123; E. T. Whittaker, Proe. R.S.(AI,cxxxiii (1931), p. 93; H.S. Coxeter, Amer. Math. Monthly, I (1943), p. 217

184

GRAVITATION

interesting to inquire what space-time as a whole is like when wesimplyignore them.l

The answer must evidently be, that it is a manifold of constantcurvature. This means that it is isotropic (i.e. the curvature is thesamefor all orientations at the same point) and is also homogeneous.As a matter of fact, there is a well-known theorem that any manifoldwhich is isotropic in this sense is necessarily also homogeneous, sothat the two properties are connected. A manifold of constantcurvature is a projective manifold, i.e. ordinary projective geometryis valid in it when we regard geodesics as straight lines: and it ispossibleto move about in it any system of points, discrete or con-tinuous, rigidly, i.e. so that the mutual distances are unaltered.

The simplest example of a manifold of constant curvature isthe surface of a sphere in ordinary three-dimensional Euclideanspace: and the easiest way of constructing a model of the de Sitterworld is to take a pseudo-Euclidean manifold of five dimensions inwhich the line-element is specified by the equation

_ (ds)1 = (dx)1 + (dy)1 + (dz) I - (dU)1 + (dV)I,

and in this manifold to consider the four-dimensional hyper-pseudo-sphere2 whose equation is

Xl +yl + Zl - ul + Vi =RI.

The pseudospherical world thus defined has a constant Riemannianmeasureof curvature - I/R2.s

The de Sitter world may be regarded from a slightly differentmathematical standpoint as havin~ a Cayley-Klein metric, governedby an Absolute whose equation In four-dimensional homogeneousco-ordinates is

Xl +y + Zl - ul + Vi =0

where u is time. Hyperplanes which do not intersect the Absoluteare spatial, so spatial measurements are elliptic, i.e. the three-dimensional world of space has the same kind of geometry as thesurfaceof a sphere, differing from it only in being three-dimensionalinstead of two-dimensional. In such a geometry there is a naturalunit of length, namely the length of the complete straight line,

I The curvature of space at any particular place due to the general curvature of theuniverse is quite small compared to the curvature that may be imposed on it locallyby the presence of energy. By a strong magnetic field we can produce a curvaturewith a radius of less than 100 light-years, and of course in the presence of matter thecurvature is stronger still. So the universe is like the earth, on which the local curvatureof hills and valleys is far greater than the general curvature of the terrestrial globe.

I The prefix hyper- indicates that we are dealing with geometry of more than theusual three dimensions, and the prefix pseudo- refers to the occurrence of negative signsin the equation.

I The world of the Poincare-Lorentz theory of relativity can be regarded as a four-dimensional hyperplane in the five-dimensional hyperspace.

185

AETHER ANb ELECTRICITY

just as on the surface of a sphere there is a natural unit of length,namely the length of a complete great circle.

We are thus brought to the question of the dimensions of theuniverse: what is the length of the complete straight line, thecircuit of all space? Since 1917 there has seemed to be a possibilitythat, by the combination of theory with astrophysical observation,this question might be answered.

Different investigations of the de Sitter world, however, reachedconclusions which apparently were not concordant. The origin ofsome of the discordances could be traced to the ambiguities whichwere involved in the use of the terms' time,' , spatial distance' and, velocity,' when applied by an observer to an object which is remotefrom him in curved space-time. The' interval' which is defined by(ds)2 = 2: gpqdxpdxq involves space and time blended together:

p,qand although any particular observer at any instant perceives inhis immediate neighbourhood an ' instantaneous three-dimensionalspace' consisting of world-points which he regards as simultaneous,and within which the formulae of the Poincare-Lorentz relativitytheory are valid, yet this space cannot be defined beyond his immediateneighbourhood: for with a general metric defined by a quadraticdifferential form, it is not in general possible to define simultaneity(with respect to a particular observer) over any finite extent of space-time.

The concept of' spatial distance between two material particles'is, however, not really dependent on the concept of ' simultaneity.' 1

When the astronomer asserts that ' the distance of the Andromedanebula is a million light-years,' he is stating a relation between theworld-point occupied by ourselves at the present instant and theworld-point occupied by the Andromeda nebula at the instantwhen the light left it which arrives here now, that is, he is assertinga relation between two world-points such that a light-pulse, emittedat one, arrives at the other; or, in geometrical language, betweentwo world-points which lie on the same null geodesic. The spatialdistance of two material particles in a general space-time may,then, be thought of as a relation between two world-points which are onthe same null geodesic. It is obviously right that 'spatial distance'should exist only between two world-points which are on the samenull geodesic, for it is only then that the particles at these pointsare in direct physical relation with each other. This statementbrings out into sharp relief the contrast between' spatial distance'and the ' interval' defined by (ds) I = 2: gpqdxpdxq : for between two

P. qpoints on the same null geodesic the 'interval' is always zero.Thus' spatial distance' exists when, and only when, the' interval' lS zero.

In order to define' spatial distance' conformably to these ideas,with a general metric for space-time, it is necessary to translate

1 E. T. Whittaker, Proc. R.S.(A), cxxxiii (1931), p. 93186

GRAVITATION

into the language of differential geometry the principle by whichastronomersactually calculate the' distance' of very remote objectssuchas the spiral nebulae. The principle is this: first, the absolutebrightnessof the object (the' star' as we may call it) is determined 1 :

then this is compared with the apparent brightness (i.e. the bri~htnessas actually seen by the observer). The distance of the star IS thendefined to be proportional to the square root of the ratio of theabsolute brightness to the apparent brightness.

In adopting this principle into differential geometry, we takea ' star' A and an observer B, which are on the same null geodesic,and we consider a thin pencil of null geodesics (rays of light)which issue from A and pass near B. This pencil intersects theobserverB's ' instantaneous three-dimensional space,' giving a two-dimensional cross-section: the 'spatial distance AB' is thendefined to be proportional to the square root of this cross-section.Distance, as thus defined, is an invariant, i.e. it is independent ofthe choice of the co-ordinate system. This invariant, however,involvesnot only the position of the star and the position of theobserver,but also the motion of the observer, since his' instantaneousthree-dimensional space' is determinate only when his motion isknown. Thus the' spatial distance' of a star from an observerdepends on the motion of the observer: but this is quite as it shouldbe, and indeed had always been recognised in the relativity theoryofPoincare and Lorentz: for in that theory the spatial distance of astar from an observer is (X2 +y2 +Z2)i, where (T, X, Y, Z) areco-ordinates referred to that particular inertial system with respect towhich the observer is at rest: the necessity for the words in italicsshowsthat the distance depends on the observer's motion.

When the de Sitter world is studied in the light of this definitionof distance (the mass of any material particles concerned beingsupposed to be so small that they do not sensibly affect the geo-metrical character of the universe), some remarkable results arefound. Thus a freely moving star and a freely moving observercannot remain at a constant spatial distance apart in the de Sitterworld. When the observer first sees the star, Its spatial distance isequal to the radius of curvature of the universe. Mter this thestar is continuously visible, the distance passing through a minimumvalue, after which it increases again indefinitely: that is; the star'sapparent brightness ultimately decreases to zero, it becomes too faintto be seen. When this happens the star is at a point which is notthe terminus of its own world-line, so the star continues to existafter it has ceased to have any relations with this particularobserver.

The Einstein world, which as we have seen is a statical solutionof the gravitational equations, spherical in the spatial co-ordinates,

1 For this purpose, astronomers in practice use the known relation between the periodand absolute magnitude of Cepheid variables, or (in the method of spectroscopic parall-axes) the known relations between absolute magnitude and spectral behaviour.

187


was generalised by A. Friedman 1 into a solution in which thecurvature depends on the time-what in fact came to be knownlater as an expanding universe. Not much notice was taken of thispaper until, five years afterwards, the Abbe Georges Lemaitre,2

a Belgian priest who had been a research student of Eddington'sat Cambridge, proved that the Einstein universe is unstable, andthat when its equilibrium is disturbed, the world progresses througha continuous series of intermediate states, towards a limit which isno other than the de Sitter universe.

Eddington, who at the beginning of 1930 was investigating, inconjunction with his research student G. C. McVittie, the stabilityof the Einstein universe, found in Lemaitre's paper the solution ofthe problem, and at once 3 saw that it provided an explanationof the observed scattering apart of the spiral nebulae. Since 1930the theory of the expanding universe has been of central importancein cosmology.

The scheme of general relativity, as put forward by Einstein in1915, met with some criticism as regards the unsatisfactory positionoccupied in it by electrical phenomena. While gravitation wascompletely fused with metric, so that the notion of a mechanicalforce on ponderable bodies due to gravitational attraction wasentirely abolished, the notion of a mechanical force acting onelectrified or magnetised bodies placed in an electric or magneticfield still persisted as in the old physics. This seemed, at any ratefrom the aesthetic point of view, to be an imperfection, and it wasfelt that sooner or later everything, including electromagnetism,would be re-interpreted and represented in some way as consequencesof the pure geometry of space and time. In 1918 Weyl' proposedto effect this by rebuilding geometry once more on a new foundation,which we must now examine.

Weyl fixed attention in the first place on the 'light-cone,' oraggregate of directions issuing from a world-point P, in which light-signals can go out from it. The light-cone separates those world-points which can be affected by happenings at P, from those pointswhose happenings can affect P: it, so to speak, separates past fromfuture. Now the light-cone is represented by the equation (ds)2=0,where ds is the element of interval, and Weyl argued that this equation,rather than the quantity (ds)2 itself, must be taken as the starting-point of the subject: in other words, it is the ratios of the tencoefficient~gpq in (ds)2, and not the actual values of these coefficients,

1 ZS. f. P. x (1922), p. 377. Aberration and parallax in the universes of Einstein,de Sitter and Friedman, were calculated by V. Freedericksz and A. Schechter, ZS.f. P.Ii (1928), p. 584.

• Ann. de la Soc. sc. de Bnvcelles,xlvii'" (April 1927), p. 49• Mon Not. R.A.S. xc (1930), p. 668• Berl. Sitz. 1918,Part I, p. 465; Math. ZS. ii (1918), p. 384; Ann. d. Phys. lix (1919),

p. 101; Phys. ZS. xxi (1920), p. 649, xxii (1921), p. 473; Nature, cvi (1921), p. 781 ;cf. A. Einstein, Bert. Sitz. 1918,p. 478; W. Pauli. Phys. ZS. xx (1919), p. 457; Verh.d.phys. Ges. xxi (1919), p. 742; A. Einstein, Bert. Sitz. 1921, p. 261; L. P. Eisenhart,Proc.N.A.S. ix (1923), p. 175

188

GRAVITATION

which are to be taken as determined by our most fundamentalphysical experiences. This leads to the conclusion that comparisonsof length at different times and places may yield discordant resultsaccording to the route followed in making the comparison.

Following up this principle, Weyl devised a geometry moregeneral than the Riemannian geometry that had been adopted byEinstein: instead of being specified, like the Riemannian geometry,by a single quadratic differential form

L gpqdxPdxqp, q

it is specified by a quadratic differential form L gpqdxpdxq and ap, q

linear differential form L ePpdxP together. The coefficients gpq ofp

the quadratic form can be interpreted, as in Einstein's theory, asthe potentials of gravitation, while the four coefficients ePP of thelinear form can be interpreted as the four components of the electro-magnetic potential-vector. Thus Weyl succeecjed in exhibiting bothgravitation and electricity as effects of the metric of the world.l

The enlargement of geometrical ideas thus achieved was soonfollowed by a still wider extension due to Eddington.2 This, andmost subsequent constructions in the same field, are based on ananalysis of the notion of parallelism, which must now be considered.

The question of parallelism in curved spaces was raised in anacute form by the discovery of General Relativity theory: for itnow became necessary for the purposes of physics to create a theoryof vectors in curved space, and of their variation from point topoint of the space. Now in Euclidean space if U and V are twovectors at the same point, P, we can find the vector which is theirdifference by using the triangle of vectors. But if U is a vectorat a point P and V is a vector at a different point Q, and if we wantto find the vector which is the difference of U and V, it is necessary(in principle) first to transfer U parallel to itself from the point Pto the point Q, and then to find the difference of the two vectorsat Q Thus a process of parallel transport is necessary for finding thedifference of vectors at different points, and hence for the spatialdifferentiation of vectors: and this is true whether the space isEuclidean or non-Euclidean.

The spatial differentiation of vectors in curved space has alreadybeen discussed 3 analytically under the name of covariant differentiation.Evidently this covariant differentiation must really be based ona parallel transport of vectors in the curved space: and in 1917thIS particular form of parallel transport was definitely formulated

1 It does not seem necessary to describe this geometry in detail, since Weyl himselflater expressed the opinion (Amer. J. Math. Ixvi (1944), p. 591) that it does not (at leastin its onginal form) provide a satisfactory unification of electromagnetism and gravitation.

I Proc. R.S.(A), xcix (1921), p. 104 I cr. p. 161r89


by T. Levi-Civita.1 It may be regarded as providing a geometricalinterpretation for the Christoffel symbols.

After this the idea 2 was developed that a space may be regardedas formed of a great number of small pieces cemented together,so to speak, by a parallel transport, which states the conditionsunder which a vector in one small piece is to be regarded as parallelto a vector in a neighbouring small piece. Thus every type of differ-ential geometry must have at its basis a definite parallel-transportor ' connection.'

To illustrate these points, let us consider geometry on the surfaceof the earth, regarded as a sphere. It is obvious that directions Uand V at two different points cannot (unless one of the pointshappens to be the antipodes of the other) be parallel in the ordinaryEuclidean sense, i.e. parallel in the three-dimensional space inwhich the earth is immersed: but a new kind of parallelism can bedefined, and that in many different ways. We mis-ht, for instance,say that V is derived by parallel transport from U If V and U havethe same compass-bearing, so that, e.g. the north-east direction atone place on the earth's surface would be said to be parallel to thenorth-east direction at any other place.

Having defined parallel-transport on the earth in this way,there are now two different kinds of curve on the earth's surfacewhich may be regarded as analogous to the straight line in theEuclidean plane. If we define a straight line by the property thatit is the shortest distance between two points, then its analogue onthe earth is a great circle, since this gives the shortest distance betweentwo points on the spherical surface. But if we define a straight lineby the property that it preserves the same direction along its wholelength, or, more precisely, that its successive elements are derivedfrom each other by parallel transport, then its analogue on theearth is the track of a ship whose compass-bearing is constantthroughout her voyage: this is the curve called a loxodrome. Theexistence of the two families of curves, the s-reat circles and theloxodromes, may be assimilated to the fact that If an electrostatic anda gravitational field coexist, the lines of electric force and the lines ofgravitational force are two families of curves in space; and this roughanalogy may serve to suggest how different physical phenomena maybe represented simultaneously in terms of geometrical conceptions.3

Weyl's proposal for a unified theory of gravitation and electro-magnetism was followed up in another direction by Th. Kaluza,4

1 PalerrrwRend. xlii (1917), p. 173• This idea is due essentially to G. Hessenberg, Math. Ann. !xxviii (1917), p. 187,

though he did not explicitly introduce Levi-Civita's notion of parallel-transport.• Eddington's generalisation of Weyl's theory was the starting-point of important

papers by A. Einstein, Bert. Sitz. (1923), pp. 32,76, 137, and by Jan Arnoldus Schouten ofDelft (b. 1883), Arnst. Proc. xxvi (1923), p. 850. The latter represented the electromagneticfield as a vortex connected with the torsion (in the sense of Cartan) of four-dimensionalspace; on this cf. H. Eyraud, Comptes Rendus, c1xxx (1925), p. 127.

• Berlin Sitz. (1921), p. 966190

GRAVITATION

in whose theory the ten gravitational potentials gpq of Einstein andthe four components cPP of the electromagnetic potential were expressedin terms of the line-element in a space of five dimensions, in such away that the equations of motion of electrified particles in an electro-magnetic field became the equations of geodesics. This five-dimensional theory of relativity was afterwards developed by OskarKlein 1 and others,2 To ordinary space-time a fifth dimension isadjoined, but the curves in this dimension are very small, of theorder of 10-30 em., so that the universe is cylindrical, and indeedfiliform, with respect to the fifth dimension, and the variations of thefifth variable are not perceptible to us, its non-appearance in ordinaryexperiments being due to a kind of averaging over it. A connectionwith quantum theory was made by assuming that Planck's quantumof Action h is due to periodicity in the fifth dimension: and theatomicity of electricity was presented as a quantum law, the momen-tum conjugate to the fifth co-ordinate having the two values eand -e.

In 1930 Oswald Veblen and Banesh Hoffman of Princeton 3

showed that the Kaluza-Klein theory may be interpreted as a four-dimensional theory based on projective instead of affine geometry.

Weyl's geometry, and espeCIally his type of parallel-transport,suggested to W. Wirtinger 4 philosophical considerations which ledhim to a very original form of infinitesimal geometry: this, how-ever, perhaps on account of its extreme generality, has not as yetfound applications in theoretical physics.

Besides the investigations that have been described, the workof Wey1 and Eddington has led, during the last thirty years, to avast number of other investigations aimed at expressing gravitationaland electromagnetic theory together in terms of a system of dif-ferential geometry. Some of them, such as those of the Princetonschool 5 in America, led by O. Veblen and L. P. Eisenhart, the

1 ZS.f. P. xxxvi (1926), p. 835; xxxvii (1926), p. 895; xlv (1927), p. 285; xlvi(1927), p. 188; J. de Phys. et Ie Rad. viii (1927), p. 242; Ark. f. Mat. Ast. Fys xxxiv(1946),No. I

I A. Einstein, Berlin Sitz. (1927), pp. 23, 26; L. de Broglie, J. de Phys. et Ie Rad.viii (1927), p. 65; G. DarriellS,J. Ik Phys. et Ie Rad. viii (1927), p. 444; F. Gonseth etG. Juvet, ComptesRendus, clxxxv (1927), p. 341; H. Mandel, <'-S.f. P. xxxix (1926),p. 136; xlv (1927), p. 285; xlix (1928), p. 697; liv (1929), p. 564; Ix (1930), p. 782 ;W. Wilson, Prot. R.S.(A), cxviii (1928), p. 441 ; J. W. Fisher, Prot. R.S.(A), cxxiii (1929),p. 489; H. T. Flint, Prot. R.S.(A), cxxiv (1929), p. 143; J. W. Fisher and H. T. Flint,Prot. R.S.(A), cxxvi (1930), p. 644

I Phys. Rev. xxxvi (1930), p. 810; cf. alsoJ. A. Schouten and D. van Dantzig, Proc.Arnst. At. ~ (1932), pp. 642, 843; ZS.f P. \xxviii (1932), p. 63~; J. A. Schoute~~ZS.f. P. \xxxi (1933), pp. 129, 405; \xxxiv (1933), p. 92; W. Pauh, Ann. d. Phys. XVlll(1933),p. 305.

• Trans. Camb. Phil. Soc. xxii (1922), p. 439; Abh. aus Ikm math. Sem. der Hamb. iv(1926), p. 178.

• cf. L. P. Eisenhart and O. Veblen, Prot. N.A.S. viii (1922), p. 19; O. Veblen,Prot. N.A:S. viii (1922),p. 19.2; ix (1923), p. 3; L. P. Eisenhart, ~rot. N.A.S. viii (1922),p. 207; IX (1923), p. 4; XI (1925), p. 246; Annals of Math. XXIV (1923), p. 367; O.Veblen andJ. M. Thomas, Prot. N.A.S. xi (1925), p. 204; T. Y. Thomas, Prot. N.A.S.xi (1925), p. 199; xiv (19213),p. 728; Math. zS. xxv (1926), p. 723, and many papersin the succeedingyears, in the samejournals, by these authors and their associates

191


Dutch school at Delft, founded by J. A. Schouten,! and the Frenchschool whose most distin~uished representative was E. Cartan 2

(1869-1951), have made discoveries which have great potentialities,but the significance of which at present appears to lie in puremathematics rather than in physics, and which are therefore notdescribed in detail here. The work of E. Bortolotti 3 of Cagliarishould also be referred to. The outlook of the Germans, and ofsome of the Americans, has been, broadly speaking, more physical. 4

It must be said, however, that, elegant though the mathematicaldevelorments have undoubtedly been, their relevance to funda-menta physical theory must for the present be regarded as hypo-thetical.

This chapter has been concerned, for the most part, with GeneralRelativity, which is essentially a geometrisation of physics. It maybe closed with some account of a movement in the opposite direction,seeking to abolish the privileged position of geometry in physics,and indeed inquiring how far it may be possible to construct aphysics independent of geometry. Since the notion of metric is acomplicated one, which requires measurements with clocks andscales, generally with rigid bodies, which themselves are systems ofgreat complexity, it seems undesirable to take metric as fundamental,particularly for phenomena which are simpler and actually inde-pendent of it.

The movement was initiated by Friedrich Kottler of Vienna,who in 1922 published two papers Newton'sches Gesetz und Metrik 5

and Maxwell'sche Gleichungen und Metrik.6 Kottler first remarked on

, cf.J. A. Schouten, Proe.Arnst.Ae. xxvii (1924), p. 407; xxix (1926), p. 334; PalermoRend. I (1926), p. 142; J. A. Schouten and I?: van Dantzig, Proe.A~t. Ae. xxxiv (1932),p. 1398; xxxv (1932), p. 642 ; ZS.j. P.IxxVIll (1932), p. 639; !xxxi (1933), pp. 129,405

I cr. E. Cartan, Ann. Be. Norm. xl (1923), p. 325; xli (1923), p. I; Bull. Soc. M.France,Iii (1924), p. 205

I Alii lnst. Veneto,!xxxvi (1926-7), p. 459; Rend. Lincei, ix (1929), p. 530• cf. R. Weitzenbock, Wien. BeT. cxxix 2a (1920), pp. 683, 697; cxxx 2a (1921),

p. 15; H. Weyl, Gott. Nach. (1921), p. 99; F.Jiittner, Math. Ann.lxxxvii (1922), p. 270;E. Reiche~biicher, ZS.1: P. xiii (1923), p. 221.~ A. Einstein, Berlin S~tz. (1925)'J' 41~.!D.]. Strwk and O. Wiener, J. Math. Phys. VlI (1927), p. I; E. Wigner, ZS .. P. hn(1929), p. 592; A. Einstein, Berlin Sitz. (1928), pp. 217, 224; (1929), pp. 2,156;N. Wiener and M. S. Vallarta, Proe. N.A.S. xv (1929), pp. 353, 802; M. S. Vallarta,Proe. N.A.S. xv (1929), p. 784; H. Weyl, Bull. Am. M.S. xxxv (1929), p. 716; N. Rosenand M. S. Vallarta, Phys. Rev. xxxvi (1930), p. 110; A. Einstein, Berlin Sitz. ~1930~,p. 18; Math. Ann. cii (1930), p. 685; A. Einstein and W. Mayer, Berlin Sitz. 1930,p. 110; (1931), pp ..257,541; (1932), p. 130; W. Pauli, Ami. d. Phys.(5) .xviii 1933,p. 305; ~. Schrodmger, Proe. R./.A. xlIX (1943), pp. 43,.135; G. I? B,rkhoff, Proc.N.A.S. XXIX (1943), p. 231 ; xxx (1944), p. 324; A. Barajas, G. D. Blrkhoff, C. Graefand M. S. Vallarta, Phys. Rev.(2) !xvi (!~44), p. 138; H. Weyl, Am.. 7- Math. !xvi(1944), p. 591 ; E. SchrOdinger,Nature, elm (1944), p. 572; Proe.R./.A. xfix (A) (1944),pp. 225, 237, 275; A. Einstein and V. Bargmann, Annals qf M. xlv (1944), p. I; AEinstein Annals qf M. xlv (1944), p. 15; E. SchrOdinger,Proe. R.I.A. I (1945), pp. 143,223; Ii' (1946), p. 41; Ii (1947), .pp. Ig 163; .Ii (1948), p. 205; Ii! P948~, p. I;A. Einstein Appendix II in the th,rd editlon of hIS The Meamng qf Relatlvlly (Pnnceton,1949); or: the further development of this, cr. A. Papapetr01;1and E. SchrOdinger,Nature, c!xvii (1951), p. 40, and W. B. Bonnor, Proe. R.S.(A), CCIX (1951), p. 353; ccx(1952), p. 427

• Wien. Sitz., Abt. IIa, cxxxi (1922), p. 1 • ibid., p. 119192

GRAVITATION

the independence of metric which characterises the analysis of skewtensors (i.e. tensors for which an interchange of any two indicesproduces a reversal of sign): thus the divergence of a six-vector(Xpq

) (which is a skew tensor of rank 2) is

~g ~ o~q (Vg Xpq

)

and this is clearly unchanged if the metric

(lilY = L gpg dxP dxq",q

is replaced by(ds)1= AIL gpq dxP dxq

",q

where Ais any constant. While he recognised that it is impossibleto abolish metric from physics entirely, he aimed at expressing thelaws of nature as far as possible in terms of skew tensors, and ascer-taining in each case the precise point where they cease to be adequateand the introduction of a metric becomes inevitable. In the firstpaper he considered the Newtonian theory of gravitation: Poisson'sequation for the gravitational potential is

Olg, Olg, Olg,!li+~+!li= -47TP(IX! (lXI (IX.

or, if the metric is given by

(ds)1= L apqdx"dxq", q

it is

(p, q= 1,2, 3),

_1_L ~(vaLa"qog,)= -47Tp.va P oxP q oxq

But L apq~~ is the contravariant vector corresponding to the'I (IX

covariant vector og,/oxP: call it (U). Then Poisson's equation is

~ 0:"( va. L")= -47TPVa.i

In order to transform this into an equation depending on a skewtensor, Kottler made use of a theorem of tensor-calculus whichmay be stated thus: in space of three dimensions, where the metricis given by

(p, q= 1,2, 3),",q

193

AJ;;TItER AND J;;LJ;;CTRICITY

let ;)pqr have the value 1 or - 1 according as we obtain (pqr) from(123) by an even or an odd substitution: and when p, q, r are notall different, let ;)pqr be defined to be zero. Then va . ;)pqr is a skewtensor of rank 3.1 We shall denote it by Gpqr.

Now writeFpq= L GpqrU

so F pq is a covariant skew tensor of rank 2: its three numericallydistinct components are

Fu= va. L3, FSl= va. U, F18= va. U.

Thus Poisson's equation may be written

of 23 + of 31 + of 12 _oxl OXS ox3 - 1-'123

where I-'U3 = - 41TpG123.Now if S be any simple surface enclosing a volume V, then by

Green's theorem (which is independent of any system of metric)we have

J JJ(O~~3+ O~~l+ O~~I)dxldx2dx3

~JJ{F230(X2' X3) +F3l~(X3' Xl) +F120(Xh.!.J}dudVo(u, v) o(u, v) o(u, v)

,;

where (u, v) are any parameters fixing position on the surface. Thuswe obtain the equation

Kottler interpreted the left-hand side of this equation as the totalflux, through the surface S, of the skew tensor F pq: and he regardedthe equation as representing the relation between the occurrencesinside S and the field (expressed by a skew tensor) acting on S, ina form independent {)f metric. It is of course essentially a form of Gauss'stheorem.- Kottler regarded this equation as the starting-point of

1 It is generally called the skew fundamental tensor of Ricci and Levi-Civita.• On Gauss's theorem and the concept of mass in General Relativity, cf: E. T.

Whittaker, Proc. R.S.(A), cxlix (1935), p. 384; H. S. Ruse, Proc. Edin. Math. Soc,(2)iv (1935), p. 144; J. T. Cambridge, Phil. Mag.(7) xx (1935), p. 971; G. Temple,Proc. R.S.(A), cliv (1936), p. 354; J. L. Synge, Proc. R.S.(A), clvii (1936), p. 434;Proc. Edin. Math. Soc.(2) v (1937), p. 93; A. Lichnerowicz, Comptes Rendus, ccv (1937),p. 25

194

GRAVITATION

a fundamental non-metrical formulation of gravitational theory :metric is introduced, at a later stage, only when the concept of work,as distinguished from that of flux, is introduced: that is, when theforce actmg on a particle comes to be considered.

His second paper, on Maxwell's equations, presented an easierproblem, since the electric and magnetic forces together constitutea six-vector, which is the kind of tensor required for a non-metricalpresentation of the subject. Moreover there were available the resultsof papers written in 1910 by E. Cunningham 1 and H. Bateman,Swho, starting from the proposition that the fundamental equationsof electrodynamics are invariant with respect to Lorentz transforma-tions, i.e. to rotations in the four-dimensional world of space-time,had remarked that these electrodynamic equations are invariantwith respect to a much wider group, namely all the transformationsfor which the equation

(dx2+ (qy)2+ (dz) 2 -c2(dt)2=O

is invariant. By writing V - I . cdt = du, this last equation becomes

The transformations which leave this latter equation invariant arewhat are called the conformal group of transformations in space offour dimensions: they had been studied long before by Sophus Lie,who had shown that they are composed of reflections, translations,rotations, magnifications and inversions with respect to the hyper-spheres of the four-dimensional space. Types of motion of anelectromagnetic system may be derived from one another by suchtransformations: they are in general more complicated than thoseconsidered in the relativity theory of Poincare and Lorentz: for inthe latter case a fixed three-dimensional configuration is transformedinto one, every point of which has the same velocity of translation:but in the conformal case, under the simplest operation of the ~roup,a fixed system becomes one in which the whole is expandmg orcontracting in a certain way.3

Cunningham and Bateman's work was now developed byKottler,' H. Weyl,6J. A. Schouten and J. Haantjes,6 and particularlyby D. van Dantzig,7 into a complete theory of electromagnetism, independent

1 Proc. L.M.S.(2) viii, p. 77I ibid., p. 223; cf. also Bateman's book, Electrical and Optical Wave-motions (Camb.,

1915); H. Bateman, Phys. Rev. xii (1918), p. 459; Proc. L.M.S.(2) xxi (1920), p. 256;E. Bessel-Hagen, Math. Ann. !xxxiv (1921), p. 258; F. D. Murnaghan, Phys. Rev. xvii(1921), p. 73; G. Kowalewski, J.fiir Math. elvii (1927), p. 193

• It is characteristic of these transformations that a sphere which is expanding withthe velocity of light transforms into a sphere expanding with the same velocity.

• loc. cit. ' Raum, Zeit, Materie, 4th Aufl., p. 260 • Physictl I (1934), p. 8697 Proc. Camb. Phil. Soc. xxx (1934), p. 421; Proc. Arnst. Ac. xxxvii (1934), pp. 521,

526,644,825; xxxix (1936), pp. 126,785; Congo Int. des Math. Oslo (1936). II, p. 225 ;cf. alsoJ. A. Schouten, Tensor CalculusfoT Physicists, Oxford, 1951

195


of metrical geometry, and in fact needing the ideas neither of metric nor ojparallelism. It is characteristic of theories such as this that differentialrelations are generally replaced by integral relations: thus forthe statement that under certain circumstances ' the divergence ofthe flux of energy vanishes' is substituted the statement that 'theintegral of the flux over a closed surface vanishes,' which is 1 amathematical form of the physical statement that 'the algebraicsum of the energies of all the particles crossing through a closedsurface vanishes.'

1 cf. D. van Dantzig, Erkenntnis, vii (1938), p. 142

196

Chapter VIRADIATION AND ATOMS IN THE OLDER QUANTUM

THEORY

In 1916 Einstein 1 published a new and extremely simple proof ofPlanck's law of radiation, and at the same time obtained someimportant results regarding the emission and absorption of lightby molecules. The train of thought followed was more or lesssimilar to that adopted by Wien in the derivation 2 of his 1aw ofradiation, but Einstein now adapted it to the new situation createdby Bohr's theory of spectra.

Consider a molecule of a definite kind, disregarding its orienta-tion and translational motion: according to quantum theory, itcan take only a discrete set of states Zl, Z2, ... Zn, ... whose internalenergies may be denoted by El, E2, . . . En, . .. If molecules ofthis kind belong to a gas at temperature T, then the relative fre-quency Wn of the state Zn is given by the formula of Gibbs's canonicaldistribution as modified for discrete states,3 namely,4

Now suppose that the probability of a single molecule in thestate Zm passing in time dt spontaneously, i.e., without excitation byexternal agencies (as in the emission ofy rays by radio-active bodies)to the state of lower energy Zn with emission of radiant energyEm - En is

(A)

Suppose also that the probability of a molecule under the influenceof radiation of frequency v and energy-density p passing in time dtfrom the state of lower energy Zn to the state of higher energy Zm,by absorbing the radiant energy Em - En, is

B:p dt (B)

and suppose that the probability of a molecule under the influenceof this radiation-field passing in time dt from the state of higher

1 Mitt. d. phys. Ges. Zurich, No. 18 (1916); Phys. ZS. xviii (1917), p. 121; cr. A. S.Eddington, Phil. Mag. I (1925), p. 803

• cf. Vol. I, p. 382 3 cr. supra, p. 87• For simplicity we omit consideration of the statistical weight' of the state.

197


energy Zm to the state of lower energy Zn, with errusslOn of theradiant energy Em - En, is

(B')

This is called stimulated emission; its existence was recognised herefor the first time.

Now the exchanges of energy between radiation and moleculesmust not disturb the canonical distribution of states as given above.So in unit time, on the average, as many elementary processes oftype (B) must take place as of types (A) and (B') together. Wemust therefore have

e-Pt B:p=e-~i-' (B:p+A:).We assume that p increases to infinity with T, so this equation gives

Bm =Bnn m'

and the preceding equation may therefore be written

(A:/B~)p =e(Em-Efll//,:'l' _ 1

(1)

This is evidently Planck's law of radiation: in order that it maypass asymptotically into Rayleigh's law for long wave-lengths, andWien's law for short wave-lengths, we must have

Em - En=hvand

(2)

The two equations (1) and (2), first given in this paper of Einstein's,are fundamental in the theory of the exchanges qf energy between matter andradiation, and have been extensively used in the later developmentof quantum theory.l It may be remarked that as there is spon-taneous emission, but not spontaneous absorption, there is asym-metry as between past and future: but so far as transitions stimulatedby radiation are concerned, there is a symmetrical probability,Bn = Bm

.2m n

1 If the weights of the energy-levels are gm, gn, the relation (I) must be replacedby gn B:;'=gm B;:.. Relation (2) is unaffected.

• Einstein's formulae were extended to the case of non-sharp energy-levels byR. Becker, <S.j. P. xxvii (1924), p. 173, and to the laws of interaction between radiationand free electrons by A. Einstein and P. Ehrenfest, <S.j. P. xix (1923), p. 301.

Ig8

RADIATION AND ATOMS IN THE OLDER QUANTUM THEORY

One of the chief problems of quantum theory is to computecoefficients, such as these Einstein coefficients, from data regardingatoms and molecules. The relation (2) has been verified experi-mentally by a comparison of the strengths of absorption and emissionlines. The B's have been found from measures of the intensitiesof the components of multiplets in spectra by L. S. Ornstein andH. C. Burger.l

Another important result established in this paper related toexchanges of momentum between molecules and radiation. Einsteinshowed that when a molecule, in making a transition from the state Zn toZm, receives the energy Em-En, it receives also momentum of amount (Em -En)/Cin a definite direction,. and, moreover, that when a molecule, in the transitionfrom Zm to the state of lower energy Zn, emits radiant energy of amount(Em - En), it acquires momentum of amount (Em - En)/C in the opposite direction.Thus the processes of emission and absorption are directed processes:there seems to be no emission or absorption of spherical waves.

