Case Study 5

Quantisation and Quanta up to 1905

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Case Study 5

Quantisation and Quanta up to 1905

The discovery of the quantum nature of the world we live in was arguably the greatestdiscovery of 20th century physics. It is a story which should be better known. In thespace of 25 years, physicists’ concept of nature changed completely. It is one of thegreatest stories in intellectual history.

Malcolm Longair has written up this story in some detail in five chapters in TheoreticalConcepts in Physics. I will only summarise some of the key points of the story. Youmight also want to consult the excellent book Black-Body Radiation and the QuantumDiscontinuity, 1894-1912 by Thomas S. Kuhn.

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Physics and Theoretical Physics in 1890

Unquestionably, the 19th century had seen a number of great triumphs for experimentaland theoretical physics.

• Lagrangian and Hamiltonian mechanics and dynamics.

• The foundations of thermodynamics – the first and second laws.

• Maxwell’s theory of the electromagnetic field, fully confirmed by Hertz’sexperiments.

• The Unification of light and electromagnetism.

• The beginnings of statistical mechanics.

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Physics and Theoretical Physics in 1890

There were, however, a number of serious problems which did not fit naturally into thescheme of classical physics.

• The null result of Michelson-Morley experiment - the aether and the propagation oflight.

• The ambiguous status of the kinetic theory of gases as propounded by Clausius,Maxwell and Boltzmann.

• The origin of the spectral lines in atomic and molecular spectra and Balmer’sformula.

• The origin of the photoelectric effect.

• The spectrum of black-body radiation.

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The Status of Black-body Radiation

At the Royal Institution in London, John Tyndall studied the intensity of radiation fromplatinum strip heated to different temperatures. The experiments were repeated byJosef Stefan who showed empirically that

I ∝ T4

where I is the radiant energy emitted per second per unit area by the hot body.

In 1884, Ludwig Boltzmann deduced this law from the thermodynamics of a cavity filledwith radiation and maintained at a temperature T . Hence the Stefan-Boltzmann law.

By 1897 Lummer and Pringsheim had shown by very careful measurements that thelaw was indeed correct with high precision.

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Lummer and Pringsheim’s Apparatusfor the Verification of Stefan’s Law

A is a vessel kept at 100C by beingimmersed in boiling water. B isimmersed in a bath of nitre and itstemperature could be varied between200 C and 600 C. Above thattemperature an iron cylinder was heatedto 1300 C by a gas furnace. G was aLummer-Kurlbaum bolometer.

To measure the spectrum of theradiation, the radiation from B waspassed first through a fluor-spar prismand the spectral energy distributionmeasured by a linear bolometer.

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The Classical Nature of the Stefan-Boltzmann Law

Notice that the law is founded upon precise experiment and Boltzmann’s derivation wasentirely classical using the thermodynamics of electromagnetic radiation.

From classical thermodynamics, he could show that

T

(

∂p

∂T

)

V=

(

∂U

∂V

)

T+ p

Thus, all Boltzmann needed was the relation between the pressure and energy densityfor electromagnetic radiation. From Maxwell’s equations, it is straightforward (seeMaxwell’s Theory of Electricity and Magnetism 1872) to show that

p = 13ε.

Inserting this result into the above relation and using U = V ε, we find

ε ∝ I ∝ T4.

Note: no mention of the spectrum of the radiation.

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Wien’s Displacement Law

The spectrum of black-body radiation was poorly known, but Wien used theoreticalarguments to derive its general form. He needed two key results:

• From the Stefan-Boltzmann law, the change in the radiation energy density in anadiabatic expansion is

ε ∝ V −4/3 and so T ∝ r−1

• In such an adiabatic expansion, the wavelength increases precisely propotional tothe size of the enclosure, λ ∝ r.

Wien considered what happens to the black-body energy density in the wavelengthinterval λ to λ+ dλ, u(λ) dλ when it is expanded adiabatically.

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Wien’s Displacement Law

By definition, in an adiabatic expansion, the equilibrium radiation spectrum must end upas an equilibrium spectrum at the lower temperature. Therefore, according to theStefan-Boltzmann law

u(λ1) dλ1

u(λ2) dλ2=

(

T41

T42

)

But, λ1T1 = λ2T2 and so dλ1 = (T2/T1) dλ2. Therefore,

u(λ1)

T51

=u(λ2)

T52

u(λ)T−5 = constant

But λT = constant and so

u(λ)λ5 = constant

Now, the equilibrium spectrum can only depend on T and λ and we know that λT =constant. Therefore, the spectrum of the radiation can only have the form

u(λ)λ5 = f(λT)

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Wien’s Displacement law in Frequency Form

We will need the expression for black-body radiation in frequency form and so we write

u(λ) dλ = u(ν) dν

λ =c

νdλ = −

c

ν2dν

and so

u(ν) dν = ν3f

(

ν

T

)

dν

This really is rather clever. Notice how far Wien was able to get, without knowinganything about the actual form of the spectrum. This was all new when Max Planckbegan his studies of the spectrum of black-body radiation.

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Max Planck and Black-body Radiation

Max Planck was an expert on classical thermo-dynamics and succeeded to Kirchhoff’s chair at Berlinin 1889. He turned his attention to black-body radiationin 1894, immediately following the publication of Wien’spaper.