Einstein's theory of the coefficients of emission and absorptionenabled W. Bothe 2 to give an instructive fresh calculation of thenumbers of quanta hv in black-body radiation which are associatedas ' photo-molecules' in pairs 2hv, trios 3hv, etc. He considered acavity filled with black-body radiation at temperature T, in whichthere were a number of gas-molecules, each of which was either in thestate of energy Zl or the state of energy Z2' where Z2 - Zl =hv, theiraverage relative numbers being given by the law of canonical distri-bution at temperature T. He assumed that when a single quanthv of the radiation causes a stimulated emission, the emitted quantmoves with the same velocity and in the same direction as thequant that causes it, so they become a quant-pair 2hv; if the excitingquant itself already belongs to a pair, then there arises a trio 3hv,and so on. The absorption of a quant from a quant-pair leavesa single quant, and a spontaneous emission produces a single quant.Writing down the conditions that the average numbers of singlequants, of quant-pairs, etc. do not change in time, and usingEinstein's relations between the coefficients of spontaneous emission,stimulated emission and absorption, we obtain a set of equationsfrom which it follows that in unit volume and in the frequency-range dv the average number of single quants which are united intos-fold quant-molecules shy is

in agreement with the result previously obtained.3 The averagetotal energy per unit volume in the range dv is therefore

8rrhvs (00 _ 'hV)-s- L:e kT dv

c ,=1

1 -?,S.j. P. xxiv (1924), p. 41 • -?,S.j. P. xx (1923), p. 145 • cr. p. 103 supra199


or

in agreement with Planck's law of radiation.Einstein's theory of the coefficients of emission and absorption

also enabled theoretical physicists within a few years to create asatisfactory quantum theory of the scattering, refraction and dis-persion of light.! In 1921 Rudolf Ladenburg 2 (b. 1882), aformer pupil of Rontgen's at Munich, who afterwards settledin America at Princeton, introduced quantum concepts into thetheory.

It was necessary first for him to find a quantum-theoretic inter-pretation for the number that in classical theory represented thenumber of electrons bound to atoms by forces of restitution, for itwas these electrons which were responsible for light-scatterin~,refraction, and dispersion. Let us call them dispersion-electrons. ThIShe achieved by calculating the emitted and absorbed energy of aset of molecules in temperature-equilibrium with radiation, on thebasis of classical theory on the one hand and of quantum theoryon the other.

Suppose then that there are m dispersion-electrons per em.a,capable of oscillating freely with frequency VI. Now for theharmonic oscillator of frequency VI if the displacement x at theinstant tis Xo cos 21TVlt, the mean value of the energy !m {(dx/dt)2+41T2VI2X2} is U = 21T2mvI2xo2, and therefore 3 the average energyradiated per second is

or

and therefore the energy radiated in one second by the in dispersion-electrohs is

If the molecules are in equilibrium with radiation at temperature T,and if we regard the electrons as spatial oscillators with three degrees

1 The classical theory of the scattering of light by small particles had been givenby Lord Rayleigh in 1871 (Phil. Mag.(4) xli, pp. 107,274,447) on the basis of the elastic-solid theory of light, and in 1881 (Phil. Mag.(5) xii, p. 81) and 1899 (Nature, lx, p. 64 ;Phil. Mag.(5) xlvii, p. 375) on the basis of Maxwell's electromagnetic theory; cr. alsoJ. J. Thomson, CondlJl:tionof Electricity through Gases, 2nd edn. (1906), p. 321.

2 ZS. f. P. iv (1921), p. 451; cr. also R. Ladenburg and F. Reiche, Naturwiss, xi(1923), p. 584; R. Ladenburg, ZS.f P. xlviii (1928), p. 15

• cr. Vol. I, p. 326200


of freedom, 1 then 2 we have between U and the radiation-densityp the relation

Thus

The energy absorbed in equilibrium is of course equal to the energyradiated.

Now in quantum-theory the emission from a molecule is due totransitions from a state of higher energy (2) to a state of lowerenergy (1). The number of spontaneous transitions per second is,in the notation we have already used,

N2A~

where N2 is the number of molecules in the state (2). The numberof transitions from the lower to the higher state per second is

2N1B1P,

where N1 is the number of molecules in the state (I) : and thereforethe absorbed energy is

By Einstein's relations we have 3

Thus we have

Equating JQ to Jel' we have

1 For a three-dimensional oscillator in an isotropic radiation-field we obtain threetimes the value for a linear oscillator in the same field; on this point, cr. F. Reiche,Ann. d. Phys. Iviii (1919), p. 693.

• cf. Planck, p. 79 supra, Supposing for simplicity that the statistical weights of the energy-levels (1) and

(2) are equal.201


Thisformula expresses the constant ~n(which may be derived experi-mentally from measurements of emission, absorption, anomalousdispersion and magnetic rotation, and which in classical theory isinterpreted as the number of dispersion-electrons) in terms of thequantum-theoretic quantities N I (the number of molecules in the lowerof the two states) and A~ (the probability of the spontaneous transi-tions which give rise to radiation of frequency VI)' Thus frommeasurements of e.g. anomalous dispersion at different lines of aseries in a spectrum, we can make mferences regarding the prob-ability of the various transitions.

Ladenburg's result enables us to replace the dispersion-formulafound in Vol. I 1 by

where (i) denotes the lower level and (k) the upper level of energyin the transition corresponding to the frequency Vi.

Now consider the scattering of light by an atom. We supposethe wave-length of the light to be much greater than the dimensionsof the atom, so that at any instant the field of force is practicallyuniform over these dimensions. We suppose also that the atomcontains an electron of charge - e and mass m, which is bound to theatom so that when the electron is displaced a distance t parallel tothe x-axis from its equilibrium position, a force of restitution 47T!mvI2tis called into play. Then when the atom is irradiated by a light-wave whose electric field is

it can easily be shown that according to classical theory it radiatessecondary wavelets such as would be produced by an electric doubletof moment

e2Ee21Tivt

47T2m(v,2 - v2)"

These secondary spherical wavelets, which are coherent with theincident wave, constitute the scattered radiation.

In quantum theory we are concerned with scattering not by asingle bound electron but by an atom, and therefore in order totransfer this expression to quantum theory we must first multiplyit by ~/N1> the number of classical dispersion-electrons per atom(the atom being supposed to be in the state (I) when scattering).

1 At p. 401: there is a change of notation, the N, k, n of Vol. I being here repre-sented by "Ai, 21TViVm, and 21TV respectively :1 and we suppose now that in the classicalcase there are several kinds of dispersion-electrons with different natural periods. Theformula had been confirmed by the experiments of R. W. Wood, Phil. Mag.(6) viii (1904),p. 293 and P. V. Bevan, Proc. R.S. !xxxiv (1910), p. 209; !xxxv (1911), p. 58, on thedispersion of light in the vapours of the alkali metals.

202


Thus we obtain for the amplitude of the doublet-moment

m Ee2

N141TSm(v12-v2)"

Applying Ladenburg's result, this becomes

c3A~E321T4V12(V12-v2)"

We must sum over all higher states (2) : and thus we have the resultthat an atom in state (I), when irradiated by a light-wave whose electricfield is Ee27Tivt,emits secondary wavelets like an electric oscillator of frequencyv, whose electric moment has the amplitude

c3E A~P=-2---321T4 i Vi2(Vi2 - v2)'

where A~ denotes the probability of the atom performing spontaneously inunit time the transition to the state (I) of energy E 1from a higher state (i)of energy Ei, and where

Ei-E1·V,= --h-'

This result was further modified by Henrik Antony Kramers 1

(1894-1952), who, taking the above Ladenburg formula as correctwhen the atom was in the normal state, took into consideration thecase when the atom is excited, and proposed to deal with it bytaking the summation not only over the stationary states i whichhave higher energy-levels than the state (I), but also over the states jwhich have lower energy-levels than the state (I), so that the formulabecomes

where 2

This formula of course relates to a single atom, and a factor mustbe adjoined representing the number of atoms in this state.

Now a3 we have seen, in the classical theory of scattering, theI Nature, cxiii (1924), p. 673; cxiv (1924), p. 310, The formula as given by Kramers contained an additional factor 3; this was a

consequence of his assumption that the free oscillations are parallel to the incident field,whereas the above formula assumes all orientations of the atom to be equally probable.

203


atom behaves like an electric doublet, the amplitude of whosemoment is

e2E47T2m( V12 - v2)'

Comparing the above Ladenburg-Kramers formula with this,we see that according to quantum theory the atom behaves withrespect to the incident radiation as ifit contained a number of boundelectric charges constituting harmonic oscillators as in the classicaltheory, one of these oscillators corresponding to each possible transi-tion between the state of the atom and another stationary state.Thus we can describe the behaviour of an atom in dispersion bymeans of a doubly-infinite (i.e. depending on two quantum numbersm and n) set of virtual harmonic oscillators, the displacement in theoscillator (m, n) being represented by

q(m, n) =Q(m, n)e27T.>(m •.••)1

where v(m, n) denotes the frequency of this oscillator. The aggregateof these virtual harmonic oscillators was called by A. Lande 1 thevirtual orchestra. The virtual orchestra is thus a classical substitution-formalism for the radiation, and so indirectly becomes a representativeof the quantum radiator itself.

In place of the classical e2/m we have the value c3A}j87T2Vi3 forone of the 'absorption oscillators,' Le. those corresponding totransitions between the state (1) and higher states, but we havethe value -C3AU87T2Vl for one of the 'emission oscillators,' i.e. thosecorresponding to transitions between the state (1) and lower states:so that there is a kind of ' negative dispersion' arising from theemission oscillators, which may be regarded as analogous to the'negative absorption' represented by Einstein's coefficient B~.Another way of putting the matter is to say that a quantum-oscillatorwhich is in the higher state, when irradiated by a light-wave whichis not markedly absorbed by it, emits a spherical wave, which differsfrom that emitted in the lower state only by being displaced 1800

in phase.Almost immediately after the appearance in May 1924 of

Kramers's paper, Max Born 2 gave a general method, to which hegave the name quantum mechanics,3 for translating the classical theoryof the perturbations of a vibrating system into the correspondingquantum formulae. In particular he studied the problem of anoscillator exposed to an external field of radiation: his method wasbased on the correspondence-principle, the frequency belonging toa transition between states characterised by the quantum numbers

1 Nalurwiss. xiv (1926), p. 455• ZS.f. P. xxvi (1924), p. 379, communicated 13June 1924• This was the first occurrence of the term in the literature of theoretical physics.

204

RADIATION AND ATOMS IN THE OLDER QUANTUM THEORYnand n' corresponding to the overtone In- n'lv of the classical solution:but the results he obtained were afterwards shown to be correct inthe light of the later form of quantum theory.

A theory of scattering is also essentially a theory of refraction.For when (in classical theory) light is scattered by atoms, thescattered light has the same frequency as the incident light andinterferes with it: the total effect produced is the same as if theprimary wave alone existed, but was propagated with a differentvelocity: and this change of velocity is the essential feature inrefraction. In the Ladenburg-Kramers formula, the terms

AI-2--1-

j v/(v/-v2)

affect the refraction in the opposite sense to the other terms. Thiswas verified in 1928 by R. Ladenburg 1 and by H. Kopfermannand R. Ladenburg,2 who studied the refractive index, in the neigh-bourhood of a certain spectral line of neon. When a current waspassed through the gas, many of the atoms were thereby raised to anexcited level, and it was found that the refractive index dropped.the refraction due to the ordinary fall in energy-level being partlycounterbalanced by that due to the rise.

Even before the appearance of Kramers's paper, new possibilitiesin regard to scattering had been indicated by A. Smeka1.3 It mayhappen that the emission of scattered radiation is associated witha quantum transition in the scattering structure, in which casethere will be a difference of frequency between the scattered andprimary rays of the same order of magnitude as the spectral fre-quencies of the scattering atomic system; if v denotes the frequencyof the primary radiation, and hVk or hvz denotes the change of energyof the atom in the transition, according as this change happens in thepositive or negative direction, then there may be scattered radiation(of low intensity) of frequency v + Vk or v - v,, At the time ofSmekal's paper, his conjecture of anomalous scattering, as he called it,had not been verified experimentally: but in 1928 Sir ChandrasakaraV. Raman 4 showed that light scattered in water and other trans-parent substances contains radiations of frequency quite differentfrom that of the incident light, and it was at once seen that thiswas the effect predicted by Smekal: almost simultaneouslyG. Landsberg and L. Mandelstam 5 found the same effect experi-mentally, working with quartz.

In general, radiation of a definite frequency v generates scattered

1 ZS.f. P. xlviii (1928), p. 15I ibid., pp. 26, 51 ; ZS.f. phys. Chern.cxxxix (1928), p. 375• Naturwiss. xi (1923), .p. 873 . . .• Ind. ]auro. of Phys. 11 (1928), p. 387; C. V. Raman and K. S. KrIShnan, ibid.,

p. 399• Naturwiss. xvi (1928), p. 557; ZS.f. P. I (1928), p. 769

205


radiation of several modified frequencies, all of the form v ± V',where v' is either an infra-red frequency in the absorption spectrumof the scattering material, or a difference of such frequencies.l

In the year following the publication of Smekal's paper,H. A. Kramers and W. Heisenberg 2 made an exhaustive study of theradiation which an atom emits when irradiated by incident light.Their method, as in Born's paper published a few months earlier,was first to study by classical theory the perturbation by incidentradiation of an atom regarded as a multiply-periodic dynamicalsystem, and then to employ the correspondence-principle in orderto translate the results into their quantum-theoretIc form.

According to classical theory, under the irradiation the systememits a scattered radiation, whose intensity is proportional to theintensity of the incident light: when it is analysed into harmoniccomponents, a component involves the sum or difference of thefrequency of the incident light and frequencies occurring in theundisturbed motion (as in the Smekal-Raman effect). The quantum-theoretic formulae must satisfy the condition that in the region ofgreat quantum numbers, where successive stationary states differcomparatively little from each other, the quantum scattering musttend asymptotically to coincide with the classical scattering. Thiscondition is satisfied by interpreting certain derivatives whichoccur in the classical formulae as differences of two quantities:in this way, Kramers and Heisenberg obtained equations whichinvolved only the frequencies characteristic of transitions, while allsymbols relating to the mathematical theory of multiply-periodicdynamical systems disappeared.

It was shown that when irradiated with monochromatic light,an atom emits not only spherical waves of the same frequency asthe incident light, and coherent with it, but also systems of non-coherent spherical waves, whose frequencies are combinations of thatfrequency with other frequencies, which correspond to thinkabletransitions to other stationary states. These additional systems ofspherical waves occur as scattered light, but they do not contributeto the phenomena of dispersion and absorption of the incidentradiation.

An interesting comment on the Kramers-Heisenberg formula wasmade by P.Jordan,3 who remarked that it remained valid even whenthe incident radiation consisted of very long electromagnetic waves,which in the limit of zero frequency tend to a field-strength constantin time: and that in this limiting case, the formula actually yieldsthe changes of frequency in spectral lines which are observed in theStark effect. He drew the moral that discontinuous jumps must

1 A review of literature on the Smekal-Raman effect to Feb. 1931 was given byK. W. F. Kohlrausch, Phys. ZS. xxxii (1931), p. 385. On the Smekal-Raman effect inmoleculesand crystals, cf. E. Fermi, Mem. Ace. Ital. Fis. Nr. 3 (1932), p. 1.

t ZS.j. P. xxxi (1925), p. 681I P. Jordan, Anschauliche Quantentheorie (Berlin, 1936),p. 85

206

RADIATION AND ATOMS IN THE OLDER QUANTUM THEORYnot be regarded as the essential characteristic of quantum theory :for phenomena in which they occur can be connected continuouslyby mtermediate types with phenomena in which there is no dis-continuity.

About this time much attention was given to scattering of adifferent type. In 1912 C. A. Sadler and P. Mesham/ working inL. R. Wilberforce's laboratory at Liverpool University, showed thatwhen a homogeneous beam of X-rays is scattered by a substanceof low atomic weight, the scattered rays are of a softer type (i.e. oflonger wave-length).2 Moreover, in the case of the y-rays producedby a radium salt, it was shown by J. A. Gray 3 that the secondaryor scattered y-rays were less penetrating than the primary, that this, softening' was due to a real change in the character of the primaryrays when the secondary rays were formed, and that it increasedwith the angle between the primary and secondary rays (generallycalled the angle of scattering).' An explanation of these phenomenafavoured by physicists at the time was that the primary beam wasnot truly homegeneous, and that its softer components were morestrongly scattered than the harder ones: this, however, was explicitlydenied by Sadler and Mesham. In 1917 C. G. Barkla 5 propoundedthe hypothesis, that there existed a new series of characteristicradiations which were of shorter wave-length than the K- and L-series which he had discovered previously, and which he named the]-series: and that the softening observed in the scattered radiationfrom light elements was due to the admixture of this J-radiationwith radiation of the same hardness as the primary.6 This explana-tion, however, lost credit when it was found 1 that the J-serieshad no critical absorption limit similar to the absorption limit ofthe K- and L-series, and that it showed under spectroscopic observa-tion 8 no spectral lines such as might have been expected. Whenthe J -series explanation was dismissed,g there still seemed to be a

1 Phil. Mag. xxiv (1912), p. 138; cr. J. Laub, Ann. d. Phys. xlvi (1915), p. 785, andJ. A. Gray, J. Frank. Inst. cxc (1920), p. 633

• It will be remembered that the secondary X-rays in general consist of a mixtureof the characteristic rays discovered by Barkla (K, L, etc.) and of truly scattered rays.The wave-length of the former depends solely on the chemical nature of the scatteringsubstance, but the wave-length of the latter is the same (or nearly the same) as thatof the primary rays. For the heavier elements the characteristic rays predominate, butfor elements of low atomic weight at that time only the truly scattered rays were normallyobservable. • Phil. Mag. xxvi (1913), p. 611

• These results were confirmed as regards y-rays by D. C. H. Florance (working atManchester under Rutherford) in Phil. Mag. xxvii (1914), p. 225, and Arthur H. Compton,Phil. Mag. xli (1921), p. 749.

• Phil. Trans. ccxvii (1917), p. 315• cr. C. G. Barkla and Margaret P. White, Phil. Mag. xxxiv (1917), p. 270. Theyfound an abnormally great mass absorption coefficient for aluminium at 0·37A, andregarded this as additional proof of the reality of J-radiation.

7 cf. F. K. Richtmyer and Kerr Grant, Phys. Rev. xv (1920), p. 547; C. W. Hewlett,Phys. Rev. xvii (1921), p. 284

• W. Duane and T. Shimizu, Phys. Rev. xiii (1919), p. 289; xiv (1919), p. 389• Further proof of this, depending on the polarisation of the scattered rays, was

given by A. H. Compton and C. F. Hagenow, J. Opt. Soc. Amer. viii (1924), p. 487.207


possibility of explaining the increase of wave-length in the scatteredlight without departing from classical theory: for 1 radiation-pressure (which had not been taken into account in the previousclassical theories of scattering) might set the electrons in motion inthe direction of the primary radiation, and the wave-length of thescattered light would then be increased by the Doppler effect. Onperforming the calculations, it was found that the Increase in wave-length should follow the law actually verified by experiment, namelythat it would be proportional to sin2 to where 0 denotes the angleof scattering: but its magnitude as calculated was not in agreementwith observation.

There seemed to be a likelihood therefore that some new typeof physical explanation was required. In October 1922 a Bulletin 2

of the National Research Council of the U.S.A. appeared, writtenby A. H. Compton and containing a full discussion of secondaryradiations produced by X-rays: in this the author suggested thatwhen an X-ray quantum is scattered, it spends all its energy andmomentum upon some particular electron. This electron in turnre-radiates the whole quantum (degraded in frequency) in somedefinite direction; the change in momentum of the X-ray quantumdue to the change in its frequency and direction of propagation isassociated with a recoil of the scattering electron. The ordinaryconservation laws of energy and momentum are obeyed,S so thatthe energy of the recoil electron accounts for the difference betweenthe energy of the incident photon and the energy of the scatteredphoton.

Compton's theory was communicated to a meeting of theAmerican Physical Society on 1-2 December 1922,4 and publishedmore fully in May of the following year.5 On examining the scatteredrays e from light elements spectroscopically, he found that theirspectra showed lines corresponding to those in the primary rays,and also other lines corresponding to these but displaced slightlytowards the longer wave-lengths; and that the difference in wave-length increased rapidly at large angles of scattering. It is thesedisplaced lines which represent the Compton effect.

In the Compton effect the electron is effectively free, i.e. it is sofeebly bound to the nucleus of the atom that the binding-energycan be neglected in comparison with the energy hv of the incidentquant; this condition is realised in the scattering of hard X-raysby elements oflow atomic number. When the frequency is decreased,or the atomic number is increased, the binding forces can no longerbe neglected in comparison with the energy of the incident photon,

1 cr. O. Halpern, ZS.j. P. xxx (1924), p. 153; G. Wentzel, Phys. ZS. xxvi (1925),p.436 • Vol. IV, No. 20

• It may be noted that while the photo-electric effect showsthat the energy of radiationis transferred in quanta, the Compton effect shows that momentum also is transferred inquanta. • Phys. Rev. xxi (1923), p. 207

• Phys. Rev. xxi (1923), p. 483 • Phys. Rev. xxii (Nov. 1923), p. 409208


and the phenomenon passes over into the photo-electric effect (inthe case when the energy of the incident photon is transferred to theelectron) or ordinary scattering (in the case when the photon retainsits energy and changes only its direction).

The Compton effect may be discussed in an elementary way bylight-quantum methods as follows: 1

Let the incident light-quantum, of frequency v, propagated inthe positive direction of the axis of x, encounter at a the electron,which recoils in a direction making an angle 8 with the axis Ox,with velocity v, while the light-quantum, degraded to frequency v',is scattered in a direction making an angle - ep with Ox. Then(using the relativist formulae for energy and momentum), theequation of conservation of energy is

hv = hv' + mc2 { ( I-~)-t - I},

while the equations of conservation of momentum are

hv hv' A. mv- = - cos 'f' + ( 2) cos 8,c c V l-~

cl

hv' . A. mv . 8- SIn 'f' = ( 2) sm .c V l-~

cl

From these equations we have to calculate v' in terms of v and ep,which are supposed given. We readily find

, vv =-------.

1+ h~ (l - cos ep)me

The increment of wave-length LlA or clv' - clv is

LlA = 2h sin2 P.me 2'

a formula which was definitely confirmed by observation.

1 In addition to the papers quoted above and below, cr. the following papers of1923and 1924: P. Debye, Phys. ZS. xxiv (April 1923),p. 161,who discovered the theoryindependently. A. H. Compton, Phil. Mag. xlvi (1923), p. 897; J. FranJc. Inst. cxcviri(1924), p 57. G. E. M. Jauncey, Phys. Rev. xxii (1923), P 233. P. A. Ross, Proc. N.A.S.ix (1923), p. 24.6; x (1924), p. 304; Phys. Rev. xxii (1923), p. ;;24. M. de Broglie, Pro:.:Phys. Soc. XXXVI (1924), p. 423. D. Skobelzyn, ZS. f. P. XXIV (1924), p. 393; XXVlll(1924), p. 278. A. H. Compton and A. W. Simon, Phys. Rev. xxv (1925), p. 306

209


Eliminating v' and v, we have

( hv) ep1+ mc2 tan 2" = cot 8.

The kinetic energy of the recoiling electron is

hv 2hv cos2 8mc2

(1 +~)2 _ h2v

2 cos28'mc2 m2c4

The quantity (h/mc), which has the dimensions of a length, is calledthe Compton wfwe-length. Its value is 0·0242 x 10-8 em. Since themass of a quantum is hv/c2 or h/AC, it is seen at once that a quantumof radiation, whose wave-length is the Compton wave-length, has a massequal to the mass of the electron.

The recoil electrons of the Compton effect were studied byC. T. R. Wilson,l using his cloud-expansion method. He found thatX-radiation of wave-length less than about o·sA in air producedtwo classes of f3-ray tracks, namely (a), those of electrons ejectedWith initial kinetic energy comparable to a quantum of the incidentradiation: these were photo-electrons: and (b), tracks of veryshort range, which were the recoil electrons. A. H. Compton andJ. C. Hubbard,2 discussing Wilson's results, showed that the motionof the recoil electrons corresponds precisely to Compton's theory.

We have seen that in the spectrum of scattered X-rays there are,in addition to the lines corresponding to the Compton effect, linescorresponding exactly to the primary X-rays. These unshiftedlines (which are the only lines to appear when the primary rays arethose of visible light) may be explained by supposing that someelectrons are closely attached to the nucleus and must scatter whilenearly at rest. The theory of this state of affairs was investigatedby Compton,3 who also explained certain results which had beenobtained by Duane and his collaborators,4 and which appeared tobe inconsistent with the original Compton theory.

A more accurate treatment of the Compton effect, making useof later developments in general theory, was given in 1929 by

1 Proc. R.S.(A), civ (1923), p. 1 ; cr. also W. Bothe or Charlottenburg, ZS. f. P. xvi(1923), p. 319; xx (1923), p. 237

• Phys. Rev. xxiii (1924), p. 439• Phys. Rev. xxiv (1924), p. 168; Nature, cxiv (1924), p. 627• G. L. Clark and W. Duane, Proc. N.A.S. ix (1923), pp. 413, 419; x (1924), pp. 41,

92; xi (1925), p. 173. G. L. Clark, W. W. Stifter and W. Duane, Phys. Rev. xxiii (1924),p.551. A. H. Armstrong, W. Duane and W. W. Stifter, Proc. N.A.S. x (1924), p. 374.S. K. Allison, G. L. Clark and W. Duane, Proc. N.A.S. x (1924), p. 379. J. A. Becker,Proc. N.A.S. x (1924), p. 342. S. K. Allison and W. Duane, Proc. N.A.S. xi (1925), p. 25.cr. P. A. Ross and D. L. Webster, Proc. N.A.S. xi (1925), pp. 56, 61. A. H. ComptonandJ. A. Bearden, Proc. N.A.S. xi (1925), p. 117

210

RADIATION AND ATOMS IN THE OLDER QUANTUM THEORYO. Klein and Y. Nishina.1 The formulae they obtained have beenfound experimentally to be accurate even with the hardest type ofradiation. 2

The Compton effect raised in an acute form the controversyregarding the reality of light-darts as contrasted with sphericalwaves of light: for in Compton's explanation, the incident anddiffracted X-ray quanta were supposed to have definite directionsof propagation. Opinion among theoretical physicists was divided :, In a recent letter to me,' wrote A. H. Compton in 19243 ' Sommer-feld has expressed the opinion that this discovery of the change ofwave-length of radiation, due to scattering, sounds the death knellof the wave theory. On the other hand, the truth of the sphericalwave hypothesis indicated by interference experiments has ledDarwin and Bohr, in conversation with me, to choose rather theabandonment of the principles of conservation of energy andmomentum.' The latter policy was embodied in a hypothesis putforward in 1924 by N. Bohr, H. A. Kramers and J. C. Slater,4 inwhich it was accepted, that in atomic processes,energy and momentumare only statistically conserved.

They abandoned the principle, common to all previous physicaltheories, that an atom which is emitting or absorbing radiation mustbe losing or gaining energy: in its place they introduced the notionof virtual radiation, which is propagated in spreading waves as inthe electromagnetic theory of light, but which does not transmitenergy or momentum: it has indeed no connection with physicalreality except the capacity to generate in atoms a probability for theoccurrence of transitions: and transitions of the atoms are the onlyphenomena actually observable. A transition of an atom from onestate to another is accompanied by changes of energy and momentum,but is not accompanied by radiation: thus the part played by theatom in its relatIOns with radiation reduces to interaction with thefield of virtual radiation, while the atom remains in a stationarystate. An atom in a stationary state is continually emitting virtualradiation, compounded of all the frequencies corresponding to possibletransitions between this state and lower states: this radiation isemitted both spontaneously and by stimulation (in accordance withEinstein's principles of 1917). While in this state, the atom is alsocapable of absorbing radiation corresponding to transitions to statesof higher energy. The absorption is performed by virtual oscillatorssituated in the atoms, the frequencies of these oscillators correspondingto the energy-differences between the state of the atom and all

1 ZS./. P. Iii (1929), p. 853• cf. H. C. Trueblood and D. H. Loughridge, Phys. &v. liv (1938), p. 545; Z. Bay

and Z. Szepesi, ZS./. P. cxii (1939), p. 20• J. Frank. Ins!. cxcviii (1924), p. 57• J. C. Slater, Nature, CX1ii(1924), p. 307. N. Bohr, H. A. Kramers, andJ. C. Slater,

Phil. Mag. xlvii (1924), p. 785; ZS./. P. xxiv (1924), p. 69. J. H. Van Vleck, Phys.Rev. xxiv (1924), p. 330. R. Becker, ZS./. P. xxvii (1924), p. 173. J. C. Slater, Phys.Rev. xxv (1925), p. 395

2 I I


higher states. When a virtual oscillator is absorbing virtual radia-tion, the atom to which it belongs has a certain probability of makinga transition to the higher state corresponding to the frequencyof this virtual oscillator. A transition marks the change from the'continuous radiation appropriate to the old state, to the continuousradiation appropriate to the new state: simultaneously with thetransition, some virtual oscillators disappear and others come intobeing: the transition has no other influence on the radiation. Theoccurrence of a transition in a given atom depends on the initialstate of this atom and on the states of the atoms with which it isin communication through the field of virtual radiation, but not ontransition processes in the latter atoms: so there is no direct con-nection between the transition of one atom from a higher to a lowerstate, and the transition of another atom from a lower to a higherstate: the principles of energy and momentum are retained In astatistical sense, though not in individual interactions of atoms withradiation. The atoms scatter radiation which is incident on them,acting as secondary sources of virtual radiation which interfereswith the incident radiation. In any transition, say between states(p) and (q), the energy of the atom changes by hllpq and its momentumby hllpq/C. If the transition is a spontaneous one, the direction ofthis momentum is random: but if it is stimulated, Le. inducedby the surrounding virtual radiation, the direction of themomentum is the same as that of the wave propagation in thisvirtual field.

The Bohr-Kramers-Slater theory was wrecked when it was shownto be inconsistent with the results of more refined experiments relatingto the Compton effect. One of these was performed by W. Botheand Hans Geiger.l According to Compton's theory, a recoil electronis emitted simultaneously with the scattering of every quantum; whileaccording to the Bohr-Kramers-Slater theory, the connection wasmuch less close, the recoil electrons being emitted only occasionally,while the scattering of virtual radiation is continuous. In Botheand Geiger's experiment two different Geiger counters countedrespectively the recoil electrons, and the photo-electrons {>roducedby the scattered photons. A great many coincidences in time wereobserved, so many that the probability of their occurrence on theBohr-Kramers-Slater theory was only 1/400,000.. It was thereforeconcluded that the conservation of energy and momentum holds inindividual encounters, and the Bohr-Kramers-Slater theory couldnot be true.

Another experiment was performed by A. H. Compton andA. W. Simon,2 who remarked that if in a Wilson cloud-experimentthe quantum of scattered X-rays produces a photo-electron in thechamber, then a line drawn from the beginning of the recoil trackto the beginning of the track of the photo-electron gives the direction

1 ZS.f. P. xxvi (1924), p. 44; xxxii (1925), p. 639• Phys. Rev. xxvi (1925), p. 289

212


of the quantum after scattering. It was therefore possible to testthe truth of the equation

(1 + hv) tan P.=cot 8mc2 2

which connects the directions of the scattered quantum and therecoil electron. If the energy of the scattered X-rays were propagatedin spreading waves of the classical type, there would be no correlationwhatever between the directions in which the recoil electrons proceedand the directions of the points at which the photo-electrons areejected by the scattered photons. The results of Compton andSimon's experiment showed that the scattered photons proceed indefinite directions and that the above equation connecting c/> and 8is true.l And this result, like that of Bothe and Geiger, is fatal tothe Bohr-Kramers-Slater hypothesis.

The discovery of the Compton effect opened a new prospect ofsolving a problem which had for some years baffled theoreticalphysicists. The thermal equilibrium between radiation and electronsIII a reflecting enclosure had been investigated by H. A. Lorentz 2

and A. D. Fokker 3 on the basis of classical electrodynamics: andthey had shown that Planck's law of spectral distribution of radiationcould not persist in such a reflecting enclosure, if an electron werepresent: and, moreover, that if the Planck distribution were artifi-cially maintained, the electron could not maintain the amount !kT ofmean translational kinetic energy required by the statistical theoryof heat: so the classical theory failed to account for the interactionof pure-temperature radiation with free electrons. W. Pauli now,4basing his investigation on the work of Compton and Debye, attackedthe problem of finding a quantum-theoretic mechanism for theinteraction of radiation with free electrons, which should satisfy thethermodynamic requirement that electrons with the Maxwelliandistribution of velocities can be in equilibrium with radiation whosespectral distribution is determined by Planck's radiation-formula.He found that the probability of a Compton interaction betweena photon hv and an electron could be represented as the sum of twoexpressions, one of which was proportional to the radiation-densityof the primary frequency v, while the other was proportional to theproduct of this radiation-density and the radiation-density of thefrequency V' which arises through the Compton process. The latterteFm was puzzling from the philosophic point of view, since it seemedto imply that the probability of something happening depended onsomething that had not yet happened. However, it was shown byW. Bothe 5 that Pauli's second term was a mistake, arising from

1 ef. R. S. Shankland, Phys. Rev. Iii (1937), p. 414, At the Solvay conference in Brussels in 1911• Diss., Leiden, 1913; Arch. Nlerl, iv (1918), p. 379• ZS.f. P. xviii (1923), p. 272 • ZS.f. P. xxiii (1924), p. 214

213


Pauli's assumption that the photons scattered all have the energyhv: whereas, as we have seen earlier,l in pure-temperature radiationthere are also definite proportions of photons having the energies2hv, 3hv, etc.: if it is assumed that each of these is scattered as awhole, in exactly the same way as the photons of energy hv, thenPauli's second term falls out, and the theory becomes much simpler.2

In the latter half of 1923 Louis de Broglie introduced a newconception which proved to be of great importance in quantumtheory. The analogy of Fermat's Principle in Optics with thePrinciple of Least Action in Dynamics suggested to him the desira-bility of studying more profoundly the parallelism between corpus-cular dynamics and the propagation of waves, and attaching to .ita physical meaning. He developed this idea first in a series of notesin the Comptes Rendus,3 then in a doctorate thesis sustained in 1924,4and in other papers.5

In de Broglie's theory, with the motion of any electron or materialparticle there is associated a system of plane waves, such that thevelocity of the electron is equal to the group-velocity 6 of the waves.Let m be the mass and v the velocity of the particle. It is assumedthat the frequency v of the waves is given by Planck's relation

E=hv,when

is the kinetic energy of the particle, so

hv=-~' (1)v(1-~~)

Since v is equal to the group-velocity of the waves, we have

dv (2)v= d(lJ"A)

where "A is the wave-length of the waves, so that "A = V Jv where V isthe phase-velocity of the waves.

1 cr. p. 103 .. . . .. .• A simple treatment of the eqUilibnum between a Maxwellian dlstnbutlOn of atoms,

and radiation obeying Planck's law, is given by P.Jordan, ZS.f. P. xxx (1924), p. 297.• Comptes Rendus, c1xxvii (10 Sept. 1923), p. 507; ibid. (24 Sept. 1923), p. 548;

ibid. (8 Oct. 1923), p. 630; c1xxix(7 July 1924), p. 39; ibid. (13 Oct. 1924), p. 676 ;ibid. (17 Nov. 1924),p. 1039

• These, Paris, Edit. Musson et Cie., 1924 ...6 Phil. Mag. xlvii (Feb. 1924), p. 445; Annales de phys.(IO) III (1925), p. 22.• cr. Vol. I, p. 253, note 4. For the purpose of calculating group-velocity, v is regarded

as a function of>' with c and m/h as fixed constants.214

RADIATION AND ATOMS IN THE OLDER QUANTUM THEORYFrom (1) and (2) we have

(1) mc2 ( v2)-l m ( v2)-! 1d ~ = hv d 1 - C2 = It 1 - C2 dv = l/Pwhere

is the momentum of the particle. Integrating, we have I/A=P/h, orA = hlp. This equation gives the wave-length of the de Broglie wave associatedwith a particle of momentum p.

The phase-velocity V of the de Broglie wave is

h E EV=AV=- . -=-P h Por

an equation which gives the phase-velocity of the de Broglie wave.1Now a wave-motion of frequency E/27Tn and of wave-length

27Tfi/P, where p has the components (px, Pv, pz), is represented bya wave-function

mc'~hvo

The particle now rests with respect to a Galilean system K', in which we imaginean oscillation of frequency Vo everywhere synchronous. Relative to a system K', withrespect to which K' with the mass m is moved with velocity v along the positive X-axis,th~re exists a wave-like process of the kind

1 The following derivation of de Broglie's result was given by Einstein, Berlin Sitz.(1925), p. 3.

A material particle of mass m is first correlated to a frequency Vo conformably to theequation

The frequency v and phase-velocity V of this process are thus given by

"V~-.v

The relation(phase velocity) x (group velocity)~,I

was shown by R. W. Ditchburn, Revue optique xxvii (1948), p. 4, andJ. L. Synge, Rev. Opt.xxxi (J 952), p. 121, to be a necessary consequence of relativity theory.

215


This is the analytical expression of the de Broglie wave associated with aparticle whose kinetic energy is E and whose momentum is (pr, Pv, p.).

But a wave of ordinary light of frequency v in the direction(t, m, n) is represented by a wave-function

!Ji=exp. [21Tiv{t-G) (lx+my+nz)}]or

!Ji = expo {~(Et - p.x - Pvy - p,z) }

when E denotes the energy hv and (pr, Pv, p.) the momentum hv/cof the corresponding photon. Comparing this with the aboveexpression for the de Broglie wave, we see that an ordinary wave oflight is simply the de Broglie wave belonging to the associated photon.It follows that if a de Broglie wave is regarded as a quantum effect,then the interference and diffraction of light must be regarded as essentiallyquantum effects. It is, in fact, a mistake to speak of the wave-theoryof light as the' classical' theory: that it is usually so called is dueto the historical accident that the wave-theory of light happened tobe discovered before the photon theory, which is the corpusculartheory. When interference is treated by the corpuscular theory(Duane's method, p. 142), then its quantum character is shown bythe fact that quantum jumps of momentum make their appearance.Moreover, the interference and diffraction of light are evidentlyof the same nature as the interference and diffraction of electronbeams and of molecular rays, and the latter phenomena areundoubtedly quantum effects.

The principle of Fermat applied to the wave may be shown tobe identical with the principle of Least Action applied to the particle;in fact, 'Bfds/A = 0 is equivalent to 'Bfpds= 0 if P is a constant multipleof l/A.

De Broglie now showed that his theory provided a very simpleinterpretation of Bohr's quantum condition for stationary statesof the hydrogen atom. That condition was, that the angular momen-tum of the atom should be a whole-number multiple of Ii or h/21T, saynh/21T. But if r denotes the radius of the orbit and p the linearmomentum of the electron, the angular momentum is rp: so thecondition becomes

or21Tr= nt..

where A is the wave-length of the de Broglie wave associated withthe electron. This equation, however, means simply that the circum-ference of the orbit of the electron must be a whole-number multiple of thewave-length of the de Broglie wave.

216

RADIATION AND ATOMS IN THE OLDER Q.UANTUM THEORY

More generally, the Wilson-Sommerfeld quantum condition isthat the Action must be a multiple of h. But the Action is fpdswhere p denotes the momentum and ds the element of the path:so hfds/"A must be a multiple of h, where "Ais the de Broglie wave-length: or, fds/"A must be a whole number. We can express thisby saying that the de Broglie wave must return to the same phase when theelectron completes one revolution of its orbit.

The connection between the Wilson-Sommerfeld condition andthe wave-theory can be seen also in the case of the diffraction oflight or electron beams or corpuscular rays by an infinite reflectingplane grating. The solution of this problem by the corpusculartheory (Duane's method) depends on the Wilson-Sommerfeldcondition, which yields (cr. p. 143)

(pd/h) (sin i-sin r) = a whole number,

where p now denotes the total momentum of the quantum oflight. The solution by the wave-theory of Young and Fresnel,on the other hand, depends on the principle that (1/27T) timesthe difference in phase between the rays reflected from adjacentspacing-intervals must be a whole number, and this at oncegives

(d/"A) (sin i-sin r) = a whole number.

These two equations are identical if (P/h) = (l/"A),which is assuredby de Broglie's relation.

In July 1925 a research student named Walter Elsasser, workingunder James Franck at Gottingen, made an important contribution 1

to the theory. Franck had been told by his colleague Max Bornof an investigation made in America by Clinton J. Davisson andC. H. Kunsman,2 who had studied the angular distribution ofelectrons reflected at a platinum plate, and had found at certainangles strong maxima of the intensity of the electronic beam. Born,who knew of de Broglie's theory, mentioned it in this connection,and Franck proposed to Elsasser that he should examine the questionwhether Davisson's maxima could be explained in some way by deBroglie's waves. Elsasser showed that they could in fact be inter-preted as an effect due to interference of the waves. There werestrong maxima, which with increasing electron-velocity approxi-mated to the positions of the maxima which would be observed iflight of the wave-length given by de Broglie's law, namely, "A= h/mvwere diffracted at an optical plane grating, the constants of thegratin~ being those of platinum crystals. This was the first con-firmatIOn of de Broglie's theory which was based on c0mparison

, NatUTWiss, xiii (1925), p. 711217

• Phys. Rev. xxii (1923), p. 242


with experiment. Elsasser remarked further that de Broglie's theoryprovided solutions for some other puzzles of current physics. Hediscussed an effect discovered by C. Ramsauer 1 in Germany, andindependently but later by J. S. Townsend and V. A. Bailey2 atOxford, and by R. N. Chaudhuri 3 working in O. W. Richardson'slaboratory in London, namely, that the mean free path of an electronin the inert gases becomes exceedingly long when the velocity of theelectron is reduced: thus when an electron moving with a velocityof about 108 cm./sec. collides with a molecule of a non-inert gas,in general it loses more than one per cent of its energy, but when itcollides with a molecule of argon, it loses only about one ten-thousandth part of its energy: and the mean free path of such anelectron in argon is about ten times as long as that calculated fromthe kinetic theory. Elsasser explained this by showing that when slowelectrons are scattered by atoms of the inert gases, the effect followsthe same laws as the classical scattering of radiation, of the associatedde Broglie wave-length, by small spheres whose radius is the same asthat of the atom.

Elsasser further suggested that electrons reflected at a singlelarge crystal of some substance might show the diffraction-effect of thede Broglie waves decisively.4 The phenomenon thus predicted wasfound experimentally in 1927 by Clinton J. Davisson and Lester H.Germer 6 of the Bell Telephone Co., who found that a beam of slowelectrons, reflected from the face of a target cut from a single crystalof nickel, gave well-defined beams of scattered electrons in variousdirections in front of the target: in fact, diffraction phenomena wereobserved precisely similar to those obtained with X-rays, of a wave-length cQnnected with the momentum of the electrons by de Broglie'sformula.

In Nov.-Dec. 1927, George Paget Thomson 6 examined thescattering of cathode rays by a very thin metallic film, which couldbe regarded as a microcrystallic aggregate, and confirmed the factthat a beam of electrons behaves like a wave: from the size of therings in the diffraction-pattern it was possible to deduce the wave-length of the waves causing them, and in all cases he obtained thevalue A =hIP. If a stream of electrons is directed at a screen in which

1 Ann. d. Phys.lxiv (1921), p. 451; !xvi (1921), p. 546; !xxii (1923), p. 345I Phil. Mag. xliii (1922), p. 593 I Phil. Mag. xlvi (1923), p. 461• Another experimental investigation which could be explained by de Broglie's

theory was that of E. G. Dymond, Nature, cxviii (1926), p. 336, on the scattering ofelectrons in helium; the moving electrons could be associated with plane de Brogliewaves, whose interference governed the scattering. I. Langmuir, Phys. Rev. xxvii (1906),p. 806 had shown that inelastic collisions in several gases lead to very small angles ofscattering.

• Phys. &v. xxx (1927), p. 707; Nature, cxix (16 April 1927), p. 558; Proc. N.A.S.xiv (1928), p. 317; cf. K. Schaposchnikow, ZS.f. P. Iii (1928), p. 151; E. Rupp, Ann.d. Phys. i (1929), p. 801; C.]. Davisson. J. Frank. Inst., ccv (1928), p. 597

• G. P. Thomson and A. Reid, Nature, cxix (1927), p. 890. G. P. Thomson, Proc.R.S.(A), cxvii (1928), p. 600; cxxviii (1930), p. 641. A. Reid, Proc. R.S.(A), cxix(1928), p. 663. R. Ironside, ibid., p. 668. S. Kikuchi, Proc. Tokyo Ac. iv (1928), pp. 271,354,471

218


there are two small holes close together, an interference-pattern isproduced just as with light.

The experimental verification of de Broglie's formula wasfurther extended by E. RUpp,l who succeeded in obtaining electronicdiffraction by a ruled grating.

The electron-energies for which the formula has been verifiedrange from 50 to 106 electron-volts.2

These effects have been observed not only with streams ofelectrons (cathode rays and the fJ-rays from radio-active sources),but also with streams of material particles: 1. Estermann andO. Stern 3 in 1930 found that molecular rays of hydrogen andhelium, impinging on a crystal face of lithium fluoride, werediffracted, giving a distribution of intensity corresponding to thespectra formed by a crossed grating. The wave-length, calculatedfrom the known constants of the crystal, was in agreement with deBroglie's formula.

Theoretical p~ers on the diffraction of electrons at crystalswere published not long after the Davisson-Germer experiment byHans Bethe,4 a pupil of Sommerfeld, and C. G. Darwin.5

The next notable advance in physical theory was made bySatyandra Nath Bose6 of Dacca University, in a short paper givinga new derivation of Planck's formula of radiation: Einstein, whoseems to have translated it into German from an English manuscriptsent to him by Bose, recognised at once its importance and its con-nection with de Broglie's theory.

Bose regarded the radiation as composed of photons, which forstatistical purposes could be treated like the particles of a gas, butwith the important difference that photons are indistinguishablefrom eath other, so i1Pat instead of considering the allocation ofindividual distinguisnable photons among a set of states, he fixesattention on the number of states that contain a given number ofphotons. He assumes that the total energy E of the photons isgiven, and that they are contained in an enclosure of unit volume.

A photon hv may be specified by its co-ordinates (x, y, z) and thethree components of its momentum (px, Py, pz). Since the totalmomentum is hv/e, we have

Let volume in the six-dimensional space of (x, y, z, px, PIt, pz) be

I ZS.j. P. Iii (1928), p. 8; Phys. ZS. xxix (1928), p. 837• J. V. Hughes, Phil. Mag. xix (1935), p. 129• ZS.j. P. lxi (1930), p. 95: cf. T. H. Johnson, Phys. Rev. xxxi (1928), p. 103: F.

Knauer and O. Stern, ZS.f. P. ciii (1929), p. 779• Ann. d. Phys. Ixxxvii (1928), p. 55• Prot. R.S.(A), cxx (1928), p. 631. This paper is concerned with the polarisation

of electron-waves, on which, see also C. Davisson and L. H. Germer, Phys. Rev. xxxiii(1929), p. 760, and E. Rupp, ZS.j. P. liii (1929), p. 548.

• ZS.f. P. xxvi (1924), p. 178219


called phase-space. Then to the frequency-range from v to v + dvthere corresponds the phase-space

Jdx 4Y dz dpx dPv dp. = 41TT2dr = 47r h3v

Sdv.

c3

Bose now assumes that this phase-space is partitioned into cells ofvolume h3, so there are 47TVsdv/c3 cells. In order to take account ofpolarisation we must double the number, so we obtain 87TVsdv/c3 cells.

Now let N, be the number of photons in the frequency-range dv"and consider the number of ways in which these can be allocatedamong the cells belonging to dv,. Let po' be the number of vacantcells, Pi' the number of cells that contain one photon, ps' the numberof cells that contain two photons etc. Then the number of possibleways of choosing a set of po' cells, a set of Pi' cells, etc. out of a totalof 87TVsdv/c3 cells is

As!po' Ip/!Pa'!···'

where

and we have N,= L rpr'.

As the fundamental assumption of his statIstics, Bose assumesthat if a particular quantum state is considered, then all values forthe number of particles in that state are equally likely, so the proba-bility of any distribution specified by the pr' is measured by thenumber of different ways in which it can be realised. Hence theprobability of the state specified by the pr' (now taking into accountthe whole range of frequencies) is

A,!W = n " " ",po . Pi . ps ....

Since the pr' are large, we can use Stirling's approximation

log n! = n log n - n

so log W = L A, log A, - L 2: pr' log pr', SInce A, = L pr'.

This expression is to be a maximum subject to the condition

E= L N,hv" when N,= L rpr'.

The usual conditions become in this case

L L opr'(l + log pr') = 0,

220

L oN ,.hv, = 0,


whereSN. = 2: rSpr',

which give

where f3 and the /I.' are constants: sorhvs

pr'=B.e -T

where the B. are constants.Therefore

rhv. ( hV.)-1A.= 2: pr'= 2: B.e-P =Bs l-e-P

r r

or

while

Thus

E= 2: Nshv.= 2: 87Th;sse-;~ (l-e-i)-ldvs.sse

The entropy isS=k log W

which gives

S = k {~ - ~ A. log (1 - ei)}.Since 8Sf8E = 1fT where T denotes the absolute temperature, wehave

so1--dv.hvs

kT 1e -

which is equivalent to Planck's formula.221


Bose's paper therefore showed that in order to obtain Planck'slaw of radiation, we must assume that photons obey a particularkind of statistics. This point may be illustrated by an analogy, asfollows. Consider an empty railway train standing at a platform,with passengers on the platform who get into the train; and supposethat when they have all taken their places, po compartments arevacant, PI compartments have one passenger apiece, P~ compartmentshave two passengers apiece, and so on: so that if A is the numberof compartments in the train, we have

A=PO+PI+P2+ - .... ,

and if N is the number of passengers, we have

N = 0 . po + I . PI + 2 . P2 +. .For simplicity, we shall assume that comparatively few people aretravelling, so that the number of compartments is greater than thenumber of passengers. Let us inquire, what is the probability ofthis particular distribution specified by the numbers (po, PI, P2, ... ).Evidently the probability depends on the assumption that we makeregarding the motives which influence passengers in their choice ofa compartment. Three such assumptions are as follows:

(i) We might assume that each passenger chooses his compart-ment at random, without regard to whether there are already anyother passengers in it, or not.

(ii) We might assume that each passenger likes to have a com-partment to himself, so he refuses to enter any compartment whichalready has an occupant.

(iii) We might assume that among the passengers there are smallfamily parties whose members wish to be together in the samecompartment, so that if we know that at least one place in a com-partment is occupied, there is a certain probability (arising fromthis fact) that other seats in it will also be occupied. We mayregard each family party or unattached traveller as a unit, andassume that each unit chooses its compartment at random, withoutregard to whether there are already other passengers in it, or not.

It is evident that these three different assumptions will givequite different values for the probability of any particular distri-bution (po, Ph P2' ... ): this we express by saying that they giverise to different statistics. The difference between classical statisticsand the different kinds of quantum statistics may be illustrated by thisanalogy.

Bose's discovery was immediately extended by Einstein,! to thestudy of a monatomic ideal gas. The difference between Bose's

1 Berlin Sitz. (1924), p. 261; (1925), pp. 3, 18222

RADIATION AND ATOMS IN THE OLDER QUANTUM THEORYphotons and Einstein's gas-particles is that for photons the energyISc times the momentum, whereas for particles a different equationholds: and, moreover, in Bose's problem the total energy is fixedbut the number of photons is not fixed, whereas in Einstein'sproblem the total number of particles is definite. These differences,however, do not affect the general plan of the investigation. Theanalysis leads to the following conclusions: the average number ofpartIcles of mass m in unit volume with energies in the range € to€+d€ is

(1)

where fL is a constant: whence the total number of particles zn unitvolume is

n = 2h1Ts(2m)! Jooo-_€ ~_2d_E__ = 2

h1Ts(2mkT) !Jooo xtdx

e£/kT+1' - 1 eX+'" - 1 (2)

and the total energy in unit volume is

E=21T(2m)!Joo EtdEhS 0 e£/kT+1' - 1 2

h1T

S(2m)! (kT)! J"'o xMx

eX+'" - 1· (3)

(4)

These are the fundamental formulae of what is generally calledBose-Einstein statistics.

Now consider the relation between these formulae and theformulae of the classical (Maxwellian) kinetic theory of gases.We should expect the classical formulae to be obtained as thelimiting case when h->-O, in which case it is evident from (1) that fLmust tend to infinity in such a way that hSe'" has a finite value,say A. From (2) we then have, in this limiting case,

2 SJoo 3 3n = --.!! (2mkT) 2" x~-e-X dx = ~ (2mkT) 2-A 0 A

while (1) states in the limiting case that the total number of particlesin unit volume with energies between € and € + d€ is

231.-..!! (2m) 2" r£/kTE2 d€A

or, by (4), out of a total of n particles, the number with energiesbetween E and E + d€ is

_~r£/kT€t d€1. 31T2(kT)2"

which is precisely the Maxwellian formula.223


Moreover (3) becomes in this limiting case

E=~.(2m)!.(kT)i. T(i) =fnkT

which is the Maxwellian formula for the total energy.We see that in Bose statistics, the slower molecules are more

numerous, as compared with the faster ones, than is the case inMaxwell's theory.

The similarity in statistical behaviour between Bose's photonsand the particles of a gas, revealed by this investigation, was examinedfurther by Einstein. He pointed out 1 that de Broglie's discoverymade it possible to correlate to any system of material particles a(scalar) wave-field: and he showed the close connection betweenthe fluctuations of energy in systems of waves and in systems ofparticles. We have seen 2 that the mean-square of the fluctuationsof energy per unit volume in the frequency-range from v to v + dvin the radiation in an enclosure at temperature T is

- csE2E

2=hvE + 87TV2dv '

the first term representing the fluctuation in the number of moleculesin unit volume of an ideal gas on the classical theory, when eachmolecule has energy hv. Einstein now showed that when the gas-particles are assumed to satisfy the Bose-Einstein statistics, both termsappear. In other words, a Bose-Einstein gas differs from a Max-wellian gas in precisely the same way as radiation obeying Planck'sformula differs from radiation obeying the law of Wien.

The next advance in the theory of quantum statistics was madeby Enrico Fermi 3 (b. 1901). He remarked that in the Maxwelliankinetic theory of gases, the average kinetic energy per molecule is!kT, and hence the molecular heat at constant volume (i.e. theheat that must be communicated to one gramme-molecule in orderthat its temperature may be raised one degree, the volume remainingunchanged), calculated from this theory, is c.= fR, where R is thegas-constant. If, however, Nernst's thermodynamical law, whichrequires (dEjdT)-+O as T-+0, is applicable to an ideal gas, then c.must vanish in the limit when T-+0, and therefore (as Einstein hadremarked in 1906) the Maxwellian theory cannot be true at very lowtemperatures. The reason for this must, he argued, be sought inthe quantification of molecular motions, and this quantification bemade to depend on Pauli's exclusion principle, that one system can

1 At page 9 of the first paper in Berlin Silz.)1925). • cf. p. 101 supra• Lincei lUnd. iii (7 Feb. 1926), p. 145; ZS .. P. xxxvi (1926), p. 902. A contribution

of lP'eat importance was made by P.A.M. Dirac somewhat later in the year, Proc. R.S.(A),exit (1926), p. 661, on account of which the type of statistics introduced by Fermi is oftencalled the Fermi-Dirac statistics; but as Dirac's paper involves the ideas of wave-mechanics, its description is postponed for the present.

224


never contain two elements of equal value with exactly the same setof quantum numbers. Thus he asserted that at most one moleculewith specified quantum numbers could be present in an ideal gas:where by quantum numbers he understood not only those which relateto the internal motions of the molecules, but also those which specifythe molecule's motion of translation.

The quantification may be performed as follows: suppose thegas is contained in a cubical vessel whose edge is of length 1, sothat the possible quantum values of the components of momentumof the particle are px = (slh/I), Pv = (sih/i), p. = (s3h/I), where SI, SI) S3are whole numbers or zero; then Pauli's principle asserts that in thewhole gas there can be at most one particle with specified quantumnumbers SI, S2, S3'

The energy of the particle is

hi (I I I)E= 211m SI +SI +S3

and as in the case of Bose statistics, the number of quantum statesof the particle which correspond to kinetic energy between E, andE.• + dE, is

27TV i!R, =·V (2m) E, dE,.

Now suppose that N, particles have energies between E, and E, + dE"so of the R, states, N, are occupied (by one particle each) and therest unoccupied. The number of ways in which this can occur is

R, !N,! (R,- N,)!

so the total number of ways in which the allocation specified bythe N, can be realised is

W-Il R,!- -- ---,, N,! (R,-N,).

and we assume that this is proportional to the probability of thisallocation. Thus

10gW=~ {logR,!-logN,!-log(R.-N,)!}.

Using Stirling's approximation to the log. of a factorial, this gives

10gW = L {R.logR,-N, 10gN,- (R,-N,) log (R,-N,)} •.'

This is to be made a maximum, subject to

:2: N, = n, where n is the total number of particles

225

and


andL €.N. = E, where E is the total energy.

The usual procedure yields

N.= -~-'-e(3£·+1' + I

where {:J and fL are independent of s. Now the entropy is

S=k logW

dS IdE =1"

whence we find IT=k{:J.

Hence the number of particles per unit volume with kinetic energy between€ and €+d€ is

This is the fundamental equation if the Fermi statistics.lThe total densiry if particles is therefore

3 1= JOO ( )d = (2mkT)221TJ"" x2dxn n € € h3 x+ Io 0 e 1'+

an equation which determines fL as a function of the density andtemperature.

The parameter fL has a thermodynamical significance. If fora finite mass of gas U is the total energy, S the entropy, T thetemperature, p the pressure and v the volume then

G= U -TS+pv

is called the Gibbs's thermodynamical potential; and the Gibbs's thermo-dynamical potential per molecule is defined as rp = oG /on, where n denotesthe number of molecules, and the temperature and pressure arekept constant in the differentiation. The Fermi constant fL is nowgiven 2 by

fL= -lr1 This may be modified, e.g. when the particles concerned have different possibilities

of spin. ' W. Pauli, ;:'S.j. P. xli (1927), p. 81, at p. 91226


The total kinetic energy of the particles in unit volume is

J'" 2 7T ;t iJ'" x!dxE= en(e)de=-h (2m)2(kT) + 1.o 3 oeXIJ+

The close similarity of these equations to the corresponding equationsin the Bose statistics is evident, the only difference being in theoccurrence of + 1 instead of - 1 in the denominator. In exactlythe same way as in the case of Bose statistics, we see that in thelimit when h-+O, J-t-+ + 00, Fermi statistics pass into Maxwellianstatistics: speaking physically, a Fermi gas approximates to aclassical gas at sufficiently high temperatures and low pressures.The deviation of Fermi statistics from classical behaviour is in theopposite direction to the deviation in the case of Bose statistics.l

The pressure of a gas in Fermi statistics is related to the energy-density by the same equation as in classical theory,

2EP=3V·Classical statistics is not the only limiting case of Fermi statistics:

there is another limiting case at the opposite extreme, namely whenJ-t is very large and negative. If we write - a for J-t, the integralsoccurring in the theory are of the form

r x'PdxJo ex-a+ 1

and it can be shown that

r {a-"-lfoo x'Pdx }_ 11m a-+ oc 0 ex-a + 1 - P +1

so the total density of particles becomes

n= i;(2mkTa)i

and the total kinetic energy of the particles in unit volume is

E= t;(2m)f(kTa)f.

Eliminating a, we have

E= 4~(~}~~2 nftt approximately,

the neglected terms involving T2 and higher powers ~ofT.I The thermodynamical functions for a gas with Fermi statistics were studied by

E. C. Stoner, Phil. Mag. xxviii (1939), p. 257.• For an electron-gas, e.g. in metals, there is an extra factor 2 arising from the two

possible values of the spin of the electron.227


It is evident from this equation that at the zero of absolutetemperature the particles of a gas with Fermi statistics are not atrest, but have a definite zero-point energy.

In this limiting case we see from the above value of n that3

nh3(mkT) -2~1,

which is realised if the temperature is low, the density n is largeand the mass m of the individual particles is small. The deviationfrom classical behaviour is due fundamentally to the circumstancethat the quantum states of very small momentum are all occupiedby particles. In fact, since (in accordance with Nernst's heattheorem) the entropy S vanishes at the absolute zero of temperature,and since S=k log W, it follows that W = 1 when T=O, I.e. thereis only one way of distributing the particles: they occupy everyquantum state in the neighbourhood of the state of zero momentum.

This fact has an interesting application in astrophysics. In 1844Bessel concluded from irregularities in the motion of Sirius that itwas one component of a double star, the other member of the pairbeing invisible with the telescopes then available. Some years laterthe companion was observed telescopically, and found to be a starbetween the 8th and 9th magnitude. Its mass is not much less thanthat of the sun, and it is a ' white' star, so that its surface-brightnessmust be greater than the sun's; but its total radiation is only about3hth of that of the sun: hence the area of its surfaces must bevery much smaller than the sun's, and in fact not much larger thanthe earth's. Its density must therefore be very great-about 60,000times that of water. This surprising inference was confi'rmed in1925,when Walter S.Adams 1of Mount Wilson found in the spectrumof the companion of Sirius a displacement of the lines which mightwell be the decrease of frequency that is to be expected in linesthat have been emitted in an intense gravitational field.

The explanation 2 of this abnormal density is that matter canexist in such a dense state if it has so much energy that the electronsescape from the nuclei, so they are no longer bound in ordinaryatomic orbits but are, in the main, free. The density of the matteris then limited no longer by the size of atoms, but only by the sizesof the electrons and atomic nuclei: and as the volumes of these areperhaps 10-14 of the volumes of atoms, we may conclude thatdensities of as much as 1014 times that of terrestrial matter may bepossible: this is much greater than that of any of the 'whitedwarfs,' as stars like the companion of Sirius are called.

According to the classical theory of the relation between energyand temperature, such a star, having excessively great energy,would have a very high temperature, and would therefore radiateintensely. It was, however, shown by R. H. Fowler 3 that this is

1 Proc.N.A.S. xi (1925), p. 382 • Due chiefly to Eddington8 Mon. Not. R.A.S. !xxxvii (1926), p. 114

228


not the case. The radiation depends on the temperature, anddepends on the energy only in so far as the energy determines thetemperature: the apparent difficulty was due to the use of theclassical correlation between energy and temperature: the correctrelation, for such dense stellar matter, is that of the above-mentionedlimiting case of Fermi's statistics, when I-' is very large and negative:the energy is still very great, but the temperature is not corres-pondingly great, and indeed ultimately approaches zero: soradiation, which depends on temperature, stops when the densematter still has ample energy.

The absolutely final state (the' black dwarf') is one in whichthere is only one possible configuration left: the star is thenanalogous to one gigantic molecule in its lowest quantum state,and the temperature (which ceases to have any meaning) may besaid to be zero.

It was shown in 1927 by L. S. Ornstein and H. A. Kramers 1

of Utrecht that the formulae of Fermi statistics may be derived ina totally different way by considering the kinetics of reactions.

Let the possible values of the energy of a gas-molecule in anenclosure be El' E2, E3, ••• ; for simplicity we suppose these valuesall different. Consider two molecules in the states k' and i' respec-tively, which by their interaction are changed to the states k"and i" respectively. Let the a priori probability that this transitionshould take place in unit time be denoted by a,,'. In order thatthe transition may actually take place, it is necessary that thereshould exist a molecule in the k' state and one in the i' state, and,moreover (by the Pauli principle), that neither a molecule in thek" state nor one in the i" state is present. Denoting by iik theprobability that a molecule is in the k state, the mean frequencyfor the process in which k', i' tend to k", i", is

A,,' = a,,' iik' iiI' (1 - iik") (1 - iiI")

while the frequency of the reverse process is

A," = a," iik" iii" (1 - iik') (1 - iii')'

In thermodynamic equilibrium we must have

AJ,'=A,"

and if we postulate further, that the a priori probability for reverseprocesses is equal, i.e.

1 ZS.f. P. xlii (1927), p. 481. On the kinetic relations of quantum statistics, cf. also:P.Jordan, ZS.f. P. xxxiii (1925), p. 649; xli (1927), p. 711. W. Bothe, ZS.f. P. xlvi(1928), p. 327. L. Nordheim, Proc. R.S.(A), cxix (1928), p. 689. S. Kikuchi andL. Nordheim, ZS.f. P.1x (1930), p. 652. S. Fliigge, ZS.f. P. xciii (1935), p. 804

229


then if we write

the above equations give

This equation must hold for all the values of k', [', kif, [", for whichthe equation of conservation of energy

Ek' + El' = ETt' + EZ"

holds: whence it follows that the only possible reasonabledistribu-tion is given by

where a and f3 are constants: whence we have_ e-a-fJEk 1nk = ----- = -----1+e-a-fJEk ea+fJEk+ 1

which is Fermi's law of distribution.Another application of the Fermi statistics was to the theory

of electrons in metals.In 1927 Pauli 1succeeded in accounting for the observed character

of the paramagnetism of the alkali metals, which is feeble andnearly independent of the temperature, by assuming that theconducting electrons in a metal may be regarded as an electron-gaswhich is degenerate in the sense of Fermi's statistics (i.e. JL is largeand negative). In the following year Sommerfeld 2 discussed themain problems of the electron-theory of metals on the same assump-tion: the conducting electrons may be supposed to have free pathsof the order of 100 times the atomic distances.

To see that the assumption is justified, we have to show that

nh3(mkT)-!~ 1.

Now we may suppose that the number n of free electrons is of thesame order as the number of atoms, say 1022 per cm3• The mass of

1 ZS.f. P. xli (1927), p. 81; communicated 16 Dec. 1926; published 10 Feb. 1927• Preliminary note in Naturwiss, xv (14 Oct. 1927), p. 825; xvi (1928), p. 374;

ZS. f. P. xlvii (1928), p. I. His associate C. Eckart, Ibid. p. 38, discussed the Volta-effed, and his associate W. V. Houston, ZS.f. P. xlviii (1928), p. 449, discussedelectricconduction. Sommerfeld's theory has beendevelopedand improved by: E. Kretschmann,ZS.f. P. xlviii (1928), p. 739. F. Block,ZS.f. P. Iii (1928), p. 555; lix (1930), p. 208.L. W. Nordheim, Prot. R.S.(A), cxix (1928), p. 689. R. Peierls, ZS.f. P. Iiii (1929),p. 255; Ann. d. Phys. iv (1930), p. 121; v (1930), p. 244. L. Brillouin, us Statistiquesquantiques(Paris, 1930), Tome II. L. Nordheim, Ann. d. Phys. ix (1931), p. 607. A. H.Wilson, Prot. R.S.(A), cxxxviii (1932), p. 594. On semi-conductors, cf. A. H. Wilson,Prot. R.S.(A), cxxxiii (1931), p. 458; cxxxiv (1931), p. 277.

230


the electron m is 9 x 10-28 gr., Planck's constant h is 6·6 x 10-27

erg. sec., Boltzmann's constant k is 1·4 x 10-16 erg./deg. Thus thecondition is roughly

or10-57 (10-43 Tfi ~ 1

T~105

which is satisfied at all ordinary temperatures. Thus the electron-gasinside a metal has a degenerate Fermi distribution. At the zero of absolutetemperature, there is a finite energy given by the equation

3 (3)~h2 iE=- - - n,40 7T m

and the derivative dE/dT is zero in accordance with Nernst's theorem:the specific heat of the electron-gas is in fact proportional to T forlow temperatures, whereas the classical statistics made it a constant.Thus it IS understood why the specific heat of the electrons in metalsis exceedingly small: the specific heat of a metal is usually whatwould be expected if it were due to the metallic atoms alone.

The replacement of classical by Fermi statistics does not appreci-ably change the theoretical ratio of the thermal and electrical con-ductivities.

We have seen that in Fermi statistics, the probability that aquantum state whose energy is € is occupied

IS

and that fL = - {J/kT where {J is Gibbs's thermodynamical potentialper molecule. For the electron-gas inside a metal, this quantity{J is called the electrochemical potential. It has the property that theelectrochemical potentials of electrons in any two regions whichare in thermal equilibrium (e.g. electrons in different metals whichare in contact with each other at the same temperature) mustbe equal.

In a second paper 1 in the Zeitschrift fur Physik, Sommerfelddiscussed thermo-electric phenomena, obtaining new formulae forthe Peltier and Thomson effects, and showing that the new expressionfor the Thomson heat agreed with the experimental values muchbetter than the old. Shortly before this a new thermo-electric effecthad been discovered by P. W. Bridgman,2 namely, that when anelectric current passes across an interface where the crystal orienta-tion changes, an ' internal Peltier heat' is developed. A theoreticaldiscussion of this and other thermoelectric phenomena was given in1929 by P. Ehrenfest and A. J. Rutgens.3

1 :(S.j. P. xlvii (1928), p. 43• Proc. N.A.S. xi (1925), p. 608; Phys. Rev. xxxi (1928), p. 221• Proc. Arnst. Ac. xxxii (1929), p. 698

231


The twentieth century brought considerable developments in thesubject of thermionics. Richardson had deduced his original formula 1

from the assumption that the free electrons in a metal have the sameenergy as is attributed in the kinetic theory of gases to gas-moleculesat the same temperature as the metal. H. A. Wilson 2 showed thatthe phenomena could not be explained completely by this theory,and proposed to replace Richardson's treatment by one analogousto the theory of evaporation. Now the latent heat of vaporisationof a liquid may be shown by thermodynamical methods to be givenby what is called the equation of Clapeyron and Clausius, which maybe written in the form

where X is the latent heat of evaporation per gramme-molecule at theabsolute temperature T, R is the gas-constant per gramme-molecule( = 1·987 cal.Jdeg.), and p is the vapour-pressure at temperature T :and Wilson proposed to apply this equation. Under the influenceof these ideas, Richardson 3 suggested as an alternative to his earlierformula that the saturation emission per unit area, in amperes percm.l, should be represented by an equation

_xI=AT2e kT

where A is a constant, k is Boltzmann's constant, and X representsthe energy required to get an electron through the surface: X, whichis called the work-Junction, is analogous to the latent heat of evapora-tion of a monatomic gas: it is usually reckoned in electron-voltsand is often written ec/> where c/> is expressed in volts, and so is infact the potential in volts necessary to impart to an eleCtron thekinetic energy required for evaporation.

It may be remarked that although the conducting electronsinside a metal have a degenerate Fermi distribution, the externalelectrons produced by thermionic emission have a distribution whichis practically Maxwellian: this is explained by their much smallerconcentration.

In 1923 Saul Dushman,4 by use of the Clapeyron-Clausiusformula, and on the assumption that the electrons within the metalobey Maxwellian statistics, obtained Richardson's second formula,showing, however, that it is not rigorously true if X is a functionof temperature (though for clean metals the deviations are small) ;

1 cf. Vol. I, pp. 425-8 2 Phil. Trans.(A), ccii (1903), p. 243• Phil. Mag. xxviii (1914), p. 633• Phys. Rev. xxi (1923), p. 623; cf. earlier papers by: W. Schottky, Phys. ZS. xv (1914),

p. 872; xx (1919), pp. 49, 220. M. von Laue, Ann. d. Phys. Iviii (1919), p. 695.R. C. Tolman, J. Am. Chern. Soc. xlii (1920), p. 1185; xliii (1921), p. 866. And cf. laterpapers by: P. W. Bridgman,Phys. Rev. xxvii (1926), p. 173; xxxi (1928), p. 90. L. Tonksand I. Langmuir, Phys. Rev. xxix (1927), p. 524. L. Tonks, Phys. Rev. xxxii (1928),p.284. K. F. Herzfeld, Phys. Rev. xxxv (1930), p. 248

232


in fact X should be understood as relating to the work-function forthe zero of absolute temperature: and he showed that A is a uni-versal constant, having the value

Sommerfeld's discovery that the electrons in a metal havedegenerate Fermi statistics naturally led 1 to a new treatmentof thermionics, but the formula finally obtained had the samegeneral character as Richardson's formula of 1914. L. W. Nord-heim 2 found

the quantities in this equation being defined as follows: A has twicethe value of Dushman's universal constant A, so

A 47Tmek2 120 I 2 d 2.= -h-a- = amp. em. eg..

the factor 2 arises when we take account of the two possible valuesof the electron-spin. It might be thought easy to discriminateexperimentally between Dushman's and Nordheim's formulae for A :but small uncertainties in the values of X and T, which occur in theexponential, affect the value of i so much that the experimentaldetermination of A is very uncertain.3 r is the reflection coefficient,i.e. the ratio of the number of electrons reflected internally at thesurface to the total number reaching it, only those electrons beingconsidered that have velocity components normal to the surfacesufficient to allow them to escape. r is small and need not usuallybe considered, since the experimental value of A is so doubtful.X corresponds to Richardson's work-function: but X now has theform

where Wa is e times the difference in electrostatic potential between theinside and outside of the metal, and Wi is an energy which dependson the pressure of the electron-gas, and which assists the escape ofthe electrons from the metal: we may take Wi to be the kineticenergy of the highest filled level at the absolute zero of temperature.

Now we saw that in a Fermi distribution, the number of quantum

1 A. Sommerfeld, ZS.f. P. xlvii (1928), p. I• ZS.f. P. xlvi (1928), p. 833; Phys. ZS. xxx (1929), p. 177• L. A. du Bridge, Phys. Rev. xxxi (1928), pp. 236, 912, found that for clean platinum

the thermionic constant A has the value 14,000 amp.fcm.' deg', which is 230 timesDushman's theoretical value 60·2; cr. L. A. du Bridge, Proc. N.A.S. xiv (1928), p. 788,and R. H. Fowler, Proc. R.S.(A), cxxii (1929), p. 36.

233


states of a particle (in unit volume) which correspond to kineticenergy between € and € + d€ is

or inserting a factor 2 on account of the two values of the electron-spin, it is

47T f!h3 (2m) € d€.

At zero temperature, all the quantum states are occupied up to acertain level of kinetic energy, say Wi : so if n be the number of freeelectrons per unit volume, we have

so

It can be shown that

where I-t is the electrochemical potential of the electrons just insidethe metal, and <l>i is the electrostatic potential just inside the metal.!

The function Wa, the true total energy necessary to liberate anelectron from the metal, may be determined independently by amethod which depends on the diffraction of beams of electrons bymetallic crystals.2 For although the experiments of Davisson andGermer, already referred to, were explained qualitatively by deBroglie's theory of the wave-behaviour of electrons, there was aquantitative discrepancy, which was resolved by assuming that thewave-length of the electron-waves in the metal was different fromthe wave-length in vacuo, or in other words, that the metal had arefractive index for the electron-waves: this refractive index, whichcould be determined from the diffraction-experiments, determinedthe grating-potential and thence the function Wa: Wa was foundto be considerably greater than the work-function x.

In a complete treatment of thermionic and photoelectric pheno-mena, it is necessary to take account of factors that cannot be fullydiscussed here, e.g. the fact that an electron just outside a metal is

1 The use of the electrochemical potential in thermionics is due to W. Schottky andH. ~othe, Handbuch d. Experimentalp~)'Sik (Leipzig, 1928), XII/2. .

cr. C. Eckart, Proc. N.A.S. XII (1927), p. 460; H. Bethe, NaturwlSS, xv (1927),p. 787; L. Rosenfeld and E. E. Witmer, ZS.f. P. xlix (1928), p. 534

234


influenced by a force due to its own image in the metal:1 in thecase of a flat perfectly-conducting surface this force is of amounte'f4z2, where z is the distance of the electron from the surface.

The constants and funct~ons that occur in thermionics occur alsoin related branches of physics. If two metals A and B at the sametemperature are in contact, electrons flow from one to the otheruntil the electrochemical potentials of the electrons in the twometals are equal, and there is a difference of potential (the Voltaeffect or contact potential-difference 2) between a point just outside Aand a point just outside B. Since the work-function X is

Wa - Wi or - e<l>o+ e<l>i - (J.L + e<l>i)

where <1>0 is the electrostatic potential at a point just outside themetal, we see that the Volta effect (<1>0) A - (<1>0) B is equal to(XB - XA) fe, where XA and XB are the thermionic work-functions of the twometals: so if iA and iB are the thermionic electric saturation currentsper unit area at the same temperature, the contact potential-difference is S

Thermionics is closely connected also with photoelectricity, aswe have seen in Chapter III.4 The frequency of light which willjust eject electrons photoelectrically, but with zero velocity, is calledthe threshold frequency, and the corresponding wave-length is calledthe photoelectric long wave-length limit. R. A. Millikan 5 verified, bya series of careful experiments, that if Vo is the threshold frequency,the photoelectric quantity hvo is equal to the thermionic work-functioneef> measured at the same temperature. L. A. Du Bridge 6 foundthat for clean platinum the photoelectric long wave-length limit is1962A, which by the above equation corresponds to 6·30 volts,while the thermionic work-function is 6·35 volts: the two are thusin agreement within the limits of error.

The theory of the connection between photoelectric thresholdfrequency and thermionic work-function was studied in the light ofSommerfeld's electron theory of metals by R. H. Fowler.7

1 The theory of electric images is due to William Thomson (Kelvin). This effecthad been considered in P. Lenard's early work on the photoelectric effect, Ann. d. Phys.viii (1902), p. 149; cr. P. Debye, Ann. d. Phys. xxxiii (1910), p. 441, and Walter Schottky,Phys. ZS. xv (1914), p. 872., cr. Vol. I, p. 71

, O. W. Richardson, Phil. Mag. xxiii (1912), p. 265; verified by O. W. RichardsoD.and F. S. Robertson, Phil. Mag. xliii (1922), p. 557

• cf. p. 90 supra • Phys. Rev. xviii (1921), p. 236• Phys. Rev. xxxi (1928), pp. 236, 912; cf. also A. H. Warner, Proc. N.A.S. xiii (1927),

p. 56, who worked with tungsten.7 Proc. R.S.(A), cxviii (March, 1928), p. 229; cr. G. Wentzel, SornTTUIrfeldF,stschrijl

(1928), p. 79235


The effect of temperature on the photoelectric sensibility of aclean metal near the threshold was studied by J. A. Becker andD. W. MuellerI in 1928, by E. O. Lawrence and L. B. Linford 2

in 1930 and by R. H. Fowler in 1931.3 It was shown that thephotoelectric long wave-length limit is shifted towards the red.

Some interesting experiments on the photoelectric properties ofthin films were carried out in 1929 by H. E. Ives and A. R. Olpin.4

They found that the long-wave limit of photoelectric action in thecase of films of the alkali metals on platinum varies with the thick-ness of the film. As the film accumulates, the long-wave limit movestowards the red end of the spectrum, reaches an extreme positionand then recedes again to the final position characteristic of themetal in bulk. The wave-length of the maximum excursion of thelong-wave limit was found in every case to coincide with the firstline of the principal series of the metal in the form of vapour, i.e.the resonance potential. This seemed to suggest that photoelectricemission is caused when sufficient energy is given to the atom toproduce its first stage of excitation.

Richardson 5 in 1912 studied the photoelectric emission from asurface exposed to black-body radiation corresponding to somehigh temperature T: this has been called the complete photoelectriceffect. It was shown by A. Becker 6 that the relative distributionof the velocities of the electrons released from platinum photo-electrically under these conditions is identical with that of electronsreleased thermionically. If thereis equilibrium, the surface mustitself be at temperature T, and then the total number of electronsleaving the surface, whether as a result of thermionic or photo-electric action, must be given by the Richardson equation.

Sommerfeld's discovery that the conducting electrons in metalsobey Fermi statistics made possible a satisfactory theory of a pheno-menon which had long been known 7 but which, in some of its features,had not been explained: namely, the extraction of electrons fromcold metals by intense electric fields.8 Experimental work byR. A. Millikan and his coadjutors 9 had shown that the currentsobtained at a given voltage are independent of the temperature,provided the latter is not so high as to approach the temperature atwhich thermionic emission becomes appreciable: whence Millikan

1 Phys. Rev. xxxi (1928), p. 431 I Phys. Rev. xxxvi (1920), p. 482• Phys. Rev. xxxviii (1931), p. 45; cf. J. A. Becker and W. H. Brittain, Phys. Rev.

xlv. (1934), p. 694• Phys. Rev. xxxiv (1929), p. 117; cr. Hughes and du Bridge, Photoelectric Pherwmena

(New York, 1932), p.. 178• Phil. Mag. xxili (1912), p. 594 • Ann. d. Phys. Ix (1919), p. 307 Knowledge of it seems to have evolved gradually from R. F. Earhart's experiments

on very short sparks. Phil. Mag. i (1901), p. 147.• A first approximate theory was given by W. Schottky, ZS.f. P. xiv (1923), p. 63,

following onJ. E. Lilienfeld, Phys. ZS. xxiii (1922), p. 306.I R. A. Millikan and C. F. Eyring, Ph)'S. Rev. xxvii (1926), p. 51; R. A. Millikan

and C. C. Lauritsen, Proc. N.A.S. xiv (1928), p. 45; cr. also N. A. de Bruyne, Proc. Camb.Phil. Soc. xxiv (1928), p. 518

236

RADIATION AND ATOMS IN THE OLDER QUANTUM THEORYconcluded that the conduction electrons extracted in this way do notshare in the thermal agitation, whereas the thermions, which areexpelled at high temperatures, do.

Millikan and Lauritsen showed that if I is the current obtainedby an applied electric force F, then log I when plotted againstIfF yields approximately a straight line. In 1928 the subjectwas attacked in several theoretical papers, 1 most completely byR. H. Fowler and L. Nordheim. a They calculated the emissioncoefficient (i.e. the ratio of the number of electrons going throughthe surface, to the number of electrons incident from inside themetal) and the reflection coefficient, at the surface, and integratedover all incident electrons according to Sommerfeld's electron theoryof metals; and they showed that it is the electrons with smallenergies that are pulled out by strong fields. Now these electronswith small energies have Fermi statistics, and that is why the intensityof the emitted current is, at ordinary temperatures, independent ofthe temperature. The formula obtained theoretically by Fowler andNordheim for the current I was

where € is the electron-charge, K2 = 87T2mfh2, ifJ is Gibbs's thermo-dynamical potential per electron, X is the thermionic work-functionand F is the applied electric force. This formula agrees with theexperimental results.s

1 R. H. Fowler, Prot. R.S.(A}, cxvii (1928), p. 549; L. Nordheim, ZS. f. P. xlvig928), p ..833; W. V. Houston, ZS.f. -f.. xlvii (1928), p. 33 (working with Sommerfeld).;. W. Richardson, Prot. R.S.(A}, CXVlI (1928), p. 719; W. S. Pforte, ZS. f. P. xlix

(1928), p. 46• Prot. R.S.(A), cxix (1928), p. 173; cf. also J. R. Oppenheimer, Phys. &v. xxxi

(1928), p. 66• cf. also: O. W. Richardson, Prot. R.S.(A), cxix (1928), p. 531; N. A. de Bruyne,

Prot. R.S.(A), cxx (1928), p. 423; A. T. Waterman, Prot. R.S.(A), cxxi (1928), p. 28;L. W. Nordheim, Prot. R.S.(A}, cxxi (1928), p. 626; T. E. Stern, B. S. Gossling andR. H. Fowler, Prot. R.S.(A}, cxxiv (1929), p. 699; W. V. Houston, Phys. Rev. xxxiii(1929), p. 361

237

Chapter VIIMAGNETISM AND ELECTROMAGNETISM, 1900-26

At the end of the nineteenth century the classical theory of electronswas well established, and one magnetic phenomenon, namely, theZeeman effect, had been explained in terms of it by Lorentz andLarmor. The time was evidently ripe for the consideration of dia-magnetic, paramagnetic and ferromagnetic phenomena in the lightof electron-theory.

It will be remembered 1 that Weber had explained diamagnetism(the magnetic polarisation of bodies induced in a direction oppositeto that of the magnetising field) by postulating the existence inmolecules of circuits whose eleGi:ricresistance is zero (but whoseself-induction is not zero), so that the creation of an external fieldcauses induced currents in them: the total magnetic flux throughthe circuits remains zero, and therefore by Lenz's law the directionof the induced currents corresponds to diamagnetism. Para-magnetism was explained by postulating Amperean electric currentsin the molecules, whose planes were orientated by the magnetisingfield: and Ewing had developed on this basis an explanation offerromagnetism.

The re-statement of these ideas in terms of the theory ofelectrons was undertaken in 1901-3 by W. Voigt 2 andJ.]. Thomson.3They studied the effect of an external magnetic field on themotion of a number of electrons, which are situated at equal intervalsround the circumference of a circle, and are rotating in its planewith uniform velocity round its centre; and they found that if asubstance contained a uniform distribution of such systems, thecoefficient of magnetisation of the substance would be zero; sothat it would be impossible to explain the magnetic or diamagneticproperties of bodies by supposing that the atoms contain chargedparticles circulating in closed periodic orbits under the action ofcentral forces.

The origin of the difference between the effects produced bycharged particles freely describing orbits, and those produced byconstant electric currents flowing in circular circuits, as in Ampere'stheory of magnetism, is that in the case of the particles describingtheir orbits we get, in addition to the effects due to the constantelectric currents, effects of the same character as those due to theinduction of currents in conductors by the variation of the magneticfield: these induced currents tend to make the body diamagnetic,I cr. Vol. I, pp. 208-11 • Gott. Nach. (1901), p. 169; Ann. d. Phys. ix (1902), p. 115a Phil. Mag.(6) vi (1903), p. 673

238

MAGNETISM AND ELECTROMAGNETISM, 1900-26

while the Amperean currents tend to make it magnetic; and inthe case of the particles describing free orbits, these tendencies balanceeach other.

Voi~t showed that spinning electrons, obstructed in their motionby contmual impacts, would lead to paramagnetism or diamagnetismaccording as they possessed, immediately after the impact, anaverage excess of potential or kinetic energy. However, besides thecomplexity of thIs representation, there is the objection that itattributes to the same cause phenomena so different as parama~netismand diamagnetism, and that it offers no interpretation of certam laws,established experimentally by P. Curie,! namely, that the para-magnetic susceptibility varies inversely as the absolute temperature,whereas diamagnetism is in all observed cases except bismuth,rigorously independent of T.

The first successful application of electron-theory to the generalproblem of magnetism was made in 1905 by Paul Langevin.2 Heaccepted Weber's view that diamagnetism is really a propertypossessed by all bodies, and that the so-called paramagnetic andferromagnetic substances are those in which the diamagnetism ismasked by vastly greater paramagnetic and ferromagnetic effects.The condition for the absence of paramagnetism and ferromagnetismis that the molecules of the substance should have no magneticmoment, so that they have no tendency to orient themselves in anexternal magnetic field.

In order to explain diamagnetism, we observe that the externalapplied magnetic field H creates a Larmor precession,3each electronacquiring an additional angular velocity eH/2mc in its orbit 4 ;

whence it follows, as Langevin showed, that the increase of themagnetic moment of a molecule due to one particular electroncirculating in it is

where r is the distance of the electron from the atomic nucleus,projected in a plane perpendicular to H, and?! is an average extendedover the duration of several revolutions. The negative sign showsthat the effect is diamagnetic. In order to obtain the total effect ona molecule we must sum over all the electrons in it. Thus if Ndenotes Avogadro's number, i.e. the number of molecules in agramme-molecule (which is the same for all elements and compounds),then the diamagnetic susceptibility per gramme-molecule is

e2N _---2: r24mc2

I Ann. chim. phys. v (1895), p. 289• Comptes Rendus, cxxxix (26 Dec. 1904), p. 1204; Soc. Franfaise de phys., Resumes

(1905), p. 13 • ; Ann. chim. phys.(8) v (1905), p. 70 • ef. Vol. I, pp. 415-6• As usual, H and M are in electromagnetic units, and e in electrostatic units.

239


where the summation is taken over all the electrons in the atomor molecule, and (r, z, c/» are the cylindrical co-ordinates of theelectron when the z-axis is taken in the direction of the magneticfield H.I Since atoms which exhibit diamagnetism without para-magnetism have spherical symmetry, 2: ~ can be replaced by -i2: R2,where R denotes the distance of the electron from the centre of theatom.

The smallness of the diamagnetic effects is accounted for by thesmallness of the radius r, which is necessarily less than the moleculardimensions.

The diamagnetic prorerty is acquired instantaneously, at themoment of the creation 0 the external field.

The intramolecular motions of the electrons depend very littleon the temperature, as is shown by the fixity of spectral lines: sothe diamagnetic susceptibility should vary very little with thetemperature, in agreement with Curie's experimental result. Itvaries also very little with the physical or chemical state: dia-magnetism is an atomic property.

The exception presented by solid bismuth, whose diamagneticsusceptibility diminishes approximately linearly when the tempera-ture rises, was attributed by Langevin (following J. J. Thomson)to the presence in the metal of free conduction-electrons.

The fact that diamagnetism and the Zeeman effect both dependon the Larmor precession shows that the two phenomena (at anyrate when considered classically) are very closely connected, andindeed may be regarded as different aspects of the same phenomenon.

If the intrinsic magnetic moment of a molecule is not null, thereis superposed on the diamagnetic effect another phenomenon, dueto the orientation of the molecular magnets by the external field;this paramagnetic effect, when it exists, is large compared with thediamagnetic, and completely masks it. In discussing it we shallassume that the molecular magnets are not associated in aggregateswithin which their mutual actions are of importance (this is thecase with ferromagnetic substances, which will be considered later) :and in fact we shall suppose that they can be treated in the same wayas the molecules in the kinetic theory of gases. In an externalmagnetic field H, a molecule whose magnetic moment is M hasa potential energy - MH cos a, where a is the angle between Mand H; the potential energy being thus least when the molecularmagnet is parallel to the external field. The situation may nowbe compared with that of a gas composed of heavy molecules andacted on by gravity, where the ascent of a molecule causes an increasein its gravitational potential energy and a decrease in its kineticenergy; similarly the magnetic molecule experiences a changeof kinetic energy when a changes, and the partition of kineticenergy between the molecules becomes incompatible with thermal

1 W. Pauli, ZS.f P. ii (1920), p. 201240


equilibrium. Through the mediation of collisions, a rearrangementis brought about, by which the mean kinetic energy of a mole-cule is made independent of its orientation: the temperaturebecomes uniform, and the molecular magnets are directed pre-ferentially in the direetion of H; thou~h this orientation is notuniversal, on account of the thermal agItation and the collisions.Unlike diamagnetism, paramagnetism does not appear instan-taneously, since collisions are required to create it.

Langevin found that if I-' denotes the intrinsic magnetic momentof a molecule, T the absolute temperature, k Boltzmann's constant,N Avogadro's number, M the magnetisation in one gramme-molecule,then

M = N I-' ( coth X - ~) where X = r~.When X is small, so that we need retain only the first term In(coth X - l/X), this gives for the molecular susceptibility x,

M N1-'2.x= B= 3k1' ,

this formula involves Curie's law, that the paramagnetic susceptibility variesinversely as the absolute temperature.1

From the point of view of the Rutherford-Bohr theory of atomicstructure, paramagnetism, being due to the possession of intrinsicmagnetic moment, is associated with incomplete shells of electrons 2 ;

for every closed shell has spherical symmetry. This explains whythe alkaline and alkaline-earth elements in the metallic state areparamagnetic, while their salts are diamagnetic: for the electronswhich do not belong to closed shells in the metals are taken intogaps in the shells of the elements with which they are combined.In the case of the incomplete inner shells of the rare earths, however,this kind of compensation does not take place, so the salts are para-magnetic.a

In 1907 Langevin's theory was extended so as to give an accountof ferromagnetism, by Pierre Weiss' (b. 1865). Ferromagnetism isa property of molecular aggregates (crystals), so Weiss took into con-sideration the internal magnetic field, assuming that each moleculeis acted on by the surrounding molecules with a force equal to thatwhich it would experience in a uniform field proportional to theintensity of the magnetisation and in the same direction. On this

1 On the magnetic susceptibility of oxygen, hydrogen and helium, cf. A. P. Wills andL. G. Hector, Phys. Rev., xxiii (1924), p. 209; for helium, neon, argon and nitrogen,cr. L. G. Hector, Phys. Rev., xxiv (1924), p. 418. For theoretical predictions regardingthese gases, cf. G. Joos, ZS.f. P. xix (1923), p. 347.

• cr. N. W. Taylor and G. N. Lewis, Proc. N.A.S., xi (1925), p. 456• F. Hund, ZS./. P. xxxiii (1925), p. 855• Bull. des stances de La soc. Jr. de phys., Annie 1907, p. 95; Journ. de phys. vi (1907),

p.661


assumption Weiss showed that a ferromagnetic body when heatedto a certain critical temperature, the Curie point,! ceases to be ferro-magnetic, and that for higher temperatures the inverse of the sus-ceptibility is a linear function of the excess of the temperature abovethe Curie point.

It follows from Weiss's work that in a ferromagnetic substancethere must be domains (i.e. regions larger than a single atom ormolecule) which are inherently magnetic, although if the magneticmoments of the domains are not oriented in any preferential direction,a finite block of the substance may show no magnetisation.

Weiss found 2 that the magnetic moments of all known moleculeswere multiples of a certain greatest common divisor, which he calleda magneton. This is called Weiss's magneton in order to distinguishit from the natural unit of magnetic moment to which W. Pauli 3

in 1920 gave the name Bohr magneton, and which has the valuehe

41Tmc·

The Bohr magneton is nearly five times the Weiss magneton. Forthe vapours of the elements in the first column of the Newlands-Mendeleev table (i.e. the alkalis and Cu, Ag, Au), it was shown byW. Gerlach and O. Stern,' by W. Gerlach and A. C. Cilliers,5andby J. B. Taylor,S that the intrinsic magnetic moment is one Bohrmagneton. As we have seen,7 the Bohr magneton was found in1925 to be the magnetic moment of an electron; and from thistime the Weiss magneton ceased to figure in physical theory.8

The researches of Langevin and Weiss represented notableadvances in the theory of magnetism, and the agreement of theirresults with experimental data was striking: yet it was shown inNiels Bohr's inaugural-dissertation 9 in 1911 and by Miss It. J. vanLeeuwen,lo a pupil of Lorentz's, in 1919, that the validity of theseresults could be explained only by supposing that Langevin andWeiss had not consistently applied classical statistics to all thedegrees of freedom concerned: in other words, they had madeassumptions of a quantistic character. The difficulty was eventuallyremoved only by the development of quantum mechanics.

An account must now be given of some questions which had been1 The name was introduced by P. Weiss and H. Kamerlingh Onnes, Comm. phys.

Labor. !Liden, No. 114 (1910), p. 3.• Jouro. de phys. i (1911), pp. 900, 965; Phys. ZS. xii (1911), p. 935; cr. B. Cabrera,

Jouro. dephys. iii (1922), p. 443• Phys. ZS. XXI (1920), p. 615 • Ann. d. Phys. !xxiv (1924), p. 673• ZS.f. P. xxvi (1924), p. 106 • Phys. Rev. xxviii (1926), p. 576• cr. p. 136supra • cr. W. Gerlach, Phys. ZS. xxiv (1923), p. 275• N. Bohr, Studier over Metallernes Elektronteori (120 pp.), Copenhagen, 1911

'0 Inaugural dissertation, Leiden, 1919: her argument, which was based on Boltzmann'sH.theorem, was substantially reproduced in her paper J. de phys. et le radium, ii (1921),p. 361 ; cf. alsoJ. N. Kroo, Ann. d. Phys. xlii (1913), p. 1354,who argued that the electrontheory could explain only diamagnetism; E. Holm, Ann. d. Phys. xliv (1914), p. 241 ;R. Gans, Ann. d. Phys. xlix (1916), p. 149.

24Q

MAGNETISM AND ELECTROMAGNETISM, 1900-26left unsettled in the nineteenth century and were definitely answeredin the twentieth.

Maxwell had suggested 1 the possibility that an electromotiveforce might be produced by simply altering the velocity of a con-ductor. This effect was not known to exist until 1916, when it wasobserved by Richard C. Tolman and T. Dale Stewart. 2 Theyrotated a coil of copper wire about its axis at a high speed andthen suddenly brought it to rest, so that a pulse of current wasproduced at the instant of stopping by the tendency of the freeelectrons to continue in motion. The ends of the coil were connectedto a sensitive ballistic galvanometer, which enabled the experi-menters to measure the pulse of current thus produced. Denotingby R the total resistance In the circuit, by 1 the length of the rotatingcoil, by v the rim speed of the coil, by Q the pulse of electricitywhich passes through the galvanometer at the instant of stopping,by F the charge carried in electrolysis by one gramme-ion, that isNele electromagnetic units, where N is Avogadro's number, and byM the eff~ctivemass of the carrier of the current (the electron), theyfound the equation

Q=MvlFR'by use of which they inferred from the experiments that the carriersof electric current in metals have approximately the same ratio ofmass to charge as an electron.

Maxwell3 mentioned another possible effect which might becaused by the carriers of electricity in conductors. He proposedto take a circular coil of a great many windings and suspend it bya fine vertical wire, so that the windings are horizontal and thecoil is capable of rotating about a vertical axis. A current is supposedto be conveyed into the coil by means of the suspending wire, and,after passing round the windings, to complete its circuit by passingdownwards through a wire which is in the same line with the sus-pending wire and dips into a cup of mercury. If a current is sentthrough the coil, then at the moment of starting it, a force wouldrequire to be supplied in order to produce the angular momentumof the carriers of electricity passing round the coil; and as thismust be supplied by the elasticity of the suspending wire, the coilmust rotate in the opposite direction.

Maxwell failed to detect this phenomenon experimentally, butit was successfully observed in 1931 by S. J. Barnett' (b. 1873).Like the preceding effects of electron-inertia, it provides a measureof mle.

1 In § 577 of his Treatise• Phys. Rev.(2) viii (1916), p. 97.; ix (1917), p. 164; cf. R. C. Tolman, S. Karrer

and E. W. Guernsey, Phys. Rev. XXI (1923), p. 525; R. C. Tolman and L. M. Mott-Smith, Phys. Rev. xxviii (1926), p. 794

• Treatise, § 574 • Phil. Mag. xii (1931), p. 349243


In 1908 O. W. Richardson 1 suggested the existence of amechanical effect accompanying the act of magnetisation. Heimagined a long thin cylindrical bar of iron suspended by a fibre,so that it is capable of small rotations about a vertical axis. Whenthe bar is not magnetised, the electrons which are moving in closedorbits in the molecules (Ampere's molecular currents) will notpossess any resultant angular momentum, since one azimuth is asprobable as another for the orbits. Now consider the effect ofsuddenly applying a vertical magnetic field: the orbits will orientthemselves so as to leave a balance in favour of the plane per-pendicular to the direction of the field, and thus an angular momen-tum of electrons will be created about the axis of suspension. Thismust be balanced by an equal reaction elsewhere, and therefore atwisting of the suspended system as a whole is to be expected.

Richardson himself did not succeed in obtaining the effect,which was first observed by A. Einstein and W. J. de Haas 2 in1915. They were followed by many other experimenters, par-ticularly J. Q. Stewart 3 of Princeton and W. Sucksmith andL. F. Bates;' working with Professor A. P. Chattock in Bristol.Sucksmith and Bates concluded that the gyromagnetic ratio, i.e. theratio of the angular momentum of an elementary magnet to itsmagnetic moment, has a value only slightly greater than me/e.This is only half the value of the ratio for an electron circulating inan orbit, and has been taken to imply that the magnetic elementsresponsible for the phenomenon are chiefly not orbital electrons butare electrons spinning on their own diameters.s

The converse of the Richardson effect has also been observed,namely, it has been found possible to magnetise iron rods by spinningthem about their axes. John Perry 6 said in 1890 ' Rotating a largemass of iron rapidly in one direction and then in the other in theneighbourhood of a delicately-suspended magnetic needle ought,I think, to give rise to magnetic phenomena. I have hitherto failedto obtain any trace of magnetic action, but I attribute my failure tothe comparatively slow speed of rotation which I have employed,and to the want of delicacy of my magnetometer.'

The effect predicted by Perry was anticipated also by Schuster 7

in 1912, but was first observed in 1914-15 by S. J. Barnett 8: theAmperean molecular currents, since they possessangular momentum,

1 Phys. Rev. xxvi (1908), p. 248I Verh. d. deutsch. phys. Ges. xvii (1915), p. 152; cr. A. Einstein, ibid. xviii (1916),

p. 173 and W.J. de Haas, ibid. xviii (1916), p. 423I Phys. Rev. xi (1918), p. 100• Proc. R.S.(A), civ (1923), p. 499; cr. W. Sucksmith, Proc. R.S.(A), cviii (1925), p. 638;

also Emil Beck, Ann. d. Phys. Ix (1919), p. 109; and G. Arvidsson, Phys. ZS. xxi (1920),p.88

• On the gyromagnetic effect for paramagnetic substances, cr. W. Sucksmith, Proc.R.S.(A), cxxviii (1930), p. 276; cxxxv (1932), p. 276

• J. Perry, Spinning Tops, p. 112 ' Proc. Phys. Soc. xxiv (1911-12), p. 121• Phys. Rev.(2) vi (1915), p. 239; Bull. Nat. Res. Council, iii (192~), p. 235; cr.

S. J. Barnett and L. J. H. Barnett, Phys. Rev. xx (1922), p. 90; Nature, CX11 (1923), p. 186244

MAGNETISM AND ELECTROMAGNETISM, 19°0-20

behave like the wheels of gyroscopes, changing their orientation,with a tendency to make their rotations become parallel to theimpressed rotation. Thus a preponderance of magnetic moment iscaused along the axis of the impressed rotation.

From time to time papers appeared on questions belonging tothe same class as the problem of unipolar induction. This problem,which had first been considered by Faraday,l may be stated asfollows. A magnet, symmetrical with regard to an axis aboutwhich it rotates, has sliding contact with the ends A and B of astationary wire ACB, at two points A and B not in the same equa-torial plane of the magnet. As Faraday found, a steady currentflows through the wire. The question is, do the lines of magneticinduction rotate with the magnet, so that the electromotive forceis produced when they cut the stationary wire ACB: or do thelines of magnetic induction remain fixed, so that the electromotiveforce is produced when the moving part of the circuit (the magnet)rotates through them? Faraday believed the latter explanation tobe the true one: but Weber, who introduced the name unipolarinduction, took the contrary view.2 It was known in the nineteenthcentury that the experimental results could be explained equallywell on either hypothesis.

More generally, we can consider a magnet which is capable ofbeing rotated about its axis of symmetry (whether it is actually sorotated or not) and a conducting circuit composed of two parts,one of which is rotated about the axis of the magnet (in the caseconsidered above it is the magnet itself) while the other, ACB,remains fixed. It is found that if the moving part of the circuitis distinct from the magnet, the electromotive force is independentof whether the magnet is rotating or not. The electromotive forceround the circuit is in all cases, if w denotes the angular velocityof the moving part of the circuit, (Wj27TC) times the flux of magneticinduction through any cylindrical surface having as boundariesthe circles described round the axis of revolution by the twocontact points A and B of the fixed with the rotating part of thecircuit.

In the twentieth century some new types of experiment weredevised. In 1913 Marjorie Wilson and H. A. Wilson 3 constructeda non-conducting magnet by embedding a large number of smallsteel spheres in a matrix of wax, and rotated this in a magneticfield whose direction was parallel to the axis of rotation. Theirmeasures of the induced electromotive force were in satisfactoryagreement with the predictions of the electromagnetic theoryof moving bodies published by A. Einstein and J. Laub 4 in1908.

Further contributions to problems of this class were made by

1 d. Vol. I, pp. 173-4• Prot. R.S.(A), lxxxix (1913), p. 99

245

• Joe. cit., Vol. I• Ann. d. Phys. xxvi (1908), p. 532

where


S. J. Barnett,l E. H. Kennard,2 G. B. Pegram,8 and W. F. G.Swann,' and in 1922 a comprehensive review of the whole subjectwas published by J. T. Tate.5

The problems of classical electromagnetic theory which werestudied in the early years of the twentieth century related chieflyto the expression of the field due to a moving electron, and therate at which energy is radiated from it outwards. Formulae forthe electric and magnetic vectors of the field were given by manywriters.8 The most interesting terms are those which involve theacceleration of the electron. Denote by (vx, VII, v.) the velocityand by (wx, WII w.) the acceleration of the electron at the instant twhen it emits the radiation which reaches the point (x, y, z) at theinstant t: let the co-ordinates of the electron at time t' be (x',y', z'),and let x' (1),y' (1), ;:,'(1), be denoted by (x', j', z'); and let

ill = (x' _x)lI+ (j' _y)lI+ (:t _;:,)1,

so t=t-ifc.

Then the terms in the x-component of the electric force whichinvolve the acceleration are

1 (x'-x)Vx+ (j'-y)vlI+ (z'-z)v.s= + ci •

From this we have at once

Dr (x'-x)+Dy (j'-y)+D. (z'-z)=0

so the vector (Dx, Dy, D.) is perpendicular to r. Moreover, the part ofthe magnetic vector which depends on the acceleration of the electron(call it H) is perpendicular to both D and r, and equal in magnitude to D.SO the wave of acceleration, as Langevin 7 called the field specified byD and H, has all the characters of a wave of light.

At great distances from the electron, the intensity of D and Hdecreases like l/r, whereas the intensity of the terms in the electric

1 Phys. ZS. xiii (1912), p. 803; Phys. Rev. xxxv (1912), p. 323• Phys. ZS. xiii (1912), p. 1155; Phys. Rev.(2) i (1913), p. 355• Phys. Rev. x (1917). p. 591 • Phys. Rev. xv (1920), p. 365• Bull. Nat. Res. Council, iv, Part 6 (1922), p. 75• K. Schwarzschild, Gott. Nach. (1903), p. 132. G. Herglotz, Gott. Nach. (1903),

p. 357. H. A. Lorentz, Proc. Arnst. Acad. v (1903), p. 608. A. W. Conway, Proc. Lond.M.S. (2) i (1903), p. 154. A. Sommerfeld, Gott. Nach. (1904), p. 99. P. Langevin,J. de phys. iv (1905),p. 165. H. Poincare, Palermo Rend. xxi (1906), p. 129. G. A. Schott,Ann. d. Phys. xxiv (1907), p. 637; xxv (1908), p. 63. F. R. Sharpe, Bull. Amer. Math.Soc. xiv (l908~, p. 330. A. Sommerfeld, Munich .Sitz. (1911), p. 51. A. W. Conway,Proc. R./.A. XXlX, A (1911), p. 1 ; Proc. R.S.(A), XCIV (1918), p. 436

, loc. cit.246

When vic is no longer neglected, it was shown by Heaviside 2 thatthe energy radiated per second is

1- (V2jc2) sin2 v'W(l_v2Ic2)3 •

The CI.uestionas to a reaction or 'back pressure' experiencedby a mOVIngmass on account of its own emission of radiant energywas discussed by O. Heaviside,3 M. Abraham,4 J. H. Poynting,SA. W. Conway,6 J. Larmor? and Leigh Page.s

A striking unification of electromagnetic theory was publishedin 1912 by Leigh Page 9 (b. 1884). It had been realised long beforeby Priestley 10 that from the experimental fact that there is no electricforce in the space inside a charged closed hollow conductor, it ispossible to deduce the law of the inverse square between electriccharges, and so the whole science of electrostatics. I t was nowshown by Page that if a knowledge of the relativity theory of Poincareand Lorentz is assumed, the effect of electric charges in motion canbe deduced from a knowledge of their behaviour when at rest, andthus the existence of magnetic force may be inferred from electro-statics: magnetic force is in fact merely a name introduced in orderto describe those terms in the ponderomotive force on an electronwhich depend on its velocity. In this way Page showed that Ampere'slaw for the force between current-elements, Faraday's law of theinduction of currents and the whole of the Maxwellian electro-magnetic theory, can be derived from the simple assertion of theabsence of electric effects within a charged closed hollow conductor.

In 1914 two new representations of electromagnetic actions wereintroduced, both of which were evidently inspired by Maxwell'sEncyclopaedia Britannica article on the aether, in which it was regardedas composed of corpuscles, moving in all directions with the velocity

• cr. Vol. I, p. 396 • Nature, !xvii (1902), p. 6, Nature, lxvii (1902), p. 6• Ann. d. Phys. x (1903), p. 156; Brit. Ass. Rep., Cambridge 1904, p. 436• Phil. Trans. ccii (1904), p. 525 • Prot. R.I.A. xxvii (1908), p. 1697 Proc. Int. Congo Math., Cambridge 1912, Vol. I, p. 213; Nature, xcix (1917), p. 404• Ph)'S. Rev. xi (1918), p. 376• Amer. J. Sci. xxxiv (1912), p. 57 ; Phys. <S. xiii (1912), p. 609,. cf. Vol. I, p. 53

247


of light, never colliding with each other, and possessing some vectorquality such as rotation.

The first of these representations was due to Ebenezer Cunning-ham of Cambridgel (b. 1881). Cunningham remarked that if theaether is supposed to be at rest, as it is in Lorentz's theory, thenthe transfer of energy represented by the Poynting vector cannotbe identified with the rate of work of the stress in the aether. This,indeed, can be done only if the aether is supposed to be in motion;for a stress on a stationary element of area does not transmit anyenergy across that element. He therefore proposed to assign avelocity to the aether at every point, such that a state of stress inthe medium would account both for the transference of momentumand for the flow of energy.

He found that the component of the aether-velocity which isin the direction of the Poynting vector must be the smaller root ofthe quadratic equation (e2 + x2)g = 2Wx where g is the value of themomentum (lie) [E. H], and W is the density of energy: that theother component lies in a definite direction in the plane of E andH: and that the total velocity of the aether at every point must beequal to e: so the direction and magnitude of the aethereal velocityare completely determined. The scheme is relativistically invariant:so that an objective aether is not necessarily foreign to the point ofview of the principle of relativity: and the mechanical categoriesof momentum, energy, and stress can also be maintained in theirentirety. The only essential modification of the ordinary theory ofmaterial media is that (as in relativity generally) we no longer takemomentum to be in the direction of, or proportional to, the velocity.

The true stress in the aether may be defined as differing fromthe Faraday-Maxwell stress by an amount representing the rate atwhich momentum is convected by the aether across stationary elementsof area. This' true stress' consists of a tension P, defined by theequation

in the direction of the component velocity of the aether in the planeof E and H, together with an e9ual pressure P in all directions atright angles to this. It is this true stress' which does work byacting on moving elements, and so transfers energy.

Cunningham gave examples of the determination of the aether-velocity in some simple cases. For a train of plane waves of lightthe stress P vanishes, and the system is one of pure convection: theaether moves as a whole in the direction of propagation of thewaves. For a moving point charge, the aether moves as if con-tinually emitted from the charge with velocity e, every elementtravelling uniformly in a straight line after emission.

1 E. Cunningham, The Principle of Relativity, Cambridge, 1914; Relativity and theElectron Theory, London, 1915

248


The continual emission from an electron of corpuscles movingwith velocity c in all directions is the chief feature also of an emissiontheory of electromagnetism published in the same year (1914) by LeighPage.l Page's work is in some ways reminiscent of ideas which hadbeen put forward by J. J. Thomson in an Adamson lecture 2 deliveredin Manchester University in 1907, when the concept of aether inmotion with velocity c was brought into relation with Thomson'sfavourite concept of moving lines of electric force.

Page proposed to regard an electron, when viewed in an inertialsystem in which it is at rest, as being like a sphere whose surfaceis studded with emittors distributed uniformly over it: each emittoris continually projecting into the surrounding space a stream ofcorpuscles, each corpuscle moving radially in a straight line with thevelocity of light: it is assumed that the emittors have no rotationrelative to the inertial system. When the electron is in motion inany way, the stream of corpuscles that have been ejected from anyone emittor at successive instants form a curve in space which, asPage showed, is a line of electric force in the field due to the movingelectron.

As in Thomson's theory,S each electron is supposed to possessits own system of lines of force, independently of other electrons,so that in general there will be as many lines of force crossing ata point as there are electrons in the field.

Owing to the motion of the electron, the direction of the lineof force at a point of space does not generally coincide with thedirection of motion of the corpuscles at the point. The componentof the electric vector d in any direction is measured by the numberof lines of force which cross unit area at right angles to this direction.The magnetic vector h is defined as in Thomson's papers, by theequation

where c is the (vector) velocity of the corpuscles at the point. Pageworked out the consequences of these assumptions for an electronof given velocity and acceleration, and found expressions for d and hat any point of the field at any time, which agreed exactly with thevalues deduced by previous investigators from the Maxwell-Lorentzequations.

In 1913 the existence of a hitherto unknown phenomenon wasdeduced by John Gaston Leathem 4 (1871-1923) of Cambridge,

1 Amer. J. Sci. xxxviii (1914), p. 169; L. Page, An Introduction to Electrodynamics, Boston,1922; L. Page and N. I. Adams, Electrodynamics, London, 1941

• Manchester Univ. Lectures, NO.8 (Manchester Univ. Press, 1908) : reprinted in theSmithsonian Report for 1908, p. 233; cf. also J.{. Thomson, Phil. Mag. xxxix (1920),p. 679, and Mem. and Proc. Manchester Lit. and Phi. Soc. !xxv (1930-1), p. 77

• cf. J. J. Thomson, Proc. Camb. Phil. Soc. xv (1909), p. 65• Proc. R.S.(A), !xxxix (1913), p. 31; cf. Larmor's Collected Papers, Vol. II, p. 72

249


from classical electrodynamics, namely, a minute mechanical forceexerted by a varying electric field on a magnetic dipole. On theassumption that the magnetism is due to molecular electric currents,he found that the ponderomotive force 1 on the dipole is

( 0 0 0) [oD]mxox + my 6Y + mz az H+ 47T m.'atwhere m. is the magnetic moment of the dipole, H is the ma~neticforce, and D is the electric displacement. The first term IS theordinary formula for the force exerted on a magnetic dipole,regarded as a polarised combination of positive and negative magnet-ism, by a field of magnetic force. The second term was unexl?ected :it represents a mechanical force exerted on a magnet by a dIsplace-ment-current, at right angles to the displacement-current and tothe magnetic moment, and proportional to the product of the twoand the sine of the angle between them.

It might seem possible to test the matter by hanging a smallmagnet horizontally between the horizontal plates of a chargedcondenser and then effecting a non-oscillatory discharge of thecondenser. If the upper plate were originally charged with positiveelectricity, the displacement-current on discharge would be upwards,and an eastwards impulse on the magnet might be looked for: theeffect would, however, be too small to be observed.

In the first quarter of the twentieth century several interestingresults in classical electrodynamics were discovered by RichardHargreaves (1853-1939). In 1908 he proved 2 that Lorentz'sfundamental equations (with c= 1) can be replaced by two integral-equations

J J (Hxc[yd~ + Hydzdx + H.dxC[Y + Exdxdt + Eyc[ydt + E.d~dtJ = 0 (I)

and

J J (Exc[yd~ + Eydzdx + E.dxc[y - Hxdxdt - Hyc[ydt - Hzdzdt)

= - J J J (pwxc[yd~dt + pWyd~dxdt + pWzdxc[ydt - pdxc[yd~). (II)

Here (t, x, y, ~) denotes the co-ordinates of a point in space-time,E is the electric and H the magnetic vector, and w is the velocityof the charge-density p. Let any closed two-dimensional manifoldS2 in the four-dimensional space-time be assigned, and let S2 bethe boundary of a three-dimensional manifold S3 in which theco-ordinates t, x, y, ~ are functions of three parameters a, f3, y, ofwhich y =0 on S2 and y < 0 in S3: so that on S2 the co-ordinates

1 In electromagnetic units I Trans. Camb. Phil. Soc. xxi (1908), p. 107250

taken over S3. In the general case the quantities occurring in theseequations are evaluated at different points of space at dijJerent times:the integrals are thus more general than the usual surface andvolume integrals.

Let S be an arbitrary closed surface in the (x, y, z) space, andlet t be expressed in terms of (x,y, z) by an arbitrary law t= t(x,y, z).Each particle is supposed to be within S at the moment when itscharge is evaluated, but since the charges on the particles areevaluated at different times, the particles need not all be withinthe closed surface at a given time. This explains why the total chargeon the particles is not fffpdxdydz, but is the triple integral on theright-hand' side of equation (II).

Hargreaves showed further that the equations

0<1> oAxEx= ----ox ot'Hx= oAt _ oAy te c.,

Oy cz

which express the electric and magnetic vectors in terms of thescalar and vector potentials, are equivalent to the single integralequation

J(Axdx + Aydy + Azdz - <1>dt)

= JJ(Hxdydz + Hydzdx + H.dxdy + Exdxdt + Eydydt + Ezdzdt).

Here we are considering a closed curve, and a surface bounded bythis curve, and we can suppose that t is expressed in terms of (x,y, z)by an arbitrary known law: then the line-integral may be under-stood to mean

J [ (Ax - <1>;:) dx + (Ay - <1>~) dy + (A. - <1>£) dzJ .In 1920 Hargreaves 1 generalised Maxwell's theory of light-

pressure 2 by showing that when a general electroma~etic field ispresent near the surface of any perfectly-reflecting (l.e. perfectly-conducting) body, which is in motion in any way, the pressure isnormal, and is measured by the difference between the magneticand electric energies (per unit volum\;) at the surface.

1 Phil. Mag. xxxix (1920), p. 662251

J cf. Vol. I, pp. 274-5


In 1922 Hargreaves 1 discovered remarkable expressions for thescalar and vector potentials of a moving electron. Let x'(t), y'(t),z'(t) specify the position of the electron at time l: let (t, x,y, z) bethe world-point for which the potentials are to be calculated: let t bethe value of t for the electron at the instant when it emits actionswhich reach the point (x,y, z) at the instant t: let x' denote x'(t),lety' denotey'Ct) and let z' denote z'(t). Then Hargreaves showedthat the scalar potential has the value

<1l = - -! ec (~ + ~ + ~ - !(2) (t - t)ox2 Oy2 OZ2 c2 ot2 ,

and the x-component of the vector-potential is

Ax = --!e(~+~+~_!~) (x'-x).ox2 Oy2 OZ2 c2 ot2

Other properties found by him were :

and

in fact, the harmonic operator applied twice to any function of1 yields a zero result, except at the source.

1 Mess. of Math. Iii (1922), p. 34

Chapter VIIITHE DISCOVERY OF MATRIX-MECHANICS

A young German named Werner Heisenberg (b. 1901), shortly aftertaking his doctor's degree in 1923 under Sommerfeld at Munich,moved to Niels Bohr's research school at Copenhagen. Here hebecame closely associated with H. A. Kramers, who in 1924 madeimportant contributions to the theory of dispersion,1 in the develop-ment of which Heisenberg took part.

The concepts employed in this theory suggested to Heisenberga new approach to the general problems of atomic theory. It willbe remembered that the quantum theory of dispersion had originatedin Ladenburg's successful translation into quantum language of theanalysis that was used in the classical theory. In place of classicalelectrons in motion within the atom, Ladenburg introduced intothe formulae transitions between stationary states: so that insteadof the atom being regarded as a Rutherford planetary system ofnucleus and electrons obeying the laws of classical dynamics, itsbehaviour with respect to incident radiation was predicted by meansof calculations based on the 'virtual orchestra.'2 The greatadvantage thereby gained depended on the circumstance that themotions of the electrons in the Rutherford planetary system werecompletely unobservable, and did not yield directly the frequenciesof the spectral lines emitted by the atom: whereas the virtualorchestra emitted radiations of the frequencies that were actuallyobserved, and thus was much more closely related to physicalexperiments.

Heisenberg saw that this idea of replacing the classical dynamicsof the Rutherford atom by formulae based on the virtual orchestracould be applied in a far wider connection. He took as his primaryaim to lay the foundations of a quantum-theoretic mechanics whichshould be based exclusively on relations between quantities that areactually observable. Previous investigators had found integralsof the classical equations of motion of the atomic system, and so hadobtained formulae for the co-ordinates and velocities of the electronsas functions of the time. These formulae Heisenberg now abandoned,on the ground that they do not represent anything that is accessibleto direct observation: and in their place he proposed to make thevirtual orchestra the central feature of atomic theory.

By taking this step, he made it no longer necessary to find firstthe classical solution of a problem and then translate it into quantumlanguage: he proved, in fact, that it is possible to translate the

1 cf. p. 203 supra • cf. p. 204 supra253


classical problem into a quantum problem at the very beginning,before solving it: that is to say, he translated the fundamentallaws of classical dynamics into a system of fundamental quantumlaws constituting what Born 1 had adumbrated under the namequantum mechanics.

He considered problems defined by differential equations, suchas, for instance, the problem of the anharmonic oscillator, which isdefined by the differential equation

d1x 1 ",,-I 0-dtl + Wo X + I~ =

and inquired how a solution of this equation can be obtained which,like a virtual orchestra, refers to a doubly-infinite aggregate oftransitions between stationary states. Let us suppose that x canbe represented by an aggregate of terms Xmn, where Xmn is associatedwith the transition between the stationary states m and n: and letthese terms be arranged in a double array thus:

The differential equation involves Xl, and therefore Heisenbergsuggested that Xl should be capable of being represented by asimIlar array

(x2) 11 (x2) 12 (x2) 13

(xl)21 (XI)22 (XB)29

(X2)s1 (Xl) 32 (X2)s3

The question is, what is the relation between the elements (Xl)mnand the elements Xmn? Now since Xmn refers to the transition betweenthe stationary states m and n, we may expect it to have the time-factor el1Ti>(m,n)l, where v(m, n) = (Wm - Wn)/h is the frequency of

1 cr. p. 204 supra254

THE DISCOVERY OF MATRIX-MECHANICS

the spectral line associated with this transition. Guided by theprincIple that the time-factor of (Xl)mn must be the same as thetime-factor of Xmn, Heisenberg suggested that the element (XI)mncould be expressed in terms of the elements of the x-array by theequation

(XI)mn = 2: Xmr Xrn.r

He went further and proposed that if x and yare two differentco-ordinates, then

(XY)mn = 2: Xmr Yrn.r

Since this involves the equation

(YX)mn = 2:Ymr Xrn,

it is evident that in general the products xy and yx are representedby different arrays: multiplication of the quantum-theoretic x and y isnot commutative.

During the month of June 1925 Heisenberg, then on holidayin the island ofHeligoland, worked out the solution of the anharmonicoscillator equation according to these ideas, and was delighted tofind that principles such as the conservation of energy could befitted into his system. In the first week of July 1925,1he wrote apaper embodying his new theory. Being then on Professor MaxBorn's staff at Gottingen, he took the MS to Born and asked himto read it, at the same time asking for leave of absence for the restof the term (which ended about I August), as he had been invitedto lecture at the Cavendish Laboratory in Cambridge. Born, whogranted the leave of absence and then read the MS., at oncereco~sed and identified the law of multiplication which Heisenberghad mtroduced for the virtual-orchestra arrays : for having attendedlectures on non-coip.mutative algebras by Rosanes in Breslau andthen discussed the subject with Qtto Toeplitz in Gottingen, Born sawthat Heisenberg's law was simply the law of multiplication of matrices.For the benefit of readers who are not acquainted with matrix-theory, some simple explanations may be given here. Consider anysquare array

(

' all all all'" a,n)all a.. a..... aln

.........

. . . . . . . . .. an, ani an.... ann

1 ZS.f. P. xxxiii (1925), p. 879 (received 29 July 1925)255


formed of ordinary (real or complex) numbers all, au, ••• , whichwe shall call the elements. This array we shall call a matrix,] andwe shall regard it as capable of undergoing operations such asaddition and multiplicatIOn-in fact, as a kind of generalisednumber, which can be represented by a single letter. The number nof rows or columns is called the order. Two matrices of the same order

A~ ("all •

• a•• ) B=C b". b•• )all a••. · a,,, bll b••. . b,,,. .• ..... .• . .am a", • · a"" b,,! b", b""

are said to be equal when their elements are equal, each to each:that is

(P, q= 1, 2, ... n).

The sum of A and B is defined to be the matrix

(

all + bllall +btl

am + bm

all +bll

a••+ b••

.a", + b",

· a1"+bl")· a,,,+b,,,

• a"" + b""

I t is denoted by A + B. Evidently addition so defined is com-mutative (that is, A + B= B+ A) and associative (that is, [A +B]+ C = A + [B+ C]). The null matrix is defined to be a matrix all ofwhose elements are zero: and the unit matrix is defined to be the matrix

(100 ... 0)o 1 0 . 0

o 0 1 . 0

o 0 0 1

If k denotes an ordinary real or complex number, the matrix

(

kall kall. . . ka1" )

katl ka... • . ka,,,............• .• .• .• . .• .•ka"l ka",. • . ka""

1 The term matrix was introduced in 1850 by J. J. Sylvester (Papers I, p. 145);but the theory was really founded by A. Cayley, J.flir Math. I (1855), p. 282 and Phil.Trans. cxlviii (1858), p. 17. Under the name linear vector operator the same idea had beendeveloped previously by Sir W. R. Hamilton: some of the more important theoremsare given in his Lectures on Quatemions (1852).

256


which is obtained by multiplying every element of the matrix A by k,is called the product of A by k, and denoted by kA.

We shall now define the multiplication of two matrices. Withthe above matrices A and B we can associate two linear substitutions

and

{

Y' = aux, + a"x. + .y. = a"x, + a••x. +

..............

.• .• .• .• .• .• .• .•~n = amX, + an.X. + .

{

U' = b,,y, + b,.y. +u. =b.,y, + b•.y. +

..............

.• .• .• .• .• .• .•Un= bn,y, + bn!J. +

+atnXn+a.nXn

+annXn

A matrix may in fact be regarded as a symbol of linear operation,representing the associated substitution.

The effect of performing these substitutions in succession isrepresented by the equations

f ~J = .(bu~n+.bl1~21 ~ : • .)x, + (bnall + b"a••+

{Un = (bn,an + bn.a" +. . .)x, + (bmall + bn.a ••+

.)X.+ .

.)x.+ .

This suggests that the product BA of the two matrices should be definedto be the matrix which has for its element in the p'h row and qlhcolumn the quantity

bp,a,q + bp.a.q + . . . + bpnanq,

so in multiplying matrices we multiply the rows of the first factor,element by element, into the columns of the second factor. Multiplica-tion so defined is always possible and unique, and satisfies theassociative law

A(BC) = (AB)C.

It does not, however, satisfy the commutative law: that is, AB andBA are in general two different matrices. The distributive lawconnecting multiplication with addition, namely,

A(B+C)=AB+AC(B+C)A=BA+CA

is always satisfied.The derivation, with respect to the time, ofa matrix whose elements

are Amn is the matrix whose elements are oAmnJot. The derivative257


of a function of a matrix with respect to the matrix of which it isa function may be defined by the equation

~~) = limk-+o {U(A+kl) - l(A)}

where 1 denotes the unit matrix: whence we readily see thatmatrix differentiation obeys the same formal laws as ordinarydifferentiation.

The above elementary account refers to matrices of finite order.The matrices which represent the co-ordinates in quantum theoryare of infinite order, and differ in some of their properties from finitematrices: in particular, the multiplication of infinite matrices isno.t necessarily associative (though the associative law holds forrow-finite matrices, i.e. those in which each row has only a finitenumber of non-zero elements).

Let us now return to Born's train of thought. He considereddynamical systems with one degree offreedom, represented classicallyby the differential equations

where q denotes the co-ordinate, p the momentum, and H theHamiltonian function: and following Heisenberg, he retained theform of these equations, but .assumed that q and p can be repre-sented by matrices, (representing a matrix by the element in its nIh

row and mlh column)q={q(n, m)e27Tiv(n,m)l}

p= {jJ(n, m)e27Tiv(n, m) I}

where v(n, m) denotes the frequency belonging to the transItIOnbetween the stationary states with the quantum numbers ~ and m, so

hv(n, m) = Wn- Wm

where Wn and Wm denote the values of the energy in the states.lSince v(n, n) = 0, the elements in the leading diagonals of the

matrices do not involve the time. This suggests the question, whatis the physical meaning of these diagonal-elements? Now in theclassical Fourier expansion of a variable, the constant term is equalto the average value (with respect to the time) of that variable in

1 Since v(m, n) = -v(n, m), it is natural to assume that the matrices are Hermitean,i.e. the elements obtained by interchanging rows and columns are conjugate complexquantities.

It is obvious that if two matrices which have equal frequencies in correspondingelements (m, n) are multiplied together, the frequency of the corresponding element(m, n) in the resulting product-matrix will again be the same as in the factor matrices.

258 -


the type of motion considered: so by the correspondence-principlewe infer that a non-temporal element Xnn of the matrix representing a variablex is to be interpreted as the average value of the variable x in the stationarystate corresponding to the quantum number n, when all phases are equallyprobable. The non-diagonal elements have not in general an equallydirect physical interpretation: but it is obvious that a knowledgeof the non-diagonal elements of the matrix representing a variablex would be necessary in order to calculate the diagonal elementsof (say) x2: and, moreover, we must not lose sight of the connectionof the non-diagonal element Xmn with the transition between thestationary states m and n : this will be referred to later.

Conversely, if the derivative of a matrix with respect to the timevanishes, the matrix must be a diagonal matrix. Hence the equationof conservation of energy (dH/dt) = 0, has in generall the con-sequence, that the matrix which represents the energy H is a diagonalmatrix. The diagonal element Hnn of this matrix represents theaverage value of H in the stationary state of the system for whichthe energy is Hn, i.e. it is Hn itself: that is, Hnn =Hn. So Bohr'sfrequency-condition may be written

( ) _H(n, n) -H(m, m)v n, m - h •

As might be expected in matrix-calculus, the products pq andqp are not equal. Now Heisenberg had given a formula, derivedoriginally by W. Kuhn 2 of Copenhagen and W. Thomas 3 of Breslau,whIch constituted a translation, into the new quantum theory, ofthe Wilson-Sommerfeld relation

Jpdq=nh:

and from this formula Born deduced that the terms in the leadingdiagonal of the matrix

pq-qp

must all be equal and must each have the value h/2rri. He couldnot, however, obtain the values of the non-diagonal elements in thematrix, though he suspected that they might all be zero.

At this stage (about the middle of July 1925) he called in thehelp of his other assistant, Pascual Jordan (b. 1902), who in a fewdays succeeded in showing from the canonical equations of motion

dq oHdt = op'

I The case of degenerate systems requires further consideration.• zS.f. P. xxxiii (1925), p. 408• Naturwiss, xiii (1925), p. 627

259


that the derivative of pq - qp with respect to the time must vanish:pq - qp must therefore be a diagonal matrix, and Born's guess wascorrect. The equation thus arrived at, namely,

hpq-qp=27Ti1

where 1 denotes the unit matrix, corresponds in the matrix-mechanics to theWilson-Sommerfeld quantum condition

Jpdq=nh

of the older quantum theory. It is known as the commutation rule. Aproof of it, much simpler than the method by which Born andJordan established it, is as follows: from the equation

q = {q(n, m)e'1T;r<n. m) '}

we have

=i(Hq-qH)

or1i8H

Hq-qH=1 ep

and therefore the equationiiOf'

fq-qf= rep (1)

is valid for f=H and f=q. Now suppose that (1) is valid for anytwo values of f, say f = a and f = b: then we can show easily thatit must be valid for f = a + b and for f = abo Since all matrix functionsdepend on repeated additions and multiplications, we conclude thatequation (1) IS valid when f is any function of Hand q. Now theequation H=H(q, p) determines p as a function ofH and q: andtherefore equation (1) is valid for f = p: that is, we have 1

Iipq-qp=-.-1t

It may be remarked that this relation could not hold if p and qwere finite matrices, since in that case the sum of the diagonalelements of pq - qp would be zero.

Born and Jordan published their discoveries in a paper which wasreceived by the editor of the Zeitschriftfiir Physik on 27 September.2

1 On the physical meaning of the commutation-rule. cr. H. A. Kramers, Physika, v(1925), p. 369 • ZS.f. P. xxxiv (1925), p. 858

260


In it they applied the theory to the case of the harmonic oscillator.For the harmonic oscillator the Hamiltonian function is

so the matrix equations of motion are

(1)

dq aH Idi ap = mP'

Hence

dp oH 2- .= - - = - mw q.dt oq

(2)

of which the solution isq = Aeiwt + Be-iWI (3)

where A and B are matrices not involving the time.Now in a matrix which has the time-factor eiWI

, in the systemof reference in which H is represented by a diagonal-matrix, theonly possible non-zero elements are those to which the time-factoreiWt belongs, i.e. those in the sub-diagonal immediately below theprincipal diagonal. Thus

A~C 0 0 0.") andB~ C f31 0 O. <f)al 0 0 0 · . 0 0 f32 o .

0 a2 0 0 · . 0 0 0 fJs .0 0 as 0 · . .

From (2) and (3)

From (3) and (5),

or

P = m 19.= imw (Aeiwt _ Be-iwl).dt

pq - qp = 2imw(AB - BA)

(5)


whence31iasfls = --.....,

2mw etc. (6)

Also H= 1.- (p' + m'w'q') mw' (A B+B A)2m

-=mw'

(T0 0 o. .)alfll +aJ3, 0 0 ..0 a2f1, + asf1s 0

wli C 0

0 0 (7)=T ")o 3 0 0 , .

o 0 5 0

....so H is a diagonal matrix, as it should be,

Moreover, q must be a Hermitean matrix, so a. and {J. mustbe complex-conjugates: and therefore from (6)

(Ii )i -t8.f11= 2- e ,mw (

2/i )! -Ill,fl,= 2mw e ,

(31i )! Ill, • • •

as= -- e2mw '

(3!i )~- -Ill"

{Js = 2mw f. , • •

where 3. 3" 3s, ••• are arbitrary real numbers. If we write 3, = y,-TT/2, from (3), (4) and (8) we have

.( Ii )!(O rt<Wt+y.)q=1 2mw e-/(Wl+y,) 0

o - y'2el(wHy,)o 0. . ..... .

and then

oy'2.r/(wt+y,)

o- y'3ei(wI+y,)

o .. ')o ...y'3ri(wt+y,), • .

o .. ,

ri(wt+ Yl)

oy'2ei(wt+y,)

o

oy'2e-i(W'+y,)

oy'3e/(wI+y,)

o ., ')o , . ,y'3ri(WI+y,) , , .

o ...


Thus the matrices which represent the co-ordinate and the momentum in theproblem of the harmonic oscillator are determined. Equation (7) showsthat the values of the energy, corresponding to the stationary states of theoscillator, are

3wn2'

5wnT

or in general(n + !)wn.

The squares of the moduli of the elements of the matrix whichrepresents the electric moment of an atom are in general the measureof its transition-probabilities. A connection is thus set up with Einstein'scoefficients A;:' and with Planck's theory of radiation.

Born and Jordan's paper reprp.sented a great advance: itcontained the formulation of matrix mechanics, the discovery ofthe commutation law, some simple applications to the harmonicand anharmonic oscillator and (In its last section) a discussion ofthe quantification of the electromagnetic field.

When the paper had been sent off to the Zeitschrift fiir Physik,Horn went for a holiday with his family to the Engadine. On hisreturn in September to Gottingen, there began a hectic time ofcollaboration with Jordan, and also by correspondence withHeisenberg, who was now in Copenhagen. Born had received andaccepted an invitation to lecture during the winter at the Massa-chusetts Institute of Technology, and was bound to leave at theend of October; but the joint paper was finished in time beforehis departure, and was received by the editor of the Zeitschrift jiirPhysik on 16 November.l

The first important result in it was a general method for thesolution of quantum-mechanical problems, analogous to the generalHamiltonian theory of classical dynamics. A canonical transjormationof the variables p, q to new variables P, Q was defined to be atransformation for which

pq-qp=PQ-QP= -in;

when this equation is satisfied then the canonical equations

transform into

d oHq= ..-dt (Jp'

dp aHdt=-aq

tiP aHat = - a~:1 M. Born, W. Heisenberg and P.Jordan, ;:'S.j. P. xxxv (1926), p. 557

263

dp_ 8Hdt - - 8q'


A general transformation which satisfies this condition is

P= SpS-lQ=SqS-l

where S is an arbitrary quantum-theoretic quantity. We then havefor any functionj(P, Q).

f(P, Q) = Sj(p, q)S-l.

The importance of canonical transformations depends on the follow-ing theorem: matrices p and q which satisfj the commutation-rule

pq -pq = - iii

and which, when substituted in H(p, q) make it a diagonal matrix, representsolutions oj the equations

dq_8Hdt - Bj)'

Therefore if we take any pair of matrices po, qo, which satisfy thecommutation-rule (for instance, we might take po, qo, to be thesolution of the problem of the harmonic oscillator), then we canreduce the problem of the integration of the canonical equationsfor a Hamiltonian H(p, q) to the following problem: To determinea matrix S such that when

the junctionH(p, q) =SH(po, qO)S-l

is a diagonal matrix. This last equation is analogous to the Hamilton'spartial differential equation of classical dynamics, and S correspondsIn some measure to the Action-function.

The next question taken up in the memoir of the three authorswas the theory of perturbations. Let a problem be defined by aHamiltonian function

H=Ho(p,q)+i\H,(p,q)+i\2H2(p,q)+ ...

and suppose that the solution is known of the problem defined bythe Hamiltonian function Ho(p, q), so that ma'trices Po, qo areknown which satisfy the commutation-rule and make Ho(Po, qo)a diagonal matrix. Then it is required to find a transformation-matrx S such that if

p = SpuS-1 and q = SqoS-lthen


will be a diagonal-matrix. It was shown how to solve this problemby successive approximations, and one of the formulae of Kramers'theory of dispersion was derived.

The theory was then extended to systems with more than onedegree of freedom. If the Hamiltonian equations are

dqk 8H dPk 8Hdt= 0Pk' dt = - 8q~'

then the commutation-rules are

{Pkql - qlPk = - iliSkl, where Ski = 1 or 0 according as k = l or k =l=lPkpt - ptpk = 0qkql - qtqk = O.

Among those who listened to Heisenberg's lectures at Cambridgein the summer of 1925 was a young research student named PaulAdrien Maurice Dirac (b. 1902), who by a different approach arrivedata theory essentially equivalent to that devised by Born and Jordan.On 7 November he sent to the Royal Society a paper 1 in whichHeisenber~'s ideas were developed in an original way.

Dirac Investigated the form of a quantum operation (denote itby d/dv) that satisfies the laws

d d ddV· (x+y) = dV.x+(JV.y

and

~(xy) = (~. x)y+x. (~.y).

It was found that the most general operation satisfying theselaws is

ddV·x=xy-yx

wherey is some other quantum variable. By considering the limit,when for large quantum numbers the quantum theory passes intothe classical theory, Dirac showed that the corresponding expressionin classical physics is

where the p's and q's are a set of canonical variables of the system.This is iii multiplied by the well-known Poisson-bracket expression 3

1 Proc. R.S.(A), cix (1925), p. 642I Note that the order of x andy is preserved in the second equation.I cf. Whittaker, Anarytical Dynamics, § 130

265


of the functions x and y. Thus, if in quantum theory we definea quantity [x,y] by the equation

-9'-yx= ili[x,y],

then [x, y] is analogous to the Poisson-bracket expressions of classicaltheory.

Now the Hamiltonian equations of classical theory,

dqr _ aHdt - apr' (r=I,2, .. n)

may be written in terms of the classical Poisson-brackets (x,y)

4!J!. = (qr H)dt " ~r = (pr, H).

The fundamental postulate on which Dirac built his theory was,that the whole of classical dynamics, so far as it can be expressed in termsof Poisson-brackets instead of derivatives, may be taken over immediatelyinto quantum theory. Thus, for any quantum-theoretic quantity x theequation of motion is

dx ] 1- = [x, H =~ (xH-Hx).dt lli

Moreover, if(r= 1, 2, ... n)

are any set of canonical variables of the classical system, then wehave for the classical Poisson-brackets the values

(qr, q.) = 0, (pr, p.) = 0, (qr, p.) = 0••

and therefore in quantum theory we must have

qrq. - q.qr = 0, prp. - p.pr = 0, qrp. - p.qr = iIi8 ••1.

Thus Dirac arrived at all the fundamental equations of Heisen-berg, Born and Jordan, without explicitly introducing matrices. Heintroduced the name q-numbers for the quantum-mechanical quantitieswhose multiplication is not in general commutative, and c-numbersfor ordinary numbers.

In a second paper 1 he applied the theory to the hydrogen atom,and obtained the formula for the Balmer spectrum. This was doneat about the same time by Pauli,2 who observed, moreover, theautomatic disappearance of certain difficulties, which had been

1 Proc. R.S.(A), ex (1926), p. 561I ZS.f. P. xxxvi (1926), p. 336

266

THE DISCOVERY OF WAVE-MECHANICS

found 1 to occur in examining the simultaneous action on thehydrogen atom of crossed electric and magnetic fields. Pauli alsogave a matrix-mechanical derivation of the Stark effect.

In the same year Heisenberg and Jordan 2 applied the theory(using the hypothesis of the rotating electron) to the problem of theanomalous Zeeman effect, and obtained Landes g-formula.3 Thefine-structure of spectral doublets in the absence of an external fieldwas also completely explained. Dirac 4 treated the Compton effectby quantum mechanics, and obtained formulae for the angulardistrIbution of the recoil electrons and the scattered radiation whichagreed completely with experiment.

Meanwhile Born had been in America since November 1925, andhad there met Norbert Wiener (b. 1894), who worked with him at theproblems of continuous spectra and aperiodic phenomena. InFebruary 1926 they wrote a joint paper 5 on the formulation of thelaws of quantification. As they pointed out, the representation ofthe quantum laws by matrices incurs serious difficulties in the caseof aperiodic phenomena. For example, in uniform rectilinearmotion, since no periods are present, the matrix which representsthe co-ordinate can have no element outside the principal diagonal.They therefore tried to generalise the quantum rules in such a wayas to cover these cases, and this they effected by developing a theoryof operators, representing a co-ordinate by a linear operator insteadof by a matrix; and they enunciated the general principle that toevery pfrysical quantity there corresponds an operator. The case of un-accelerated motion in one dimension, which has no periodic com-ponents, was shown to be as amenable to their methods as a periodicmotion.

The development of matrix mechanics in the year followingHeisenberg's first paper was amazingly rapid: and, only eightmonths from the date of his discovery, it was supplemented bya parallel theory, which will be described in the next chapter.6

1 O. Klein, ZS.j. P. xxii (1924), p. 109; W. Lenz, ZS.j. P. xxiv (1924), p. 197• ZS.j. P. xxxvii (1926), p. 2638 On the Zeeman effect, cr. also S. Goudsmit and G. E. Uhlenbeck, ZS. j. P. xxxv

(1926), p. 618; L. H. Thomas, Phil. Mag. iii (1927), p. 13.• Proc. R.S.(A), cxi (1926), p. 405• J. Math. Phys. Mass. Inst. Tech. v (1926), p. 84; ZS.j. P. xxxvi (1926), p. 174• On the state of matrix-mechanics at the end of 1926, cr. Dirac, Physical interpretation

qf quantum dynamics, Proc. R.S.(A), cxiii (I Jan. 1927), p. 621

Chapter IXTHE DISCOVERY OF WAVE-MECHANICS

IN 1926 a new movement began, which developed from de Broglie'sprinciple,! that with any particle of (relativistic) energy E andmomentum (pr, Pv, p.) there is associated a wave represented bya wave-function

De Broglie had not hitherto extended his theory to the point of intro-ducing a medium-a kind of aether-whose vibrations might beregarded as constituting the wave, and whose behaviour could bespecified by partial differential equations. A step equivalent tothis was now taken.

With the above value of ifJ, we have

f)2ifJ _ px2ax'- -/il ifJ,

so

and

so 1 a'ifJ a'ifJ a2ifJ a2ifJ _ (E' I) ifJ _ mlc'c' at2 - ax' - ayl - az2 - - c2 - P fi2 - - 7i2 ifJ,

where m denotes the mass of the particle. Thus the de Broglie wave-function satisfies the partial differential equation

1 a2ifJ a2ifJ a'ifJ a2ifJ mlc2Cs at2 = ax2 + Oy' + az2 -y ifJ·

This equation was discovered by L. de Broglie2 and others.8It is satisfied by all possible de Broglie waves belonging to theparticle, but it does not specify their passage into each other asthe particle moves-that is, it yields no mformation as to the varia-tion from moment to moment of the energy and momentum of

1 cr. p. 214 • Comptes Rendus, cIxxxiii (26 July 1926), p. 272• E. Schriidinger, Ann. d. Phys.lxxxi (1926), p. 109; O. Klein, ZS. f.P. xxxvii (1926),

p. 895, at p. 904; cr. also W. Gordon, <S.f. P. xl (1926), p. 117268


the particle. What was really wanted was a partial differentialequation for the de Broglie waves associated with a particle, whichwould yield the equations of motion of the particle by a limitingprocess similar to that by which geometrical optics is derivedfrom physical optics: a partial differential equation, in fact, whose

filiform solutions (i.e. solutions which are zero everywhere exceptat points very close to some curve in space-time) are the trajectoriesof the particle. Such an equation can be obtained, at any rate, inthe non-relativistic approximation. For from the above equationswe have at once

aS'" as'" as'" _ p2 02'"-+-+--- -.ox2 /!yS 0;;.1 E2 ot2

Substituting the values for p and E in terms of the velocity v. of theparticle, this becomes

02'" 0

2'" 0

2'" _ v

202'"ox2 + /!y2 + 0;;.1 - C' ots' (A)

This equation is not relativistically invariant, so we shall restrictourselves to the non-relativistic case of a particle m moving in afield of potential V, whose equation of energy is

!mv2+ V =€

where € is its total non-relativistic energy. The equation (A) hasfiliform solutions, each of which is 1 a null geodesic of the metricfor which the square of the element of interval is

(C4/V2) (dt)2 _ (dx)S _ (dy)2 _ (d;;.)s.

It can be shown that the null geodesics of this metric are thecurves which satisfYthe differential equations

d2x oVmdtS = - AX and similar equations iny and;;. ;

but these are the ordinary Newtonian equations of motion of theparticle. Thus the filiform solutions of equation (A) are precisely thetrajectories of the particle in the given potential field. The solution '" ofequation (A) is therefore the wave-function we are seeking.

The time factor in '" is exp {(i/Ii) Et}, so if'" now denotes the wave-function deprived of its time-factor, we have from (A)

02

'" 02

'" 01

'" _ V

S

£2oxl + /!y2 + 0;;.1 - - C4 """f2 '"mlvl

= -lil(l-vl/cll'

J E. T. Whittaker, Proc. Camb. Phil. Soc. xxiv (1928), p. 32269


Since we are considering only the non-relativistic approximation,we can replace the factor (1 - Vi let) in the denominator by unity,and thus obtain

or

This equation, which was published by Erwin Schr6dinger 1 inMarch 1926, gave the first impetus to the study of wave-mechanics.Schr6dinger's own approach, which was different from that givenabove, laid stress on a connection with the theory of Hamilton'sPrincipal Function in dynamics. This will now be considered.

He considered a [article of mass m with momentum P and totalenergy e in a field 0 force of potential V(x, y, z), so that the (non-relativist) equation of energy is

12mp2+ V =e, or P= y{2m(e- V)}.

If we associate with this moving particle a frequency v given bye= hv, and a de Broglie wave-length A given by A= hiP, then thephase-velocity of the de Broglie wave is

e h ea1=vA= - - =-----h • P y{2m(e- V)f

Take any phase-surface, or surface of constant phase, at the instantt=O; and suppose that the equation of the phase-surface at theinstant t, which has been derived (as in Huygens' Principle) bywave-propagation with the phase-velocity w from this originalphase-surface, has the equation

T(X, y, z) = t.

Then by elementary analytical geometry we have

and therefore

(or) I + (aT) 1+ (or) I =2m(e- V) •ax Oy OZ el

(1)

This equation can, however, be obtained in a very different way.1 Ann. d. Phys.(4) lxxix (1926), pp. 361, 489

270

and W is Hamilton's Principal Function. The Hamilton's equation istherefore in this case

a,,; +2~ {(o:r+ e;r + c:r}+ V(x,y, z) =0.

The Principal Function may be written

W= -€t+S(x,y, z)

where € denotes the total energy and S is Hamilton's CharacteristicFunction. The equation for S now becomes

1 {(oS)! (oS)! (oS)!}-€+2m ox + Oy + oz + V(x,y, z) =0

or

(oS)! (oS) 2 (oS) Iox + oy + oz = 2m(€- V).

Comparing this with equation (1), we see that the equation of thephase-surfaces of the de Broglie waves associated with a particle, namely,

T(X,y,Z)=t

is obtained by equating to zero the Hamilton's Principal Function f!f theparticle. Thus the theory of Hamilton's Principal Function in Dynamicscorrespondsto HU';ygens'Principle in Optics.

This investlgation so far belongs to what may be called the•geometrical optics' of the de Broglie waves; a similar equationexists in ordinary optics, the surfaces T(X, y, z) =constant being thewave-fronts, whose normals are the 'rays.' Schr6dinger now putforward the idea that the failure of classical physics to account forquantum phenomena isanalogous to the failure of geometrical optics toaccount for interference and diffraction: and he proposed to create inconnectionwith de Brogliewaves a theory analogous to Physical Optics.

Considering the stationary states of an atom, we have seen 2

that the de Broglie wave associated with an electron in an atom1 cf. Whittaker, Ana{}ltical Dynamics, § 142

271I cf. p. 216 supra


returns to the same phase when the electron completes one revolutionof an orbit belonging to a stationary state. Therefore at anyone pointof space the de Broglie disturbance in a stationary state is purelyperiodic, with the frequency v = €/h, where € is the energy of thestationary state; and it can be represented by a wave-junction ofthe form

.p = eCi£/1t)[ -t+r(%, ,I, ')J.

Differentiating this twice, we have

02.p €I (OT) t iE otToxt = -lP ox .p + 7i oxt.p,which gives, using (I),

~ + ot.p + ot.p = _ 2m(E - V) .p + iE.p (OIT + otT + OIT).oxt oyt ozt iiI Ii oxl Byl OZI

The ratio of the second term to the first term on the right-hand sideof this equation is excessively small if Ii is excessively small, as it is :so we may neglect the second term, and write

ot.p + ot.p + Ol.p = _ 2m( E - V) .1.oxl Byl OZI iiI 'f',

which is again SchrOdinger's equation for the wave-function .p of the particle.The potential function V which occurs in this equation will

possess singularities at certain points or at infinity, and these points(with others) will in general be singular points of the solution .pof the differential equation. But if .p is to represent the de Brogliewave of a stationary state, it must be free from singularities; andtherefore Schrodinger laid down the condition that the solution of thewave-equation corresponding to a stationary state must be one-valued, finiteand free from singularities even at the singularities of V (x, y, Z).l

Now it is known that the partial differential equation possesses a solutionof this char(J£ter onry for certain special values of the constant E, which areknown as the proper-values.2 These proper-values will be the onlyvalues of the energy E which the atom can have when it is in astationary state; and thus was justified the title which Schrodingergave to his first paper, Qyantification as a Problem of Proper-values.8

1 '!he conditions laid down in S~hrOdinger's earlier papers were unnecessarily stringent:on this, c.~.G. Jaffe, ZS.f. P. !xvI (1930), p. 770; R. Ii. Lan~er and N. Rosen, Phys.&v. XXXVII(1931), p. 658; E. H. Kennard, Nature, CXXVlI(1931), p. 892.

• The undesirable hybrid word eigenvalues has often been used, but proper-values,which is in every way preferable, is used in the English translation of Schrodinger'sColllCtedPapers on Wave MlChanics (Blackie and Son, 1928).

• 'In the winter of 1926, Born and Jordan having just announced a new developmentin quantum mechanics, I found more than twenty Americans in Gottingen at this fountof quantum wisdom. A year later they were at ZUrich, with SchrOdinger. A coupleof years later, Heisenberg at Leipzig, and then Dirac at Cambridge, held the Elijahmantle of quantum theory.' (K. T. Compton, Nature, cxxxix (1937), p. 222.)

272


The first problem that Schr6dinger investigated by his newmethod was that of the hydrogen atom, consisting of a proton andan electron; denoting their distance apart by r, so that the potentialenergy is - e2/r, and neglecting the motion of the nucleus, thewave-equation is (now writing E for the total energy)

We want the values of E for which solutions of this equation existthat are one-valued and everywhere finite.l To find these, weintroduce spherical-polar co-ordinates defined by

x=r sin fJ cos 4>, v=r sin fJ sin 4>, Z= r cos fJ,

and try to obtain a solution of the form

where R is a function of r alone, and Y is a function of fJ and 4>only. The wave-equation now becomes

~ ~(rzdR) +_1_ [_1_ .£ (sin fJ 8Y)+ _1 _ 82Y] +2m(E +~) =0rlR dr dr rZY sin fJ 8fJ 8fJ sin Z fJ 84>2 liZ r '

and this may be broken up into the two equations :

The latter is the well-known equation of surface harmonics,2 andthe requirements of one-valued ness, finiteness and continuity, makeit necessary that C should have the value n(n +1), where n is zeroor a positive whole number. The surface-harmonic Y(fJ, 4» is thena sum of terms of the form

Pnll(COS fJ)e±iIl'P

where JL is one of the numbers 0, 1, 2, ... n, and where Pnll(COS fJ)

1 The problem was first solved by Schrodinger, Ann. d. Phys. !xxix (1926), p. 361 ;!xxx (1926), p. 437. For a rigorous proof of the completeness of the set of proper functions,cf. T. H. Gronwall, Annals of Math. xxxii (1931), p. 47. It will of course be understoodthat the solution given above was improved tater when relativity, electron-spin, etc.were taken into account.

2 d. Whittaker and Watson, Modern Ana!ysis, § 18·31273

.,


is the Associated Legendre Function.l The differential equationfor R now becomes

~(r2dd~) + ~~(Er2 + e2r)R - n(n + I)R = O.

From the elementary Bohr theory of the hydrogen atom, we knowthat for the energy-levels whose differences give rise to the Balmerlines, the total energy E is negative. Suppose this to be the case:writing k for (e2/2/i) v( - 2m/E) and Z for (2/tz) v( - 2mE)r, theequation becomes

d2R+~ dR + {_! +~_ n(n+!Jl R=Odz2 Z dz 4 Z Z2 f

of which the solution that remains finite as r~ 00 isI-Wk,nH(Z)Z

where Wk, n+l(Z) is the confluent hypergeometric function.2 Thus!f;must be a constant multiple of

~ Wk.nH(Z)Pn!'(COSB)e±ip<P.

It is, however, necessary also that!f; should be finite at r = O. This con-dition requires that the asymptotic expansion OfWk,n+l(Z),3 namely,

Wk,nH(Z)- e-lzzf 1 + (n+t);; (k-t)2

+ {(n+ t)2 - (k- t)2} {(n + t)2 - (k - ~)2} ]2 !Z2 + ...

should terminate; which evidently can happen only if

±(n+t)=k-t, or k-·~, or k-tthat is, k must be a whole number: and on account of the factor z\it must be a positive whole number: thus

must be a positive whole number: so

me4

E= -21i.2f2 where k= 1,2,3,4, ....

1 d. Whittaker and Watson, Modern Anarysis, § 15·5I cf. Whittaker and Watson, Modern Anarysis, § 16·1I cf. Whittaker and Watson, Modern Anarysis, § 16·3

274


These values of E are precise!y'"the energy levels of the stationary orbits inBohr's theory,. to each value oj E there corresponds afinite number of particularsolutions,. and thus we obtain the line spectrum of the hydrogen atom.It is evident from this equation for E that k must be identified with thetotal quantum number.

The question may now be raised as to the physical significanceof the wave function if. Schrodinger at first, in a paper 1 received18 March 1926, supposed that if if* denotes the complex quantityconjugate to if, then the space-density of electricity is given by thereal part of if oif*jot. In a paper2 received on 10 May, however,he corrected this to ifif*, basmg his new result on the fact that theintegral of tPif* taken over all space is, like the charge, constant in time.This interpretation of if was, however, soon again modified. Thenotion of waves which do not transmit energy or momentum, butwhich determine probability, had become familiar to theoreticalphysicists from the Bohr-Kramers-Slater theory of 1924: and in apaper 3 received 25 June, and one received 21 July,4 both dealingwith the treatment of collisions by wave-mechanics, Max Born adoptedthis concep'tion, and proposed that ifif* should be interpreted in termsof probabIlity; to be precise, that ifif* dx dy dz should be taken to bethe probability that an electron is in the infinitesimal volume-elementdx 4Y dz. This interpretation was soon universally accepted.

It is convenient to normalise the wave-functions by the condition

JifntPn*dT= 1

where dT denotes the element of volume and the integration is takenover all space. As an example of normalisation, consider thefundamental state of the hydrogen atom, for which k = 1, n = 0,p.=0. The wave-function is now some multiple of (l/r)W1, l(Z), andsince W l' l (z) = rl'Z, this gives for the wave-function a multiple ofri', or e-(rja) where a is the radius of the first circular orbit in Bohr'stheory of the hydrogen atom. The wave-function may thereforebe written

rif= Cea where C is a constant.

The normalisation condition is

JOO J71 J271 2r

C2 0 0 0 e-" r2 sin 8 drd8dep= 1

which gives

1 Ann. d. Phys.(4) Ixxix (1926), p. 734, equation (36)2 Ann. d. Phys.(4) lxxx, p. 437, note on p. 476• <S.f. P. xxxvii (1926), p. 863• <S.f. P. xxxviii (1926), p. 803

275


and therefore the normalised wave-junction of the fundamental state of thehydrogen atom is

• , r.1. -. -I --'t' = 1T a e a.

Of course this is to be multiplied by the time-factor e-(f1h)E", whereE1 is the energy of the state.

Similarly the normalised wave-function of the hydrogen atomin the state k = 2, n = 0, p- = 0, is

l/J=2-i1T-ia-ie-ia(;a _I)e-~EJ'

where E2 is the energy of the state; and the normalised wave-functionof the hydrogen atom in the state k = 2, n = I, p- = 0, is

l/J= (321Ta5)-ie-i, r cos 8 e-~E.'

where Es is the energy of the state.Now suppose that to two different proper-values of the total

energy, say En and Em, there correspond respectively solutions l/Jnand l/Jm of the wave equation, so that

V2l/Jn + ~": (En - V)l/Jn = 0

V2l/Jm +~": (Em - V)l/Jm = O.

Multiplying these equations by l/Jm and l/Jn respectively, subtractingand integrating over all space, we have

The first integral may by Green's theorem be transformed into asurface-integral taken over the surface at infinity, which vanishes;and thus we have

that is, two wave-junctions l/Jm and l/Jn, corresponding to different valuesof E, are orthogonal to each other. •

Since in this equation we can replace l/Jm by its complex-conjugatel/Jm*, we can combine it with the normalisation-equation into theequation

Jl/Jnl/Jm*dT=3mn

276


where 8m n = 1 or 0 according as m and n are the same or differentnumbers.! •

In the case of the hydrogen atom. a single proper-value of theenergy, say

me'E= - 2/Pk2

where k is a definite whole number, corresponds to many wave-functions

n and I-' being able to take different whole-number values. Thisphenomenon-the correspondence of several different stationarystates to the same proper-value of the energy-is called degeneracy,a term already introduced in Chapter III. It is known from thegeneral theory of partial differential equations that we can alwayschoose linear combinations of the wave-functions which belon~ tothe same value of the energy, in such a way that these combinatIOnsare orthogonal among themselves 2 (and, of course, orthogonal tothe wave-functions which belong to all other proper-values of theenergy). The degeneracy of the hydrogen atom, so far as it is dueto the fact that many different values of n correspond to the samevalue of k, can be removed, exactly as in Sommerfeld's theory of1915, by taking account of the relativist increase of mass withvelocity: while the degeneracy, so far as it is due to the choiceof different values of 1-', can be removed by applying a magneticfield, as in Sommerfeld and Debye's theory of 1916. It is evidentthat n is essentially the azimuthal quantum number, and that I-' isthe magnetic quantum number.

Now consider the case when the total energy E of the electronin the hydrogen atom is positive. v'( - E) is now imaginary, andall the confluent hypergeometric functions which are solutions ofthe differential equation for rR remain finite as r-+oo, so R tendsto zero as r-+oo. Moreover, at least one solution of the differentialequation exists which is finite at r = O. Thus every positive value of Eis a proper-value, to which correspond wave-functions possessingazimuthal and magnetic quantum numbers, in the same way as thediscrete wave-functions. Physically, this case corresponds to thecomplete ionisation of the hydrogen atom, or to the reverse process,i.e. the capture of free electrons.

1 If the proper-values of Schrodin~er's wave-equation have also a continuous spectrum,the above equations persist in a modIfied form, for which, see H. Weyl, Math. Ann. !xviii(1910), p. 220, and Glitt. Nach. (1910), p. 442; E. Fues, Ann. d. Phys. !xxxi (1926), p. 281.

• There is a certain degree of arbitrariness in the choice, the arbitrariness beingrepresented by an orthogonal transformation which may be performed on the wave-functions of the set.


In a further paper, 1 Schrodinger extended his theory by find-ing the wave-equatIOn for problems more general than those con-sidered hitherto. Still restricting ourselves for simplicity to themotion of a single particle, suppose that its Hamiltonian FunctionH( ql, ql, q3, (h PI, P3) is the sum of (I) a kinetic energy representedby a ~enera quadratic form T in the momenta (Ph P2, P3), withcoefficIents whIch depend on the co-ordinates (ql, ql, q3), and (2)a potential energy V which depends only on (ql, qh q3)' Then thewave-equation may be derived by a process similar to that followedalready in the case of the particle referred to Cartesian co-ordinates.It will be evident from what was there proved that the wave-functionmust be of the form

~r/J = jJ. -E/+S(1".' ••}}

where S(x, y, z) is Hamilton's Characteristic Function, satisfyingthe equation

( as aS oS)H qh q" q" ~, ~-'!i""- = E.uql uql Uq3

Differentiating the expression for r/J, we have

02r/J _ I(oS)1 i 02Soqrl - -lil oqr r/J+ Ii oq,2 r/J.

As before, the second term on the right is very small compared withthe first, and may be neglected. Thus

(Ii a Ii a Ii 0) ( as as oS)T ql, ql, q3, .,..~, •. ~, .,..~ r/J= T ql, ql, q3, !l' , i)-' ~ cP,t uql t uql t uq3 Uql uql uq~

and therefore

or as we may write it

This is the extension of the wave-equation, applicable to the more generalform of the Hamiltonian Function.

Sinceor/J iai= -lfEr/J,

1 Ann. d. Phys.(4) !xxix (1926), p. 734278

THE DISCOVERY of WAVE-MECHANtCS

we have

H (q, ~ ;q) .p= iii ~f.This equation, which does not involve E explicitly, may be calledthe general wave-equation.l It is clearly the equation that would beobtained by replacing p by Ii/i %q and E by -li/i %t in the equationof energy, and then operating on .p.

It would seem from the foregoing derivations that Schrodinger'swave-equation is connected with Hamilton's Principal and Character-istic Functions by equations which are not exact, but only approxi-mate, involving the neglect of powers of Ii. I t was shown 2 manyyears after the ~~s~overyof the .wa~e-~quation th.at this conclu~ion ismcorrect: Schrodmger's equatIOn ISngorously equwalem:to HamIlton'spartial differential equation for the Principal Function, provided thesymbols are understood in a certain way.

Considering for simplicity a conservative system with one degreeof freedom, let the co-ordinate at the instant t be q, and let Q bethe value of the co-ordinate at a previous instant T. Then, asHamilton showed, there exists in classical dynamics a PrincipalFunction W(q, Q, t - T), which has the properties

oW(q, Q, t-T) _poq

oW (q, Q, t - T) = _ PoQ

oW(q, Q.., t-T) =-Hoi

where p and P are the values of the momentum at the instants t andT respectively, and H is the Hamiltonian Function.

Now consider the quantum-mechanical problem which is specifiedby the Hamiltonian H(q, P). As explained in Chapter VIII, thevariables q and p are no longer ordinary algebraic quantities, butare non-commuting variables satisfying the commutation-rule

Iipq - qp= •.1

The quantity Q, which represents the co-ordinate at the instantT, also does not commute with q or p. A function of q and Qis saidto be well-ordered, when it is arranged (as of course it can be, by

1 With regard to the general character of Schrodinger's theory, Heisenberg r<S.f. P.xxxviii (1926), p. 411, at p. 412] said' So far as I can see, Schrodinger's procedure doesnot represent a consistent wave-theory of matter in de Broglie's sense. The necessityfor waves in space off dimensions (for a system with f degrees of freedom), and thedependence of the wave-velocity on the mutual potential encrgy of particles, indicatesa loan from the conceptions of the corpuscular theory.'

• E. T. Whittaker, Proc. R.S. Edin. lxi (1940), p. 1

279


uSe of the commutation-rules) as a sum of terms, each of the formf(q)g(Q), the factors being in this order.l Then it can be shownthat there exists a well-ordered function U (q, Q, t - T), which formallysatisfies exactly the same equations as Hamilton's Principal Function,namely,2

aUaq =p,

aUaQ= -P,

U may be called the quantum-mechanical Principal Function. Althou~hit satisfies the same equations as Hamilton's Principal Function, Itsexpression in terms of the variables q, Q, t - T, is qUItedifferent fromthe expression of the classical function W: the reason being thatthe above equations for U are true only on the understanding thatall the quantities occurring in them are well-ordered in q and Q:but on substituting the well-ordered expression for p in the Hamil-tonian H(q, P), we shall need to invert the order of factors in manyterms, by use of the commutation-rules, in order to reduce H toa well-ordered function of q and Q; and this introduces new termswhich do not occur in the classical equations. This explains why,although the equations of quantum mechanics are formally identicalwith those of classical mechanics, the solutions in the two cases arealtogether different.

Now introduce a function R(q, Q, t- T), which is obtained bytaking the well-ordered function U and replacing the non-commutingvariables q and Q by ordinary algebraic quantities q and Q Itmay be called the Third Principal Function. By what has been said,it is quite different from the classical Principal Function W belong-ing to the same Hamiltonian. Then if we write

i!f;(q, Q, t - T) = e jiR(g, Q,l-T)

it may be shown that !{J (q, Q, t- T) satisfies SchrOdinger's differentialequation for the wave1"unction belonging to the Hamiltonian H(q, p). Thusthe relation between Principal Function and SchrOdinger's wave1"unction isrigorous, not requiring the neglect of any powers of Ii, provided the PrincipalFunction is understood to be the Third Principal Function R, and not Hamilton'sclassical Principal Function W.

We must now consider how the proper-values and wave-functionsof Schrodinger's equation are to be determined. The followingmethod for determming them, at least approximately, was givenin 1926 by G. Wentzel,3 H. M. Kramers4 and L. Brillouin.5

1 This conception is due to Jordan, ZS.j. P. xxxviii (1926), p. 513• The first two of these equations were given substantially by Dirac, Phys. Zeits.

Sowjetunion, iii (1933), p. 64.• ZS.j. P. xxxviii (1926), p. 518 • ZS.j. P. xxxix (1926), p. 828• Comptes Rendus, c1xxxiii (1926), p. 24; J. de phys. et ie rad. vii (1926), p. 353. The

method was to a great extent anticipated by H. Jeffreys, Proc. L.M.S.(2) xxiii (1925),p. 428; cr. also R. E. Langer, Bull. Amer. M.S. xl (1934), p. 545, who indicated a correc-tion in Jeffreys's paper.

280


Consider a one-dimensional oscillatory motion defined by aHamiltonian

P'H=-+ V(q).2m

The Schrodinger equation for the wave-function ifi isIt, d'ifi2m dq' + (E - :') ifi=0

where E is the proper-value of the energy. Suppose that the rangein which the particle would oscillate according to classical dynamics,namely, that defined by E - V = 0, is q1.L.q.L. q" so that ql and q,are roots of the equation V (q) =E; within this range put

so the wave-equation becomes

If V, and therefore " were constant, the solution would be of theform

ifi= Constant x sin ('q+ Constant):

this suggests that even when , is variable, we should try within therange ql.L. q.L.q2 to represent ifi by a sine-curve of slowly varyingamplitude and wave-length, say

ifi= A(q) sin S(q).

Substituting in the wave-equation, we have

(A" - AS" + A'S) sin S + (2A'S' + AS") cos S = O.

Let us try to satisfy this equation by imposing on A and S the twoconditions

A" - AS's + A'2 = 0, 2A'S' + AS" =O.

Now suppose that A' varies so slowly that IA"I is very small com-pared to '2:AI (it is at this stage that the approximation enters) :then we may neglect the first term in the former of these equations,which now becomes


giving

S = Jq 'dq + a, where a is a constant.q.

The second condition2A'S' +AS" =0

gives at onceA =e(S')-i where e is a constant

or

The approximate expression for if1is therefore

if1=eW-1 sin U:,'dq+ a).

Attention must be given, however, to the behaviour of if1 at thepoints for which q = ql and q = q2, say P and Q respectively, since atthese points' vanishes, and A" cannot be neglected in comparisonwith A'2. A closer consideration of this difficulty 1 shows that thefunction if1 must have at P the phase rr/4, so the approximateexpression of if1in the range PQ is

if1=em-1 sin (J: 'dq+~).

This is the Wentzel-Kramers-Brillouin approximate solution of Schrb'dinger'sequation.

Similarly the function if1 must have at Q the phase 3rr/4: sowhen q=q2,

where n is a whole number, representing the number of nodes com-prised between P and Q; that is to say,

This equation determines the proper-values E oj the energy, in the Wentzel-Kramers-Brillouin approximation. •

Between P and Q the graph of if1 oscillates like a sine-curve,whereas to the left ofP and to the right ofQit decreases exponentially.

The Wentzel-Kramers-Brillouin approximation throws light onthe connection between wave-mechamcs and the formula of quanti-fication enunciated by Wilson and Sommerfeld in the older quantum

1 For which ef. E. Persico, N. Cimento, xv (1938), p. 133282


theory, namely that fpdq, where the integration is taken round anorbit, is a multiple of Planck's constant h.

For the momentum, calculated according to classical dynamicsfrom the equation

p2 +V(q) =E2m

ISp= ±{2m(E- V)}l= ±Ii~(q)

where the upper sign must be taken for the semi-oscillation fromql to q2, and the lower sign from q2 to ql. Thus

J Jq, Jq,pdq round the orbit =2 pdq=21i ~(q)dq,

q, ql

and thus the Wentzel-Kramers-Brillouin condition becomes

21liJpdq = (n + t) 7T

or

JPdq= (n+!)h,

which is the Wilson-Sommerfeld condition, completed by the termt which appears in the more accurate theory. ~

So far we have connected Schrodinger's wave-equation only withthe stationary states of the atom, to which correspond proper-valuesof the total energy. We shal} now consider more general states.

In Bohr's theory a stationary state meant a particular kind oforbital motion, so that an atom could be in only one stationarystate at one time. In Schrodinger's theory, on the other hand,the stationary states correspond to different solutions of a linearpartial differential equation, and therefore the various stationarystates can be superposed just as overtones can be superposed on thefundamental tone of a violin string. We have to consider what isthe physical interpretation of this superposition.

Suppose that plane-polarised light whose vibrations are in adirection a passes through a Nicol prism, which resolves it intovibrations in directions f3 and y respectively parallel and perpen-dicular to the plane of polarisation of the Nicol, and permits onlythe former to pass through. Fixing our attention on a single photon,which is initially polarised in the direction a, we can regard this stateas a superposition of two states, namely, that of polarisation in thedirection f3 and polarisation in the direction y.l We can speak of the

1 The superposition here spoken of must not be confused with superposition in classicalmechanics; in classical mechanics, the superposition of a certain state of vibration onitself gives a vibration of twice the amplitude, but in quantum-mechanics it gives merelythe same state of vibration.


probability of the photon being in either of the states fl and y, theseprobabilities being such as to account for the observed intensityof light polarised in the direction fl, which emerges from theNicol.

This connection between superposition and probability will nowbe extended to wave-functions. Consider two stationary states of anatom, for which the energy has the proper-values E1 and Es re-spectively. Let rpi and rps be the corresponding normalised solutionsof the wave-equation, with their appropriate time-factors r(iH,/II)' andr(iH./1I)', and let rps= CIrpl + csrps where CI and Cs are arbitrary complexnumbers. Then the state or physical situation represented by rps issaid to be formed by the superposition of the states represented by rpiand rps. rpa of course satisfies the general wave-equatIOn

H(q, } ~)rp=ili~.

Let us find the condition that mtlst be satisfied by CI and c_ if rps isnormalised. We have

1= Jrpsrps*dT= J (Clrpi +C2rp2) (CI*rpl* +cs*rps*)dT

=CIC1* JrpIrpl*dT + C2C2* Jrpsrp2*dT

-= ICIls + Icsl-·

This equation can be interpreted to mean that there is an uncertaintyas to the value which would be found by a measurement of the energy of theatom, either of the values EI or Es being possible: and their respectiveprobabilities are ICIls and Ic_I-.

More generally, if rpo, rpl, rp2, •.• are the normalised wave-functions (with their appropriate time-factors) belonging respectivelyto the proper-values Eo, EI, E2, ••• (supposed for simplicity to beall different) of the energy of the atom, and if a normalIsed solutionrp of the general wave-equation is expanded in the form

rp = corpo + Cirpi + C2rp2 + •.. ,

then in the physical situation defined by the wave-Junction rp, the probabilitythat a measurement of the energy will yield the value En is ICn /s. On accountof the relation Jrpnrpm*dT = Dmn, it is seen at once that"the value of thecoefficient Cn is frprpn *dT.

The equations which we have found lead to a certain connection I

with the classical electromagnetic theory of the emission of radiation.Suppose that we consider an atom in a stationary state, so that thewave-function rp involves the time through a factor (say) e2fTi,'•

1 E. Fermi, &nd. Lincei, v (May 1927), p. 795284


Then r/J* will have the time-factor e-2'7Tivt, so in the product r/Jr/J* thesetime-factors will destroy each other: that is to say, the distributionof electric charge in the atom, and therefore its electric moment,will not vary with the time: and therefore according to the classicaltheory, the atom, when it is in a stationary state, will not emit radiation.

Suppose next, however, that the atom is not in a pure stationarystate, but is in a state which is represented by the superposition oftwo stationary states, for which the wave-function has the time-factors e27Tiv,t and e27Tiv,t respectively, so that

where A and B do not involve the time. Then we have

= AA * + BB* + AB*e2'7Ti(v,-.,) 1+ A *Be-2'7Ti('1-'~)I,

and hence the electric moment of the atom, which depends on anintegral involving t only in the combination r/Jr/J*, will be periodic,with frequency (VI- V2), and consequently the atom, according to theclassical theory, will emit radiation of frequency (VI-VI). This radia-tion will continue until the consequent exhaustion of energy hasagain reduced the atom to a single pure stationary state.

If E1 and E2 are the energies associated with the two stationarystates, then E1 = hVI and E2 = hV2, so the radiation is of frequency(I/h) (E1 - E2), just as in Bohr's theory, and this result has now beenobtained by what are essentially classical methods-the classicaltheory of the solutions of partial differential equations-withoutdoing violence to the electromagnetic theory of light.

Schrodinger did not, at the outset of his researches, suspect anyconnection between his theory and the theory of matrix-mechanics.lHe now, however,2 showed that the two theories are actuallyequivalent. In the first place, the commutation-rules of matrix-mechanics, which in the case of systems with one degree of freedomreduce to

qp-pq=in,

become obvious identities if we write p = (Ii/i) %q, since

q(~ ~) r/J - ~ ~ (qr/J) = ilir/J.l oq l oq

To any physical quantity ~(q, p) we can correlate a differential, He says so in Ann. d. Phys.(4) lxxix (1926), p. 734; 'I naturally knew about his

[Heisenberg's] theory, but was discouraged by what appeared to me as very difficultmethods of transcendental algebra.'

• loc. cit.: cf. also C. Eckart, Phys. Rev. xxviii (1926), p. 711285

where


operator '(q, Ii/i.dldq). Let us consider the operation of this quantityon a wave-function tfl(q). As we have seen, it is possible to expandtfl(q) as a series of normalised orthogonal wave-functions, in the form

tfl(q)=Cltfll+Cltfll+' ••

Cn=J tfl(q) tfln* (q)dq.

If {(q, lili dldq)tfl(q) has a corresponding expression

{(q, ~ ~)tfl(q)=C/tfll+CI'tfll+""

then evidently we must have

Cn'= Jtfln*(qg(q, ~ ~)tfl(q)dq

-= Jtfln*(qg(q, ~ ~)(Cltfll+Cltfl.+ ••• )dq

-= efllc, + enoCl+ enoCl+ ...

where

But this equation shows that the column-vector (c/, cl', ca', • • .) isderived from the column-vector (Ch c., C., ••. ) by operating on itwith a matrix whose element in the nth row and -rh column is enr.Thus the physical quantity '(q, P), or the operator {(q, lili dldq), may becorrelated to the matrix whose element in the nth row and -rh column is

in the sense that the performance of the operator on any wave-function tfl(q)is equivalent to the operation of the matrix (enr) on the column-vector of thecoefficients Cr, which express tfl(q) in terms of the normalised orthogonal wave-functions tfll(q), tfl2(q), ..•.

We observe that the matrix thus found for the physical quantity{(q, p) is specially associated with the set of norm-alised orthogonalwave-functions tfln(q) which belong to a particular Schr6dinger'sequation

H(q, ~ o~)tfl = ili?J.

We may call this set of wave-functions the basis of the matrix.286


In order to establish completely the identity of this correlation(between physical quantities and matrices) with matrix-mechanics,it is necessary to prove some other mathematical theorems, e.g. thatthe result of operating with the product of two operators '(q, /i/i d/dq)and 'Y}(q, /i/i d/dq) on a wave-function is the same as the result ofoperating on the column of coefficients of the wave-function with thematrix-product of the matrices corresponding to , and 'Y}. Theproofs will be omitted here.

Suppose now that the basis of the matrix representing '(q, p) isthe set of wave-functions belonging to the Hamiltonian function'(q, p), so it is the set of wave-functions of the Schrodinger equation

Then the matrix-elements are given by

e"r= J~,,*(qg(q, ~ ~)~r(q)dq

= J~,,*(q)Er~r(q)dq

where Er is the proper-value of the energy corresponding to the wave-function ~r(q). Thus

e"r=ErSr"

so the matrix is now a diagonal matrix whose elements are the proper-valuesof the energyfor the ScltrOdinger equation

,( q,~ :q)~= i/i oft.Thus the physical quantity '(q, p), when expressed as a matrix in terms ofthis basis, is a diagonal matrix whose elements are its proper-values.

Therefore the problem as formulated in matrix-mechanics,namely, to reduce the matrixfor '(q, P) to a diagonal matrix, is solved whenwe have found a solution of the problem as formulated in wave-mechanics, namely, tofind the proper values of E and the corresponding wave-functions for the Schriidinger equation

'(q, ~ :q)~=E~.

The basis constituted by the wave-functions of this equation enablesus, by the above formulae, to calculate the matrices that representq and p. Thus matrix-mechanics and wave-mechanics are equivalent.

Let us now investigate more closely the physical meaning of the287


matrix-elements. Consider the electric moment MPQ of the classicaloscillator which would emit the radiation associated with a transitionfrom a stationary state P to a stationary state Q of the atom: letM denote 2 er, the electric moment of the atom, expressed in termsof the co-ordinates of the electrons: and let iflp and iflQ be the wave-functions of the states P and Q. We may expect that MpQ willdepend in some way on M, iflp and iflQ. Now we know that the time-factor in MpQ is e-2fTivpQt, where VPQ = (llh) (Ep - EQ), the time-factor iniflp is riEpt/li, the time-factor in iflQ is riEQt/li, and M has no time-factor. We therefore expect that the expression for MpQ will involvethe product iflpMiflQ*, where iflQ* is the complex-conjugate of iflQ. Theexplicit expression of the co-ordinates has to be removed from this,which can evidently be done by integrating over space. Thus weobtain the expression

JiflpMiflQ*d'T,

and this matrix-element we shall identify with MpQ, the electric momentof the classical oscillator which would emit the radiation associated withjhetransition P-+Q. This identification is justified by comparison withthe results of experiments.

Let us now illustrate the equivalence of matrix-mechanics andwave-mechanics by considering the harmonic oscillator in onedimension, for which the Hamiltonian function is

where m and ware constants. The Schr6dinger wave-equation is

where E is the total energy of the motion.

Writing q= (1i/2mw)IR;, this may be written

d2ifl + (~ -1R;2)ifl = 0dR;2 liw

which is the well-known differential equation of the parabolic-cylinder functions.l It has a solution which is finite for all realvalues of R; only when

E-=n+tliwI cf. Whittaker and Watson, Modern Anarysis, § 16·5

288


where n is a whole number,! and the solution is then aconstant multiple of the parabolic-cylinder function Dn(z),which is a polynomial multiplied bye-pl. Thus the proper-valuesofR are

E= (n+t)nw where n=O, 1,2,3, .

and the corresponding wave-functions are

where A IS a constant to be determined by the normalisingcondition

Since

this gives

where an is real; so the normalised wave-functions are

(mw)l ( J2mw) .lfn(q) = TTn (n !)-iDn q T e'an.

The definition of the element in the l'h row and nIh columnof the matrix representing the co-ordinate q, in terms of the wave-functions, is

Using the known properties of the parabolic-cylinder functions

and

1 The Wilson-Sommerfeld quantum condition would lead us to expect the Actionto be a whole multiple of h, that is, energy x period to be a multiple of h, or E equalto a multiple of nw. The occurrence of (n +!) instead of n is characteristic of the quantum-mechanical solution.

289


the value of the matrix-element can readily be calculated, and weobtain (inserting the time-factor)

. e-i(wI+fJ\)ei(Wl+fJ\) . 2ie-i(wt+fJ.)

2Iei(wI+fJ,) )in agreement with the value found in chapter VIII.

Similarly the element in [lh row and nIh column of the matrixrepresenting the momentum p is expressed in terms of the wave·functions by

which when evaluated gives the same value as was found in ChapterVIII.

I t was shown by C. Eckart 1 by use of the wave-equation that ingeneral problems, an element of the matrix representing a co-ordinatecan be expressed as a series of the form

x(m, n) =xo(m, n) +hXl(m, n) +h2X2(m, n) + ...whereas n-+ro, Xo(m, n) tends to the coefficient of the (m-n)lhharmonic in the Fourier expansion of the co-ordinate as determinedby the classical theory of the motion. This is in accord with Bohr'scorrespondence principle.

In a later paper 2 he illustrated this theorem by calculating thematrix-elements of the radius vector for the hydrogen atom, andshowing that their limiting values, as h-+O and the quantum numberstend to infinity, coincide with the Fourier coefficients of the classicalmotion. In particular, the diagonal terms of the matrix tend tothe constant term of the Fourier expansion.

In a paper entitled' The Continuous Transition from Micro-to Macro-mechanics,' 3 Schrodinger showed how to construct for theharmonic oscillator a wave-packet, i.e. a group of wave-functions ofhigh-quantum number and small quantum-number-differencessuch that the electric density is very nearly concentrated at a singlepoint. He proved that the differential equation ofthe wave-functionsof the harmonic oscillator, namely,

.~ (-1i2 f)21/1+ m2w2q2) = iii ~~ •2m aq2 at

is satisfied by

(. _iwI J(2mw) mwq2 k2 k2 _.iWI)1/1(/)=exp -itwt+ke q -y - -27i" - :2 - 2e

\ Proc. Nat. Ac. Sci. xii (1926), p. 684; cf. P. Debye, Phys. ZS. xxviii (1927), p. 170• ZS.f. P. xlviii (1928), p. 295 • Naturwiss, xxviii (1926), p. 664

290


whenk is any constant. This gives

./../.* = [_ {q - (2/i/mw). k cos wt }2]'f"f' exp /i/mw .

Theprobability that at the instant t the co-ordinate q lies in the rangefromq to q + dq is r/Jr/J* dq: and clearly if Ii. is very small (as it is) and kisverylarge, while IWw is finite, say equal to E, so (21i./mw)!k becomes(2E/mw2)', then r/Jr/J* is negligibly small except when the numerator oftheargument of the exponential is approximately zero; that is, thewave-function represents a wave-packet whose position at time t isgivenby

q = J (~:2)cos wt,

which is precisely the equation determining the position of the particle in theclassicalproblem when the energy is E.

The quantum-mechanical theory of collisions was founded in twopapersof 1926 by Max Born.l A classical treatment of the problemhad been given in 1911 by Rutherford,2 whose investigation of thescattering of particles by a Coulomb field will first be described.

Consider the scattering of a narrow beam of a-rays by a sheetof metal foil on which the beam impinges at right angles. Thescattered particles afterwards strike a screen of zinc sulphide, andthe number of scintillations on each square millimetre of the screenisobserved.

Let the number of atoms per unit volume in the foil be n, andlet q be the thickness of the foil, so there are nq atoms per unit cross-section of the beam. The area of that part of this umt cross-sectionwhich is within a distance B of the centre of an atom is therefore7TnqB2,and out of a incident particles the number that are scatteredat distances between Band B + dB is 27TanqBdB. Now accordingto classical dynamics the path of a particle is a hyperbola havingthe centre of the atom as Its external focus: and since the perpen-dicular from the focus on an asymptote of the hyperbola IS equalto the minor semi-axis, it follows that B is the minor semi-axis ofthe hyperbola. The angle () through which the particle is scatteredis equal to the external angle between the asymptotes of the hyper-bola, so

~B Bcot 2 = A

where A is the major semi-axis. The number of particles scatteredin the annulus between angles () and B + dB is therefore

nanqA 2 cot t() cosec2 t() dB.

1 ZS.f P. xxxvii (1926), p. 863; xxxviii (1926), p. 803• Phil. Mag. xxi (1911), p. 669

291


These fall on the screen, which is supposed to be at a distance lbeyond the foil, on an annulus of area

d( TTll tanl8) or 2TTlI tan 8 secl8 d8.

The number of scintillations observed per unit area is therefore

anqA I cosS8cos i8 anqA I cosS8211 sin 8 sins i8 or 412 sin4 i8 .

Writing r = distance of the scintillation from the atom = llcos 8,we see that the number of scintillations per unit area perpendicularto ris

anqAI4r2 sin4i8·

Now if E be the charge at the centre, e the charge and m the massof the incident particle, and Vo the velocity at infinity, the usualformulae of hyperbolic motion give A = eElmvo2: so the numberof scintillations per unit area perpendicular to v is

anqe2E24mSvo4 rl sin4!lf

Thus if the beam of incident particles is of such intensity that oneparticle crosses unit area in unit time, and the beam falls on asingle scattering centre, then the probability that in unit timea particle should be scattered into the element of solid angle sinB dBdep(whose 8, ep are spherical-polar co-ordinates, the polar axis beingthe direction of the incident particle) is

e2Es4 . i8 sin 8 d8dep.

m2v04 s1n4

This formula, given in Rutherford's paper, was verified experi-mentally. It afforded a means of determining the charge E on thenucleus of an atom, and led to the conclusion that the nuclear chargeis equal to the electronic charge multiplied by the atomic number.

The quantum-mechanical treatment of collision problemsoriginated in the two papers of Max Born already referred to.Consider the scattering of a beam of particles by a fixed centreof force, the potential energy of one particle at a distance r fromthe scattering centre being V(r). Schr6dinger's wave-equation fora particle in presence of the scattering centre is

\l2t/J+ {kl- ~~V(r) }t/J=O

292

(I)

(2)

TtlE DI~COVERY OF WAvE-MEctIANtcS

where k = mvo/fi, m being the mass and Vo the initial velocity of t1t~incident particles. The general solution of the equation

'ij2,p+k2,p=f(x,y, z)

wheref is a given function, is known to be

,p= C(x,y, z) - LJe;p f(x',y', Z')dT'

where C(x, y, z) is the general solution of the equation V2,p+ kl,p= 0,and where p2= (x' - X)2+ (y' - y)2 + (z' - Z)2, and dT' is the volume-element dx'dy'dz', the integration being extended over all space.So a solution of the wave-equation for the particle is a solution ofthe integral-equation

m JeikP,p(x, y, z) = C(x, y, z) - 27Tfil P V(r'),p(x', y', Z')dT'.

Now if the incident beam is directed parallel to the z-axis, it maybe represented by a wave-function, which involves Z only: sincethis wave-function must satisfy the Schrodinger equation

V1,p+kl,p= 0,

it must satisfy (d2,p/dz2) + k2,p= 0, so it must be a linear combination ofeih and rih

: a stream moving in the positive direction along theaxis of z, with electron-density unity, will be represented by theterm eilt< alone. The incident beam is thus represented as a planemonochromatic ,p-wave, whose wave-length is inversely proportionalto the momentum of the particles.

The complete wave-function ,p of equation (I) must representthis incident beam together with the scattered wave, which is awave diverging from the scattering centre, of the form

eikr

-,- g(O),

where (r, 0, 4» are spherical-polar co-ordinates, with the scatteringcentre as origin, so that 0 is the angle of scattering. Evidently thesecond term -in (1) represents the scattered wave, and thereforeC(x,y, z) must represent the incident wave. Thus (1) becomes

J ikp

,p(x,y, z) =eikz- 2:fi2 7V(r'),p(x',y', Z')dT'.

Let us find the asymptotic form of the second term on the rightat great distances from the origin. When r is very large we haveapproximately

(996)

, ( /I ')P =r - r cos rr ,293


When the particles of the incident beam are travelling so fast thatthe scattering is not very great, the function if;(x',y', z') in the integralmay be replaced by the wave-function of the incident wave alone,so the last equation becomes

ei/i;rif;(x, y, z) = eili:z+-g( 0)

rwhere

Making use of Gegenbauer's formula,l this integral becomes

g(O) = - ~7J~si; fV(r)r1dr, (3)

where p =2kr sin to.Now denoting the electric density eif;if;* by p, and denoting

the quantity

2eli .( if;* grad if; - if; grad if;*)mt

by 5, it can be shown from the wave-equation that

op +div 5=0atand this equation suggests that 5 may be interpreted as the electriccurrent 2; and therefore the number of particles which in unit timepass through a unit cross-section is the component, normal to thecross-section, of the vector

s= J!-.-.( if;* grad if; - if; grad if;*)2mtso

Ii ( oif; oif;*) •S.,=-. if;*--if;- etc.2mt ax ax

For the incident plane wave, if; = eili:z, so S. = klilm = Vo, so the number1 G. N. Watson, Besselfunctions, p. 378I Schrodinger, Ann. d. Phys. lxxxi (1926), p. 109. Born, ZS. j. P. !xxxviii (1926),

p. 803; xl (1926), p. 167. Gordon, ZS.j. P. xl (1926), p. 117294


of particles which in unit time pass through a unit cross-section atright angles to the beam is Vo, the velocity of the particles, as it mustbe. For the scattered wave we have

eik,ap= -g(8)r

so

Thus, with an incident beam in which one particle crosses unit area transverseto the path in unit time, the number of particles scattered into the solid anglesin 8 d8dep in unit time is Ig( 8)12sin 8 d8dep, where g( 8) has the value givenby (3).

Of course in this treatment, which is called the Born approximation,the interaction V(r) has been treated as a small perturbation, andonly the first approximation has been retained.1 Born's formulawas applied by Wentzel 2 to the scattering of a beam of electrically-charged particles by an electrically-charged centre. Suppose thecharges of the centre and of a particle are Ze and Z' e, where e isthe electronic charge; we suppose the Coulomb force of the centreto be modified by , shielding,' so we can write

2ZZ' rV() e --r =--e ar

rwhere the factor ea represents the shielding, a being the effectiveradius of the atom. Evaluating (3) on this assumption, we fina:

g( 8) = _ 2me2ZZ' •

1i2{(2k sin t8)2+~}

In the usual experiments, lla2 is small compared with (2k sin !8)2,so may be omitted. Thus

me2ZZ'g( 8) = - 21i2k2 sin2 t8

e2ZZ'= - 2mvo2 sin2 !8'

and this is precisely the formula obtained classically by Rutherford.Agreement with experiment confirmed the interpretation of apap*that had been used III the derivation.

I The wave-function .p of the quantum-mechanical treatment of the problem wasderived analytically from the classical solution (by considering the Action) by W. Gordon,z:,S.j. P. xlviii (1928), p. 180. I z:,s.j. P. xl (1926), p. 590

295


Born's method was applied in 1927 and following years toinvestigate the collision of an incident beam of electrons with theatoms of a gas, the atoms bein~ possibly raised to excited states,and the electrons scattered: thIs work, however, falls outside thelimits of the present volume.!

On 10 May 1926 Schrodinger contributed another paper to theAnnalen der Physik,2 in which he set forth a general method for treat-ing perturbations (i.e. solving problems which are very closelyrelated to problems that have already been solved), and appliedhis theory to investigate the Stark effect in the Balmer lines ofhydrogen. The method was essentially the same as that given byBorn, Heisenberg and Jordan, a few months earlier, or indeed asthat used long before by Lord Rayleigh S in discussing the vibrationsof a string whose density has small inhomogeneities. The wave-equation for the unperturbed system is

Let the proper-values of E, and the corresponding normalised wave-functions l/J (which are supposed to be known) be EI, E2, Ea, ••• ;l/JI(q), l/Ja(q), l/Ja(q), • • .. Let the perturbation be represented bythe adjunction of a small term Arl/J to the left-hand side of the wave-equation, where Adenotes a small constant and r is a known functionof the q's, so that the wave-equation for the perturbed system is

Let the new proper-values of the energy be E'l, E'2, ... , where

E~= E, + A~,+ terms involving higher powers of A,

and let the new wave-functions be l/J'l, l/J'2,. . ., where

l/J~= l/J,+ AV, + terms involving higher powers of A.

Substituting in the wave-equation, and retaining only the first powerof A,we have

{H - E, + A(r- ~,)}(l/JI + AV,) = O.•

Since (H - E,)l/J, = 0, this gives

(H - E.)v. = - (r - ~.)l/J•.

1 Though Born himself obtained important results in Colt. NIJI:h. 1926,p. 146.• Ann. d. Phys.(4) !xxx (1926), p. 437• Theory of Sound, 2nd edn. (London, 1894), i, p. 115

296


Now in Rayleigh's problem of the vibrating string, if there is reson-ance between the applied force and a proper vibration of the un-perturbed system, the oscillation increases without limit, and nofinite solution exists: it can be shown that corresponding to thisin the present case, in order that there may be a finite solution,the right-hand side of the last equation must be orthogonal to thewave-function which is the solution !f;. of the equation obtained byequating the left-hand side to zero; so we must have

J(r - «:.)!f;.!f;.*dq = 0

or, since J!f;.!f;.*dq = 1,

This equation gives the new proper-value of the energy as

E~ = E.+ AJr!f;.!f;.*dq.

We now proceed to find the corresponding wave-function, the newpart of which is given by the equation

(H - E.)v. = - (r- «:,)!f;,.

The integration of this presents no difficulty, and gives for the per-turbed system the wave-function

where the prime above the L means that the term correspondingto k =s is to be omitted.

Schr6dinger then considered the case of degeneracy, when toa single proper-value E. of the energy there correspond severalwave-functions: in the perturbed system, the degeneracy will ingeneral be wholly or partly removed, so that a single original energy-level gives rise to a number of energy-levels close together, andspectral lines are split, as in the Stark and Zeeman effects.

In applying this theory to the investigation of the Stark effectin the Balmer lines of hydrogen, we suppose that there is an electricfield of intensity F in the positive z-direction, so the wave-equation is

v2!f;+ ~7(E+~-eFZ)!f;=O.

297


Schrodinger followed Epstein 1 in introducing space-parabolicco-ordinates A, /1-, c/> by the equations

for which the volume-element dxdydz is l(A + /1-)dAd/1-dc/>. The wave-equation now becomes

-~(A oif;) + il_ (/1-~)+ !(!+ l) 02if;OA OA 0/1- 0/1- A P. 0c/>2

+ 2~2{E(A + /1-) + 2e2 - leF(A2 - p.2)}if; = O.

To solve this equation we write

where A is a function of A only, M is a function of /1- only and cI>is a function of c/> only. The equation for cI> is

d2cI> = _ n 2cI>dc/>2

where n is a constant which (in order that cI> and its derivativesshould be single-valued and continuous functions of c/» must be awhole number, so n = 0, 1, 2, 3, . . .. The equations for A and Mare both of the form

The term Dg2 represents the Stark-effect perturbation. So for theunperturbed system

Put A= g-!u, 2g.y'(-A) =7}, and B( - A)-l =p. The equationbecomes

the solution of which is the confluent hypergeometric function

u= WP,l" (7}),1 cr. p. 121 supra

2g8


SO in order that the solution may be finite everywhere we must have

p= t(n+ 1) +k (k=;=O, 1,2, ... ),

and the corresponding solutions are

Uk(7]) =Wk+(n+!)/a,n/a(7]).

There are two values of k, corresponding to the equations for Aand M respectively: call them k1 and ka. Then n, k1 and ka arethe three quantum numbers of the unperturbed motion. Denotingthe values of B in the A- and M-equations respectively by B1 andBI> we have

B1= (n; 1 +k1) v( -A)

Ba= (n; 1 +ka) v( -A)

so

ButB mel dAmE

1+ Ba = 2lia an = 21i1

so the energy in the unperturbed motion isme'E=------

21i1(n+ I+k1 +ka)1

an equation from which it is evident that (n + 1+ k1 + ka) is theprincipal quantum number.

If we now carry out the process described above for finding theenergy-levelsin the perturbed motion, we obtain

E=- me' _31i2F(kl-kl)(n+l+kl+ka)21i2(n+ 1+ k1 + k2)a 2me

which is precisely the formula found by Epstein 1 for the energy-levelsin the Stark effect of the Balmer lines of hydrogen.

Schrodinger proceeded to discuss the intensity of the Starkcomponents. In order to find the intensity of a line, it is necessaryto calculate the matrix-element relating to the corresponding transi-tion. This matrix-element can be determined, as we have seen,fromthe wave-functions: and it was found to vindicate the selectionand polarisation rules given by Epstein. Some of the theoreticalresultson intensities were confirmed experimentally in the followingyear by J. S. Foster.ll

1 cf. p. 121 supra • Proc. R.S.(A), cxiv (1927), p. 47; cxvU (1927), p. 137299


The terms proportional to the square of the applied electricforce in the Stark effect were calculated in the following year byG. Wentzell and I. Waller.2 The Stark effect was investigated in1926 also by Pauli 3 from the standpoint of Heisenberg's quantum-mechanics.

Schrodinger's treatment, like that of Schwarzschild and Epstein,ignored the fine-structure of the hydrogen lines: the theory wascompleted in this respect not long afterwards by R. Schlapp,' whosework was based on the new explanation of the fine-structure whichhad been provided by the theory of electron-spin.

Another phenomenon which is to be treated by the quantum-mechanical theory of perturbations is the Zeeman effect.6 If in afirst treatment of the subject we ignore electron-spin, and considersimply the motion of an electron attracted by a fixed nucleus of chargee and under the influence of a magnetic field of intensity H parallelto the axis of z, then the Lagrangean function is

!m{ (dx) I + (dy)I + (d,z) I} _~:tI(x rJl_y~) + ~dt dt dt 2e dt dt r

and therefore the Hamiltonian function is 8

2~{ (pz - ;~ yr + ( Pr + e2~xr + P" } - ~

(measuring e in electrostatic and H in electromagnetic units, asusual) or

1 {p I pip I eH('h ,h) elHI (I .,I)} el2m z + y + • + c Xn - Y"z + 4el x +J - ,.

The corresponding wave-equation, obtained by replacing pz by(Ii/i) (a/ax) etc., is

Vlt/J+ 2m(E + ~)t/J+ ieH(x~ _ y ot/J) - eIH2_(xl +yl)t/J= O.iii r eli Oy ax 4e2/i1

Introducing spherical-polar co-ordinates (r, 8, 4», with the z-axisas polar axis, this becomes

~ ~(r2 Ot/J) + _1_ ~ (sin 8 ot/J) + 1 . 02t/J + '?-1!!:(E+ ~)t/Jrl Or or rl sin 8 08 08 rl sinl 8 04>1 liz rl

ieH ot/J e'H' I • 18 .1. 0+Cfi 84> - 4el/il r sm 'f' = .

1 ZS.f. P. xxxviii (1927), p. 518 1 ZS.f. P. xxxviii (1927), p. 6351 ZS.f. P. xxxvi (1~?6), p. 336; cr. also C. Lanczos, ZS.f. P. !xii (1930), p. 518;

\xv (1930), p. 431 ; \xviiI (1931), p. 204• Proc. R.S.(A), cxix (1928), p. 313 • cf. Schrooinger, Phys. &v. xxviii (1926), p. 1049• cr. V. Fock, ZS.f. P. xxxviii (1926), p. 242

300


We shall ignore the term 1 quadratic in the magnetic intensity H :and we shall regard the linear term in H as a perturbation: sothe wave-equation of the unperturbed system is the ordinary wave-equation of the hydrogen atom, which has already been considered:the wave-function is a constant multiple of

~Wk.nH(~)Pnl'(COS 8)e±il'</>

and the energy-levels are

metEk= -21i2k2 where k= 1,2,3,4, ....

Now in the above wave-equation the perturbation-term implies thereplacement of E by E - Arwhere

'\ ieliH 0 •I\r= - 2mc oeP·

so the term An to be added to Ek on account of the perturbation is(by the general theory)

AEk= J{ _i;~~(±iJL)}¢Jk¢Jk*dq

when ¢Jk is supposed to be normalised, so Nk¢Jk*dq = 1.displacement of the energy-level in the Zeeman eifect is

AEk = ±eJLnH2mc

Thus the

where JL is the magnetic quantum number: which is precisely the valuefound in Lorentz's classical theory.s

The above derivation accounts only for the normal Zeemaneffect; as might be expected, since we know 3 that the anomalousZeeman effect requires for its explanation the assumption of electron-spin. The problem thus presented was solved by W. Heisenbergand P. Jordan 4 by matrix-mechanical methods in 1926, and byC. G. Darwin 5 by wave-mechanics in 1927.

Darwin's model consists of a charged spinning spherical bodymoving in a central field of force. Denoting the charge by - e,

1 The effect of the quadratic term was considered by O. Halpern and Th. SexI,Ann. d. Phys.(5) iii (1929), p. 565.

• Vol. I, p. 412; cr. P. S. Epstein, Proc. X.A.S. xii (1926), p. 634; A. E. Ruark,Phys. Rev. xxxi (1928), p. 533

• p. 136supra• z:,S.j. P. xxxvii (1926), p. 263 • Proc.R.S.(A), cxv (1927), p. 1

301


the mass by m, the position of the centre by (x,y, z) and the potentialenergy at distance r by V(r), then the motion of revolutIon con-tributes terms

to the Lagrangean function. There is also a contribution

11(wzl + wvl +w.l)

where I is the moment of inertia and (Wz, Wv, w.) the componentsof spin about (x, y, z). The orbital motion in the magnetic fieldH along z gives a contribution

_eH(xtJl_y~).2c dt dt

The spin giveseHIw.---

me

since elme is the ratio of magnetic moment to angular momentum.Lastly there is the interaction of the spin and the motion. Theelectric force has components - (xlr) V' etc., so we obtain a term

_ eIV'{wz (y dz _ z dy) + Wv (z dx _ x dz) + w. (x dy _y dx)} .melr dt dt dt dt dt dt

All these terms were taken together and converted into Hamiltonianform, and the Schrodinger wave-equation was then deduced. Fromthis equation by use of spherical harmonic analysis the proper-valuesand wave-functions were calculated and Lande's g-formula wasobtained.l

Schrodinger followed up his work on perturbations and theStark effect by another paper 2 in which he extended the J?erturba-tion theory to perturbations that explicitly involve the tIme, andsucceeded in obtaining by wave-mechanical methods the Kramers-Heisenberg formula for scattering.3 Let q represent the co-ordinateswhich specify the state of the atom at the instant t, and let Ho (q, p)represent its energy, so the wave-function for the unperturbed atomis given by the equation

1 cr. K. Darwin, Proc. R.S.(A), cxviii (1928), p. 264-• Ann. d. Phys.(4) !xxxi (1926), p. 109; cr. O. Klein, ZS.f. P. xli (1927), p. 407• cr. p. 206 supra

302

Hpt

rPpo= eiilrPp(q)

and


Let the atom be irradiated by light whose wave-length is so greatcompared with atomic dimensions that we can regard the electricvector of the light as constant over the atom: let this electricvector be

E = V*e21Tivl+Ve-21Tivl

where V is a complex vector. The additional energy of the atomdue to this perturbation is - (M . E), where M denotes the electricmoment L er of the atom: so the wave-function of the perturbedatom is determined by the equation

The wave-function, when the atom is in the state P, may be denotedby

and a general solution may be represented by a series of these wave-functions. Supposing that the atom is initially in the state P, letus solve the wave-equation of the perturbed atom by writing

rP= rPpo+ rPp\ where rPpl is small.

Substituting in the differential equation, and neglecting small quan-tities of the second order, we have

(Ii 8) 8rP1Ho q, rFq rPpl- iii 8: = (M . E)rPpo.

We solve this by substituting in itHp+hv Hp-h.I

rPpl=rPp+e-ui' +rPp-etil

and equating terms which have the same time-factor: the resultingequation for rPP+ is

Ho(q, ~ ;q)rPP+ - (Hp +hv)rPp+ = (M . V)rPP'

Now expand rPP+ and MrPp as series of the wave-functions rPR, say

rPp+ = L apRrPRR

MrPp= ~ MPRrPR where MpR= JrPR*MrPpdT,

3°3


SO MpR is the element in the (matrix) electric moment of the atomthat corresponds to a transition from the level R to the level P:Thus

.~ apR{Ho(q,.~ ;q) -Hp-hv}ifiR= ~ (MpR. V)ifiJJ.

or L apR (HR - Hp - hv)ifiR = L (MpR• V)ifiJJ.JJ. R

and therefore apR = (MpR . V).h(vRp-v)

Similarly the coefficient of ifiR in the expansion of ifip- is determined:and thus

This equation gives the perturbed value of the wave-Junction. The electricmoment of the classical oscillator which would emit the radiationemitted in the scattering process of an atom in the state P, associatedwith the transition to the state Q, is the real part of

JifiQ* MifipdT, or in this case J(ifiQO+ ifiQl) *M( ifipo + ifipl)dT.

Neglecting terms of the second order, this is found to be

MpQrl7TIVPQI +1 L {MRQ(MpR • V) +MpR(MRQ • V)}rl7Tl(VPQ+V)1h R VRP-v VRQ+V

The term Mpqe-I7T1VpQ' represents the spontaneous emission associatedwith the transition P+Q, and the other terms are precisely those found byKramers and Heisenberg for the electric moment corresponding to the scatteredradiation associated with the transition P+Q.

The Compton effect was investigated quantum-mechanically in1926 by Diracl by means of his symbolic representation of matrix-mechanics, and by W. Gordon 2 by wave-mechanics. Gordon foundthat the quantum-mechanical frequency and intensity are the

1 Proc. R.S.(A), cxi (1926), p. 405• ZS.j. P. xl (1926), p. 117; cr. G. Breit, Phys. Rev. xxvii (1926), p. 362; O. Klein,

ZS. j. P. xli (1921), p.. 407, at p. 436; G. Wentzel, ZS. j. P. xliii (1927), pp. I, 779 ;G. Beck, ZS.j. P. xliii (1927), p. 658

304


geometric means of the corresponding classical quantities at thebeginning and end of the process. Breit 1 remarked that accordingto Gordon, the radiation in the case of the Compton effect may beregarded as due to a t/Jrf;* wave moving with the velocity of light.Similarly Breit showed 2 that if the motion of the centre of gravityof an atom is taken into account, emission takes place only bymeans of unidirectional quanta due to t/Jrf;* waves moving with thevelocity of light. .

In 1927 Schrodinger 3 showed that by means of the conceptionof de Broglie waves, the Compton effect may be linked up with someother investi~ations of a quite different character. He began byrecalling an Investigation of Leon Brillouin" on the way in which(according to classical physics) an elastic wave, in a transparentmedium, affects the propagation of light. Let there be an elasticcompressional or longitudinal wave (i.e. a sound-wave) of wave-length A, which is propagated in a transparent medium. It reflectsat its wave-front hght-rays which traverse the medium; but thereflected ray has negligible intensity except when there is betweenthe wave-lengths and the angles the relation

A=2A cos i

where i is the angle of incidence and A is the wave-length of thelight in the medium (so it is K-I times the wave-length of the light invacuo, where K is the dielectric constant). This is precisely theequation which had been found in 1913 by W. L. Bragg as thecondition that X-rays of wave-length A should be reflected bythe parallel planes rich in atoms in a crystal, when i was the angleof incidence and A was the distance between the parallel planesin the crystal.5 In Brillouin's theorem it is supposed that the velocityof propagation of the elastic wave is small compared with thevelocity of light: more accurately, the formula must be modifiedas in the case of reflection at a moving mirror.6

Schrodinger now showed that the Compton effect can beassimilated to the Brillouin effect, if the electron of the Comptoneffect in its initial state of rest is replaced by a ' wave of electricaldensity' and also in its final state is replaced by another wave.The two waves form by their interference a system of intensity-maxime located in parallel planes, which correspond to Bragg'splanes in a crystal. The analytical formulae are, as he showed,identical with those found by Compton in the particle-interpretation.Thus a wave-explanation is obtained for the Compton effect. Other casesare known in which a transition of a particle from one state of motion

1 Proc. N.A.S. xiv (1928), p. 553 I J. Opt. Soc. Amer. xiv (1927), p. 374• Ann. d. Phys. !xxxii (1927), p. 257 • Ann. d. Phys. xvii (1921), p. 88• cf. p. 20 supra• Brillouin's problem was treated quantum-mechanically in Phys. ZS. xxv (1924),

p. 89 by Schrodinger, who found that quantum theory led to the same formula as classicalphysics.


to another may be translated into the superposition of two interferingmatter-waves.

The year 1926 saw the publication of. some important paperson the problem of atoms with more than one electron. It hadbeen known for some years that the energy-levels or discrete station-ary states, that account for the spectrum of neutral helium (whichhas two electrons), constitute two systems,! such that the levelsbelonging to one system (the para-system) do not in general combinewith the levels of the other system (the ortho-system) in order to yieldspectral lines. This fact led spectroscopists at one time to con-jecture the existence of two different chemical constituents in helium,to which the names parhelium and orthohelium were assigned. Thepara-level system consists of singlets and the ortho-system consistsof triplets.

Heisenberg 2 and Dirac 3 now investigated the general quantum-mechanical theory of a system containing several identical particles,e.g. electrons. If the positions of two of the electrons are inter-changed, the new state of the atom is physically indistinguishablefrom the original one. In this case we should expect the wave-functions to be either symmetrical or skew in the co-ordinates of theelectrons (including the co-ordinate which represents spin). Nowa skew wave-function vanishes identically when two of the electronsare in states defined by the same quantum numbers: this meansthat in a solution of the problem specified by skew wave-functionsthere can be no stationary states for which two or more electronshave the same set of quantum numbers, which is precisely Pauli'sexclusion principle. A solution with symmetrical wave-functions,on the other hand, allows any number of electrons to have the sameset of quantum numbers, so this solution cannot be the correct onefor the problem of several electrons in one atom.

In a second paper,4 Heisenberg applied the general theory tothe case of the helium atom. There are energy-levels correspondingto wave-functions which are symmetric in the space-eo-ordinatesand skew in the spins of the electrons (i.e. the spins of the twoelectrons are antiparallel): these may be identified with the para-system. There are also wave-functions which are skew in the space-co-ordinates and symmetric in the spins (Le. the spins of the twoelectrons are parallel) : these are associated with the ortho-system.In both systems the wave-functions involving both co-ordinates andsrin change sign when the two electrons are interchanged. Nowi a wave-function is symmetrical (or skew) at one instant, it mustremain symmetrical (or skew) at all subsequent instants, and there-fore the changes in the system cannot affect the symmetry (or

1 For studies of the helium atom and the related ions Li+ and Be++ from the stand-point of the earlier quantum theory, cf. A. Lande, Phys. ;::S. xx (1919), E. 228; H. A.Kramers, ;::S.j. P. xiii (1923), p. 312; J. H. van Vleck, Phys. Rev. xxi (1923), p. 372;M. Born and W. Heisenberg, ;::S.j. P. xxvi (1924), p. 216 .

. I ;::S.j. P. xxxviii (1926), p. 411 • Proc. R.S.(A), cx.ii (1926), p. 661• ,zS.j. P. xxxix (1926), p. 499

306


skewness), so the para-levels and ortho-levels cannot have transitionsinto each other; which explains their observed property.

Heisenberg's first paper was notable for the discovery of theproperty of like particles known as exchange interaction, which hadan Important place in the physical researches of the years immed-iately succeeding: an account of these must be reserved for thenext volume.

Index of Authors Cited

137, 199126108

Abraham, M. 68, 153,247Adam, M. G. 181Adams,J. C. 147Adams, N. I. 249Adams, W. S. 228Aepinus, F. U. T. 150Alexandrow,W. 183Allen, H. S. 97Allison,S. K. 210Andrade, E. N. da C. 20Armstrong,A. H. 210Arvidsson,G. 244Aston, F. W. 13Auger, P. go

Back, E. 135Backlund,O. 149Biicklin,E. 139Bahr, E. von 108Bailey,V. A. 218Band, W. 175Barajas,A. 192Bargmann, V. 192Barkla,C. G. 14-16,23,2°7Barnett, L. J. H. 244Barnett, S. J. 243, 244, 246Bateman, H. 8,64, 76,94, 154-6, 195Bates,L. F. 244Batho, H. F. 94Bay, Z. 211Beams,J. W. 96Bearden,J. A. 210Beatty, R. T. 17Beck,E. 244Beck,G. 304Becker,J. A. 210, 236Becker,R. IgB,211Becquerel,E. IBecquerel,H. 1-3Bergmann,A. 112Bernoulli,Johann 144Bessel,F. W. 228Bessel-Hagen,E. 195Bethe, H. 219,234Bevan, P. V. 106,202Bhatnagar, P. L. 123Biesbroek,G. van 180Birge,R. T. 112Birkhoff,G. D. 175, 191Bishop,F. N. 116Bjerrum, N. 108, 109Blackett,P. M. S. 26Blake,F. C. 93

(996)

Block,F. 230Blom, C. E. 97Bohr, N. 25, 1Og-12,114, 118, 120, 130-

132, 140, 141,211,242Boltwood,B. B. 8, 10, II, 13Boltzmann, L. 82, 85, 96, 122Bonnor, W. B. 192Born, M. 100, 103, 130, 138, 204, 217,

254, 255, 258-60, 263, 267, 272, 291,292, 294-6, 306

Bortolotti, E. 192Bose,S. N. 87,219-24Bothe, W. 102, 103, 199, 210, 212, 213,

229Brace, D. B. 29Brackett, F. 112Bragg, Sir W. H. 7, 17--'20,93Bragg, Sir W. L. 19,20,3°5Brdicka, M. 183Breit, G. 304, 305Bridge, L. A. du 233, 235, 236Bridgman, P. W. 231, 232Brillouin,L. 160,230, 280, 282, 305Brinsmade,J. B. 108Brittain, W. H. 236Broek,A. van der 24Broglie,Prince L. V. P. R. de, 102, 103,

141, 143, 191,214, 216--19,268, 269,271

Broglie,M., Duc de ag, 20gBrouwer,D. 149Brown, R. 9Brunn, A. von 180Bruyne, N. A. de 236, 237Bucherer,A. H. 15,53, IIIBudde, E. 43Burger, H. C.Burgers,J. M.Burmeister,W.

Cabras, A. 160Cabrera, B. 242Campbell, N. R. 94Capon, R. S. 38Caratheodory, C. 43Cartan, E. 190, 192Cassini,G. D. 144Castelnuovo,G. 184Catalan, M. A. 136Cayley, A. 34,256Chadwick, Sir J. 25, 26Chase, C. T. 29Chattock, A. P. 244

309

INDEX OF AUTHORS

Chaudhuri, R. M. 218Chazy, J. 184Cherwell, F. A. Lindemann, Baron 13,97Christoffel, E. B. 160, 161, 166Cilliers, A. C. 242Clark, George L. 210

Clark, Gordon L. 54Clausius, R. 122Clemence, G. M. 179, ISoClifford, W. K. 156Collier, V. 16Combridge, J. T. 194Compton, A. H. 134, 142, 2°7-12, 304,

3°5Compton, K. T. 89, 90Comstock, D. F. 52Conway, A. W. 106, 128, 246, 247Copson, E. T. 158Coster, D. 140, 141Cowell, P. H. 147, 149Coxeter, H. S. 184Crommelin, A. C. D. 149Cl'ookes, Sir W. 3,5, 12

Crowther, J. A. 15Croxson, C. 121Cunningham, E. 195,248Curie, M. 2, 3Curie, P. 2, 3, 239, 241, 242Curzon, H. E. J. 182

Dantzig, D. van 191, 192, 195, 196Darrieus, G. 191Darwin, Sir C. G. 19,20,23,25,87,211,

219, 3°1Darwin, K. 302Dauvillier, A. 140, 141Davisson, C. J. 2 I 7- I 9Debierne, A. 3Debye, P. 87, 98, 99, 105, 126, 128, 2og,

235,290DemarlSay,E. A. 2Dempster, A. J. 94Dirac, P. A. M. 113, 224, 265-7, 272,

280, 304, 306Ditchburn, R. W. 215Dobronrawov, N. 17Dorgels, H. B. 137Dorn, F. E. 4Droste,J. 175Duane, W. 16,93, 142,2°7,210,216,217Dulong, P. L. 96,97Dunoyer, L. 9Dushman, S. 232, 233Dymond, E. G. 218

Earhart, R. F. 236Eckart, C. 230, 234, 285, 290Eddington, Sir A. S. 54, 87, 150, 151,

183, 188~1, 197

Ehrenfest, P. 38, 103, log, 122, 123, J28,143, 183, 19B, 231

Einstein, A. 8-10, 40, 42, 52, 87~, 91,93, 94, 96-8, JOI-4, 118, 152, J53,156-60, 166-8, J70, 175, 179, ISo,182-4, 188~2, 197~, 215, 219, 222-224, 244, 245, 263

Eisenhart, L. P. 188, 191Elsasser, W. 217,218Elster, J. 3Encke, J. F. J48, 149EOtvos,R. von 151Epstein, P. S. 55, 121, 143,298,299, 301Esc1angon,E. 43Essen, L. 43Estermann, I. 219Eucken, A. 100Euler, L. 145, 146Evans, E. J. II4, 121Ewald, P. P. 18, 19Ewing, Sir J. A. 238Eyraud, H. 190Eyring, C. F. 236

Fajans, K. 13, 14Faraday, M. 245Fermi, E. 51, 54, 206, 224-30, 233, 237,

284Finlay-Freundlich, E. ISoFisher, J. W. 191FitzGerald, G. F. 28, 37, 157, 158Fleck, A. J2Flint, H. T. 191Florance, D. C. H. 207Fliigge, S. 229Fock, V. 300Fokker, A. D. 104>153, 170, 182, 213Forsyth, A. R. 178Foster, J. S. 299Fotheringham, J. K. 148Fowler, A. 113, I IS, 136Fowler, Sir R. H. 87, 228, 233, 235-7Franck, J. 104, J IS, 116, 2J7Frandsen, M. 13Frank, P. 43, 87Franklin, B. IS0Freedericksz, V. 188Fresnel, A. 2 I 7Freundlich, E., see Finlay-Freundlich, E.Friedman, A. 188Friedrich, W. 18, 19Frisch, R. 91Fuchs, K. 103Fues, E. 277Furth, R. 9

Gans, R. 242Gehrckc, E. 94Geiger, H. 7, 8, IS, 20, 22-5, 212

310

INDEX OF AUTHORS

Kaluza, Th. 190Klirmlin, T. von 100Karrer, S. 243Kaufmann, W. 3, 53Kaye, G. W. C. ,6Keesom,W. H. 100Kelvin, W. Thomson, Baron 21, 28, 57.

66,9',235Kemble, E. C. '08, 109Kennard, E. H. 43, 94, 246, 272Kennedy, R. J. 43Kikuchi, S. 218,229Kirschbaum, H. 117Kleeman, R. 7Klein, F. 56,76, '70, 184Klein, O. 13°,191.211,267,268,3°2,3°4Kliiber, H. von 180Knauer, F. 219Knipping, P. 18, 19Koch, P. P. 18Kohl, E. 43Kohlrausch, K. W. F. 206Kopfermann, H. 205Kossel,W. 132, 137, 138, 140Kothari, D. S. 123

311

Geitel, W. 3Gerber, P. 148Gerlach, W. go,96, 130, 242Germer, L. H. 218,219Gibbs,J. Willard 8,57,86, 123,226,237Giesel,F. O. 3Gilloch,J. M. 54Giorgi, G. 28, 160Gmelin, P. III

Gomes,L. R. 43Gooseth, F. 191Gordon, W. 268,294,295, 3°4, 305Goaling, B. S. 237Goudsmit, S. 134, 137,267Graef, C. 192Grant, K. 207Grant, R. 147Gray,J. A. 20,207Grier, A. G. 5Gronwall,T. H. 273Grossmann,M. 156Grilneisen,E. 97Guernsey,E. W. 243Guye, C. 53

Haantjes,J. 195Haas, A. E. III

Haas, W. J. de 244Haga, H. 18Hagenow,C. F. 207Hagihara, Y. 178Hahn, O. II

Hall, D. B. 44Halley, E. 146, 149Halpern, O. 208,301Halsted, G. B. 30Hamilton, Sir W. R. 256. 264, 270, 27',

278-80Hapke, E. I 'IHardtke, O. 117Hargreaves, R. 64. 250-2Hasenohrl, F. 51, 54Heaviside,O. 67,247Hector, L. G. 241Heisenberg, W. 103, 137, 206, 253-5,

258, 259. 263, 265, 267, 272, 279,285,296,3°1,3°2,3°4,3°6,3°7

Herglotz,G. 246Hertz, G. 13, 1°4, 115Hertz, P. 123Herzfeld,K. F. 22, 232Hessenberg.G. 190Heuse,W. 100Hevesy,G. 140Hewlett, C. W. 207Hilberl, D. 76, 159. 170, 173, 175Hoffman,B. 182, 191Holm, E. 242Honl, H. 137

(995)

Houston, W. V. 230, 237Hubbard. J. C. 210Hughes, A. L. B9.236Hughes,J. V. 219Hund, F. 137,241Hunt, F. L. 93Huntington. E. V. 43Huygens, C. 144,271

Imes, E. S. 108Infeld, L. 182Ironside, R. 218Ishiwara, J. 118Ives, H. E. 40, 42, 43. 52. 236

Jackson, J. 180Jacobsohn, S. 103Jaffe, G. 272Jahnke, E. 78Jaos, C. de 178Jauncey, G. E. M. 209Jeans, Sir J. 38, 78, 120, 148Jeffery, G. B. 182Jeffreys, Sir H. 280Joffe, A. 17Johnson. J. B. 10Johnson, T. H. 219Jones, Sir H. Spencer 148Joos, G. 241Jordan, P. 103, 206, 214, 229. 259, 26o,

263, 267, 272, 28o, 296, 301Jiittner, F. 192Juvet, G. 191

INDEX OF AUTHORS

Val, P. du 184Vallarta, M. S. '92Veblen, O. '9'Villard, P. 3, 4ViIley,]. 43Vleck,]. H. van '32,211,3°6Voigt, W. 33, II 7, ,6o, 238, 239Volta, A. 235

Uhlenbeck, G. E.Urbain, G. '40Urey, H. C. '3

Thomsen,]. 25Thomsen, G. 182Thomson, Sir G. P. 96,2,8Thomson, Sir].]. 2, 4, '2, 14, '7. 21,

25, 26, 5', 67, 85, 9', 93, 94, '37,150,200, 238, 240, 249

Thomson, W., see KelvinThoraeus, R. 14'Thorndike, E. M. 43Toeplitz, O. 255Tolman, R. C. 38,48,55, ,84, 232, 243Tomaschek, R. 29, 39Tomlinson, G. A. 43Tonks, L. 232Townsend, Sir ]. S. 2,8Trefftz, E. 184Tresling,]. '70Trouton, F. T. 29Trueblood, H. C. 2"Trumpler, R.]. ,8o

Wadlund, A. P. R. '39Walker, A. G. ,8,Waller, 1. 300Walter, B. ,8Walters, F. M. '37Warburg, E. 9', "', 118Warner, A. H. 235Washburn, E. W. '3Wataghin, G. 38Waterman, A. T. 237Watson, Sir W. '50Weber, W. '50, 238, 245Webster, D. L. 93,2'0Weiss,P. 24',242Weitzenbock,R. 192Wendt, G. "7Wentzel,G. 208, 235, 280, 282,295, 300,

3°4Weyl, H. '53, '70, 182-4, 188-g2, '95,

277Whiddington, R. 16Whipple, F. '49White, M. P. 207Whitehead, A. N. '74Whitrow, G.]. 43

314

Shankland, R. S. 2'3Sharpe, F. R. 246Sharpless,B. P. '48Shimizu, T. ,80,2°7Siegbahn, M. 141Silberstein,L. 54, ,68, ,84Simon, A. W. 209,212Sitter, W. de 38, 151, 178, ,83, ,84Skobelzyn,D. 209Slater, ]. C. 2II

Smekal,A. 205Smith, E. R. 13Smith,]. D. Main '4'Smoluchowski,M. von 9Soddy, F. 5.6, '0, '2, 13Soldner,]. ,80Solomon,]. 103Sommerfeld, A. ]. W. ,8, 55, 68, '04,

105, 118-20, '26, '28, '32-4, '37,139, '4', ,60,211,2'7,23°,23,,233,237, 246, 282

Stackel, P. 126, 127Stark,]. 42,53,9°,91,93,95, 117,267Staude, O. '27Stefan,]. 85Stern, O. 87, 104, '3°,2'9,242Stern, T. E. 237Steubing, W. 9'Stewart,]. Q. 244-Stewart, T. D. 243Stifler, W. W. 2'0Stillwell,G. R. 42, 43Stoner, E. C. 96, '4',227Stomholm, D. 12Struik, D.]. 192Strutt,]. W., see Rayleigh, LordStrutt, R. ]., see Rayleigh, LordSucksmith,W. 244-Sutherland, W. 97Svedberg, T. 9, 12Swann, W. F. G. 246Sylvester,].]. 256Synge,]. L. 43, '74, '75, 194,2'5Szepesi,Z. 2' ,Szily,C. 122

Tait, P. G. 28Tate,]. T. 246Taylor, Sir G. 1. 94Taylor,]. B. 242Taylor, N. W. 24'Temple, G. 43, '75, '94

. Thibaud,]. 89. '39Thiesen, M. 78Thirring, H. 38Thomas,]. M. '9'Thomas, L. H. 267Thomas, T. Y. 19'Thomas, W. 259

INDEX OF AUTHORS

McCrea, W. H. 54Macdonald, R. T. 13Mack, E. '5°,151,167,168McVittie, G. C. 188Madelung, E. 97, 138Mandel, H. 19'Mandelstam, L. 205Marsden, E. 20, 22-5Mathisson, M. 74Maxwell, J. Clerk 67, 243, 247Mayer, A. M. 21, 137Mayer, W. 192Mendeleev, D. 1. II

Mesham, P. 207Metz, A. 43Meurers, J. 43Meyer, E. 90Meyer, O. E. 85Meyer, S. 3Michaud, F. 38Michelson, A. A. 38, 39Mie, G. '53, '54Miller, D. C. 175Millikan, R. A. 89, I I 1,235-7Milne, E. A. 148Mineur, H. 183Minkowski, H. 35, 64--8, 70, 71, 77, 151,

172Morgan, H. R. '79Morley, E. W. 38Morley, F. 178Morton, W. B. '23Moseley, H. G. J. 19,20,24,25Mosengeil, K. von 52Mossotti, O. F. 150Mott-Smith, L. M. 243Muller, C. I I I

Mueller, D. W. 236Murnaghan, F. D. 180, 195

Lo Surdo, A. II 7Loughridge, D. H. 211Lummer, O. 78,81,84,94Lyman,T. 112",6

Nagaoka, H. 22

Narliker, V. V. 43Nernst, W. 96, 97, 100, '04, 108, 224Neumann, C. 27Neumann, G. 53Newcomb, S. 148Newlands,J. A. R. II

Newton, Sir 1. 144Nicholson, J. W. 107Nicol,J. 16Nishina, Y. 21 I

Niven, C. 57Noble, H. R. 29Nordheim, L. W. 229, 230, 233, 237

312

'59, 2302°5

134, 137

Ladenburg, E. R. 89Ladenburg, R. 138, 200, 202, 205, 253Lagrange, J. L. '45, '46Lalen, V. 43Lanczos, C. 174, 184,300Lande, A. 96, 135, 138, '4', 204, 267,

302, 306Landsberg, G. 205Langer, R. H. 272, 280Langevin, P. 9, 29, 239, 24°-2, 246Langmuir, 1. 137, 138,218,232Laplace, P. S., Marquis de 47, 145-7, 151Laporte, O. 137Larmor, Sir J. 28, 31, 32, 37, 87, 128,

247, 249Laub, J. 207, 245Laue, M.von ,8-20,43,55,94,1°3,153,

165, 184, 232Lauritsen, C. C. 236, 237Lavanchy, C. 53Lawrence, E. O. 96, 236Leathem, J. G. 249Ledermann, W. 180Leeuwen, H. J. van 242Lehmann-Filhes, R. 149Leibnitz, G. W. 144Leithauser, G. I I I

Lemaitre, G. 188Lenard, P. 22, 235Lenz, W. 13°,267Le Sage, G. L. 149Lev~ita, T. 58, 155, 159, 178, ,82,

19°,194Lewis, G. N. 13, 48, 52, 53, 55, 88, 137,

241Lichnerowicz, A. 194Liebert, G. 117Lilienfeld, J. E. 236Lindemann, F. A., see CherwellLinford, L. B. 236Lohuizen, T. van 128Longo, C. 182Lorentz, H. A. 30-3, 36, 39, 53, 86, 93,

102, 106, '49-51, 170, 213, 242, 246,3°1

Kottler, F. '92-5Kowalewski, G. 195Kramers, H. A. 122, 132, 182, 203, 206,

2' 1,229,253,260,265,280,282,3°2,304, 306

Kretschmann, E.Krishnan, K. S.Kronig, R. de L.Kroo, J. N. 242Kuhn, W. 259Kunsman, C. H. 217

Kurlbaum, F. 78,81,84Kurth, E. H. 16

INDEX OF AUTHORS

Ricci·Curbastro, G. 58, 155, 159, ,61,194

Richardson, Sir O. W. 89, 90, 218, 232,233, 235-7, 244

Richmond, D. E. 43Richtmyer, F. K. 207Riemann, B. 166Righi, A. 43Ritz, W. 38, 106, 109Rontgen, W. K. I

Robertson, F. S. 235Robertson, H. P. 43, 74, 182, ,84Rosanes,J. 255Rosen, N. 175, ,83, 192, 272Rosenfeld,L. 234Ross, P. A. 2°9,210Ro~i, B. 44Rossi,R. I'Rothe, H. 43, 234Royds, T. 8Ruark, A. E. 301Rubens, H. 78,81Rubinowicz, A. 87, 13'Rupp, E. 218,2'9Ruse, H. S. '94Russell,A.S. II, '3Russell,H. N. '37Rutgens, A. J. 23'Rutherford, E., Baron 2-8, '0, II, '3-15,

20, 22-6, 106, 134,2°7, 291, 292

Sadler, C. A. ,6,207Sagnac, G. 43Stjohn, C. E. ,81Saunders, F. A. 137Schaefer, C. 53Schaposchnikow,K. 218Schechter, A. ,88Scheel, K. 1ooSchimizu, T. 16Schlapp, R. 300Schmidt, G. C. 2Schoenberg, M. 175Schott, G. A. 246Schottky, W. 10, 232, 234-6Schouten, J. A. 58, 182, 190-2, 195Schrodinger, E. 76, 77, 99, 192, 268,

270-3, 275, 278, 283, 285, 287, 290,294, 296-3°0, 302, 305

Schuster, Sir A. 244Schwarzschild, K. 121, 175, 177, 184,

246Schweidler, E. von 3, 8

3, Schwers,F. '00Seddig, M. 9Seeliger,H. 148Sen, N. R. 43, 184Severi, F. 43Sex.!,T. 30

313

Nordstrom, G. '53, ,82Nyquist, H. '0,' '7

Ogura, K. '78Olpin, A. R. 236Onnes, H. Kamerlingh 100,242Oppenheim, S. '48Oppenheimer,J. R. 237Ornstein, L. S. '37, 199,229Oseen, C. W. 175Ostwald, W. 9Owens,R. B. 4

Page, L. 247, 249Palatini, A. 170, '75Papapetrou, A. 175, '92Pars, L. A. 43Paschen, F. 81, "2, "5, '2', '34, '35Pauli, W. 134, '35, 142, 188, 191, 192,

213, 224, 226, 230, 240, 266, 267,300

Pegram, G. B. 246Peierls,R. 230Perrin, J. B. 9Perry,J. 244Persico,E. 282Petit, A. T. 96, 97Plcrte, W. S. 237Pickering,E. C. 113Planck, M. 44, 48, 5'-4, 68, 69, 78-86,

88, 103, '04, 118, 15', '52,201, 263Plucker,J. 34Plummer, H. G. 38Pohl, R. 18Poincare, H. 23, 30, 31, 33, 36, 37, 39,

48,5',64, 104, 151,246Poisson,S. D. 265Pons,J. L. 148Porter, H. L. 17Poynting,J. H. 67, 247, 248Priestley,J. 247Pringsheim,E. 78, 81, 84Ptolemy, Claudius, 144, 146

Rainich, G. Y. 180Raman, Sir C. V. 205Ramsauer, C. 2,8Ramsay, Sir W. 6, II

Rankine, W. J. M. 57Ray, B. B. 141Rayleigh, J. W. Strutt, third Baron

78, 97, 108,200, 296Rayleigh, R. J. Strutt, Icurth Baron,,6Reiche, F. 86, 100,200,201Reichenbacher, E. 192Reid, A. 218RelSSIler,H. 182

SUBJECT INDEX

Whittaker, Sir E. T. 146, 165, 170, 184,186, 194, 265, 269, 279

Whittaker, J. M. 153Wiechert, E. 158, 181Wien, W. 78,88, 122, 197Wiener, N. 267Wiener, O. 192Wigner, E. 192Wilberforce, L. R. 207Wilkens, A. 149Wills, A. P. 241Wioon, A. H. 230Wilson, C. T. R. 4, 18, 22, 210Wilson, H. A. 232, 245

Wilson, M. 245Wilson, W. 51, 118, 182, 191,217,282Wind, C. H. 18Wirtinger, W. 191Witmer, E. E. 234Wolfke, M. 87, 103Wood, A. B. 43Wood, R. W. 202Wyman, M. 183

Young, T. 217

Zacek, A. 141Zollner, F. 150

Subject Index

branching in radio-active series 14de Broglie waves associated with a

moving particle 214de Broglie's wave-function satisfies a

partial differential equation 268Brownian motion 9

canonical distribution, Gibbs' 86, 197canonical transformations in matrix

mechanics 263Christoffel symbols 160, 161clocks, relativist theory of rate of 42coefficients of emission and absorp-

tion, Einstein's theory of 197coherence of light 94, 95collisions, classical and quantum-

mechanical theory of 291-5cOlnlnutation-rule in matrix mechanics

260complexions 81Compton effect 208 ; quantum-mechani-

cal theory of 267, 304-5Compton wave-length 210

conduction in metals 230conformal transformations in four-

dimensional space 195its charge contact potential-diflerence between

two metals 90, 235energy-density of contracted curvature tensor 167

contraction of tensors 62contraction, FitzGerald, explained by

relativity theory 37contravariant vector or tensor 58correspondence theorem for frequencies

128

covariance of laws of physics 159

315

ballistic theory of light 38Bahner spectrum, Bohr's derivation of

110; discussed by Dirac and Pauli266

beta-particle, discovered 2 ;

and mass 3black-body radiation,

83body alpha 28Boltzmann constant k, introduced 82 ;

calculated 84Bose statistics 219

Bragg law for reflection of X-rays atcrystals 20

aberration, relativist theory of 40actinium, discovered 3adiabatic invariants 122sether, Cunningham's moving 248alpha particles, discovered 2; their

charge and mass 3; are heliumnuclei 6--8 ; scattering by matter 20 ;

produce disintegration of atomicnuclei 25-6

anomalous Zeeman effect 135; ex-planation of 136 ; quantum-mechani-cal theory of 3°1-2

aperiodic phenomena 267atom, Thomson's model 21; Lenard's

model 22; Rutherford's model 22-4average value of a variable in matrix-

mechanics 259Avogadro nwnber 8, 9, 18; calculated

by Planck 84azUnuthal quantum number 120, 131,

132

SUBJECT INDEX

electroDlllgnetisD'l, independent ofmetric geometry 195; Page's emis-sion theory of 249; fusion withgravitation 188-92

electrQIDotive force produced by alter-ing the velocity of a conductor243

electron, energy radiated by 247; fielddue to a moving 246

electronic charge, calculated by Planck85

electron-inertia, effects of 243electrons, diffraction of 218; extracted

from cold metals by electric fields236-7; in metals 230; thermalequilibrium with radiation 213

emission theory of electroll1llgnetism,Page's 249

emission and absorption, Einstein'stheory of coefficients of 197 ; they aredirected processes 199

Encke's comet, anomalous accelerationof 148

energy and temperature of a star, relationbetween 229; as observed by aparticular observer 74 ; connection ofmass with 51, 53; kinetic, relativistformula for 46, 48; conservation of68; conservation of, in impact, 51 ;of moving system 73

energy-molDentUD'l vector 69; con-nection with energy-tensor 72

energy-tensor 66, 67; connection withenergy-momentum vector 72; Min-kowski's, derived from Hilbert's world-function 172; expressed in terms ofLagrangean function 76

entropy, connected by the Boltzmann-Planck law with probability 82

equivalence, Einstein's principle of 152exchange interaction 307exchanges of energy between matter and

radiation 19B; of momentum inemission and absorption 199

exclusion-principle 142expanding universe 188

Fermi statistics 224ferromagnetism, explanation of 241-2filifonn disturbances 165; solutions of

partial differential equations 269fine-structure constant 120 ; of hydrogen

lines 120, 134Fitzgerald contraction 37five-diJDensional relativity 191flat space, relativity in 175fluctuations in radiation 101force in relativity 69fundamental teuaor 62, 63

316

eclipse values for deflection of light froma star 180

Eddington's fusion of electromagnetismwith g:<avitation 189

Einstein universe 183Eiastein's laws of motion, relation to

Newton's 168Einsteinian theory of gravitation, its

characteristics 158electric density and current, relativist

transformation of 39electrochemical potential 231electroD'lagnetic field, equations of, in

general space-time 160-4, 174electroll1llgnetic theory, classical, con-

nection with quantum theory ofemission 284-5 ; deduced from electro-statics 247

degeneracy 277diagonal matrices 259, 264diamagnetism, explanation of 238-42differentiation, covariant 161, 189diffraction, of reflected electrons 218;

of streams of material particles 219Dirac's q-nUD'lbers 266directed processes, emission and ab-

sorption are 199discontinuous jUD'lPS, not essential to

quantum theory 206-7disintegration of atomic nuclei 25-6dispersion, scattering, and refraction of

light, quantum theory of 200-6dispersion-electron 200displacement of spectral line emitted at

a place of high gravitational potential152,180

displacement-laws for radio-active dis-integrations 13

distance, spatial, between two particles186

divergence of a tensor 162, 163Doppler effect, relativist theory of 40,

92 ; transverse Doppler effect 42dual six-vectors 164dwarf stars, white 181, 228dynamics, relativist 44, 70

covariant vector or tensor 59covariant differentiation 161, 189crystals, diffraction of X-rays by 18Cunningham's stress in a moving

rether 248Curie's law of variation of paramagnetic

susceptibility with temperature, 239,241

current, electric, representation in wave-mechanics 294

curvature, scalar, 167

SUBJECT INDEX

ganuna - rays discovered 3; wave-lengths of 20

Geiger counter 7geometry, attempts to make physics

independent of 192-6Gibbs's canonical distribution 86Gibbs's thermodynamical potential

226grating, quantum theory of diffraction by

142-3gravitation, relation between Einsteinian

and Newtonian constants 170gravitational field, Einstein's equations

of 166-8, 173; potentials 157;properties essentially the same asin~rtial properties 152 ; waves 183

gyromagnetic ratio 244

hafniUDl 140Halley's comet, anomalous motion of

149HaDliIton's Principal Function, connec-

tion with Schr6dinger's equation270-2, 279--80; principle, connectionwith quantum theory 104

HaDliltonian form in relativity 47harmonic oscillator, solved by matrix

mechanics, 261-3; by wave mechanics288--g0

heliUDl, ionised, spectrum of I 14 ;nucleus 6-8; spectrum of 306

Huygens' principle in optics 271hydrogen atom, discussed by Dirac and

Pauli 266; discussed by wave-mechanics 273-5

hyperfine structure of spectral lines 142

Identical particles, quantum-mechanicaltheory of a system containing 306

inertial systems of reference 27, 35inequalities 144 ; secular, of Jupiter and

Saturn 145instantaneous three - dimensional

space of observer 159integral form of electromagnetic equa-

tions 250interference and diffraction are essen-

tially quantum effects 216interval 65ionisation-potential of a gas 91ionised heliUDl, spectrum of 114ioniUDl, parent of radium II

isotopes 11-13

kinetics of reactions 229

Lagrangean form in r.elativity 47lambda-term in the gravitational equa-

tions 183

Lande splitting-factor 135lead, end-product of radio-active series 13Leathem effect 249-50lever, Lewis and Tolman's bent 55light, deflection of a ray by a gravitating

body 180; rays are null geodesics 165;velocity of, with respect to an inertialframe of reference 159

light-darts 21 I

Lorentz's suggested explanation of gravity149-51

Lorentz transformations 33

Mach's principle 167magnetic moment of electron 134, 136magnetic quantUDl nUDlber 129magnetisation, mechanical effect pro-

duced by 244 ; produced by spinningrods about their axes 244

magneton, the VVeissand Bohr 242mass, connection with energy 51, 53;

proper 51matrices, diagonal 259, 264matrix-elements, physical meaning of

288matrix-mechanics, connection with

wave-mechanics 285-7matrix-theory 255--8Mercury, motion of perihelion of 148, 179,

180mesons, rate of disintegration 44metals, electrons in 230metric 62metric, general, introduced by Bateman

154Michelson-Morley experiment, theory

of 43mirror, light reflected from a moving 39mixed tensor 59molecular heats of gases 100momentUDl, of a photon 91 ; of moving

system 74; conservation of 68;Planck's definition of 54; relativist47,48

moon, secular acceleration of 146-8motion, relation of Einstein's to Newton's

laws 168multiplets 137multiply-periodic systems 127

needle radiation, J. J. Thomson's intro-duction of 94

Newtonian doctrine of gravitation, diffi-culties in 144; law of attraction,derived from Schwarzschild's solu-tion 179 ; mechanics, relativity of 27 ;laws of motion, relation to Einstein's168

normalisation of wave-function 275

radiation, discovery of Planck's law 81,83

radio-activity 1-8, 10-14, 20radium, discovered 2Raman effect 205Rayleigh's law of radiation, derived from

Planck's 83reaction on moving mass due to its

radiation 247; kinetics of 229rectilinear orbits in field of a single

particle 178refraction, scattering, and dispersion of

light, quantum theory of 200-6refractive index of a metal for electron-

waves 234relativity, principle of 30Ricci-tensor 167; identical relations of

173Riemann tensor 166rod, rotating, longitudinal vibrations of

43rotation of coil produced by starting

current 243Rydberg constant I I I

scalar 57scalar-curvature 167scattering, wave-mechanical derivation

of the Kramers-Heisenberg formula3°2-4

scattering of charged particles by acharged centre 295

scattering, refraction and dispersion,quantum theory of 200-6

Schrodinger's partial differentialequation for the wave-function 270,272

Schwarzschild's solution for a singlemassive particle 175-7

selection-principles 131, 132shells in atoms 137-9, 141, 142sbnultaneity 36de Sitter world 184-5, 187six-vectors 35, 60skew fundamental tensor of Ricci and

Levi-Civita 194-.kew tensor 59

q-numbers, 266quantum of action, introduced 83;

calculated 84quantum condition, expressed in terms

of de Broglie waves 216-17quantum mechanics, introduction of

term 204

SUBJECT INDEX

25~ proper-tUne 68proper-values 272proton, named 23

parallel-transport 189paraDlagnetism, explanation of 238, 242particle, free motion of, in gravitational

field 156; single massive, field of175-7

Peltier effect 23I

periodic table I I

perturbations, matrix theory of 264 ;wave-mechanical theory of 296

phase, of photons 95phase-surfaces of de Broglie waves 270-

271photo-chemical decomposition 91photo-electric phenomena 234-6; Ein-

stein's theory of 89; photo-electriceffect for gases 90 ; compound photo-electric effect 90

photo-molecules in radiation, numbersof 89, 103, 199

photons introduced 88; momentum of9IPlanck's second theory 103; third

theory 104Planck's law of pure temperature radia-

tion, discovery of 81,83Poisson-bracket, quantum analogue of

266polonium, discovered 2ponderoDlotive force in electrodyna-

mics 71positive-ray analysis 12

potential, electrochemical 231 ; Gibbs'sthermodynamical 226; gravitational157; of electromagnetic field 164;scalar and vector, Hargreaves' ex-pressions for 252

potential-vector, electric 75~precession of earth's axis, relativist 182pressure due to electromagnetic field 251principal function, connection with

Schrodinger's equation 270-2, 279-280

probability of value of energy found by ameasurement 284; that an electronis in a given volume-element 275;thermodynamic 82

proper-mas. 5I

operator corresponding to any physicalquantity 267

orchestra, virtual 204, 253orthogonal vectors 64orthogonality defined 64, 159; of wave-

functions 276outer product of tensors 61

nucleI, disintegration of atomicnull cone 65, 165null geodesics are tracks of rays of light

165

SUBJECT INDEX

Sntekal-Ra.maD eft"ect 205solutions of equatiolH of general

relativity, particular 182space quantification 130space-tilne, four-dimensional 64; as a

Cayley-Klein manifold 65spatial directions 65; spatial distance

between two particles 186specific heats of solids g6; Einstein's

formula for 97; Debye's formulafor 99

spectra, Bjerrum's theory of molecular108; Bohr's explanation of Balmerlines 110;- Conway's discovery re-garding 106; Nicholson proposes aquantum theory of 107

spin of electron 134Stark eft"ect, discovered 117; Schwarz-

schild and Epstein's theory 121 ; dis-cussed by matrix mechanics 267;wave-mechanical theory of 297

statistics, Bose 219 ; Fermi 224Stefan-BoltzD1&D law 86Stern-Gerlach eft"ect 130stinlulated emission Ig8superposition of states 283sYD1nletric tensor 59

telnperature and energy of a star,relation between 229

telnporal directions 65tensors 57therlnionic work-function gothernaionics 232

therntodynamic potential, Gibbs's 226therlnoelectric phenolnena 231

ThoD1son eft"ect 231transition of an atont, brought about

by collision I 15transition-probabilities 263transport, parellel 189transvection of tensors 62

unipolar induction 245universe, Einstein 183; expanding 188

vector, nature of ,56; contravariant 58 ;covariant 59

vectorial diveqreace of a tensor of ranktwo 163

velocity cannot exceed velocity of light38

virtual orchestra 204, 253virtual radiation, Bohr-Kramers-Slater

theory of 211Volta eft"ect 90, 235

wave-equation, general 279wave-function, physical significance of

275wave-length, Compton 210wave-Inechanics, connection with matrix

mechanics 285-7wave-packets 2go-1waves, gravitational 183Wentzel-Knuners-BrillouiD method for

approximate solution of wave-equation 280-3

Weyl's fusion of electromagnetism withgravitation 188

white dwarf stars 181, 228Whitehead's relativity theory 174Wien's displacement law 84; his law

of radiation derived from Planck's82

Wilson's cloud-eluunber 4Wilson-SoD1nlerfeld quantunl condi-

tions 118; connection with wave-mechanics 283

world, de Sitter 184-5, 187world-function, Mie's 154; Hilbert's

170; Lanczos's 174world-line of particle 158

X-rays, characteristic (K- and L-groups)16; Moseley's law of 24; diffractionof 18; polarisation of 14; spectra of138, 139; spectrometer for 20

ZeeD1&D eft"ect, anomalous 135; dis-cussed by matrix mechanics 267;Paschen-Back effect 135; Sommer-feld and Debye's quantum theoryof 128; wave-mechanical theory of300

zero-point energy 104

Whittaker History Aether Vol2

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