His first paper seemed a major diversion from hisprevious work and concerned the emission ofelectromagnetic radiation by an oscillator. He termedthis conservative damping since the energy lost is notdissipated as heat, but can be reabsorbed by otheroscillators. In his words,

‘The study of conservative damping seems tome to be of great importance, since it opensup the prospect of a possible generalexplanation of irreversible processes bymeans of conservative forces.’

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Planck’s Results (1)

Planck derived the following results:

• The average rate of loss of energy of an oscillator of charge e executing harmonicoscillations x = x0 eiω0t is

−

⟨

dE

dt

⟩

=ω40x

20e

2

12πǫ0c3= γE

where γ = ω20e

2/6πǫ0c3m.

• The absorption of the radiation can be found from the expression for the forcedoscillations of the oscillator

x+ γx+ ω20x =

eE eiωt

m

The amplitude of the oscillations under the influence of the radiation field is

x0 =eE0

m(ω20 − ω2 + iγω)

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Planck’s Results (2)

Now, he worked out the form the equilibrium spectrum had to have when the emissionrate of the oscillator was precisely balanced by the energy absorbed by the radiationfield. He found the remarkable result

u(ν) =8πν2

c3E

where E is the average energy of the oscillator in thermal equilibrium with the radiation.

Note what a remarkable formula this is.

All trace of the oscillator has disappeared except its average energy.

Although the calculation proceeded through electrodynamics, Planck ended up with aremarkably thermodynamic looking expression.

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What Planck Did Not Do

Remarkably, Planck did not set E = kT , which would have resulted in theRayleigh-Jeans law. Probably he did not do this because he had already rejected thestatistical point of view of Maxwell and Boltzmann. Instead, in his words,

‘I had no alternative than to tackle the problem once again - this time from theopposite side - namely from the side of thermodynamics, my own hometerritory where I felt myself to be on safer ground. In fact my previous studiesof the Second Law of Thermodynamics came to stand me in good stead now,for at the very outset I hit upon the idea of correlating not the temperature ofthe oscillator but its entropy with its energy . . . Nobody paid any attention to themethod which I adopted and I could work out my calculations completely at myleisure, with absolute thoroughness, without fear of interference or competition.’

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The Entropy of Black Body Radiation

Planck’s reason for taking this route was that he had already derived a generalexpression which described how the system of oscillators would evolve towardsequilibrium and it depended upon the second derivative of the entropy with respect toenergy of the system.

Wien had proposed that the form of the black body spectrum, consistent with hisdisplacement law, was

u(ν) =8πα

c3ν3 e−βν/T

(see next slide). Therefore, from his relation between u(ν) and E, Planck could find anexpression for E

E = αν e−βν/T

Planck then used the thermodynamic relation(

∂S

∂E

)

V=

1

T= −

1

βνln

(

E

αν

)

to find an expression for the entropy S.

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The Shapes of Black-body Spectraat Different Temperatures

This spectrum is drawn in terms ofpower per unit wavelength interval.Wien argued that the function f(ν/T)

should be an exponential cut-off todescribe the sharp fall-off at shortwavelengths. Wien’s law could bewritten

I(ν) ∝ ν3 exp (−βν/T) .

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The Entropy of Black Body Radiation

Taking the next derivative, Planck found

∂2S

∂E2= −

1

βν

1

E

According to Planck’s analysis this meant that, since this second derivative isnecessarily negative, the system tends towards maximum entropy. In May 1899, hestated at the Prussian Academy of Sciences

‘I believe that this must lead me to conclude that the definition of radiationentropy and therefore Wien’s energy distribution law necessarily result fromthe application of the principle of the increase of entropy to the electro-magnetic radiation theory and therefore the limits of validity of this law, in so faras they exist at all, coincide with those of the second law of thermodynamics’.

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New Experiments by Rubens and Kurlbaum

Planck presented these calculations to the Prussian Academy of Sciences in June1900.

In October 1900, Rubens and Kurlbaum showed that Wien’s law was inadequate toexplain the distribution of black body radiation at low frequencies and hightemperatures. They showed that, at low frequencies and high temperatures, theintensity of radiation was proportional to temperature, u(ν) ∝ T .

This is clearly inconsistent with Wien’s law because, if

u(ν) ∝ ν3e−βν/T ,

then, for βν/T ≪ 1, u(ν) ∝ ν3 and is independent of temperature.

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What Planck Did

Planck immediately set about incorporating this feature into his expression for theentropy. From his analysis of the equilibrium of oscillators, he knew that

u(ν) =8πν2

c3E.

He now required u(ν) ∝ T at low frequencies and hence E ∝ T . Therefore,

∂S

∂E=

1

T∝

1

E

∂2S

∂E2∝

1

E2.

He now needed to combine this with his previous result, valid in the Wien region

∂2S

∂E2= −

1

βν

1

E

The standard way of combining these is to write

∂2S

∂E2= −

a

E(b+ E)

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What Planck Did

Integrating up this expression, Planck found

E =b

(eb/aT − 1)

and so the expression for the radiation spectrum became

u(ν) =8πν2

c3E =

8πν2

c3b

(eb/aT − 1)

This expression can now be compared with Wien’s formula in the low temperature, highfrequency limit

u(ν) =8πα

c3ν3 e−βν/T

b must be proportional to ν. Therefore,

u(ν) =Aν3

(eβν/T − 1)

This is the primitive form of Planck’s Law.

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The Rayleigh-Jeans Law (1)

Meanwhile, Lord Rayleigh had published an alternative approach to the determinationof the spectrum of black-body radiation, in particular, he aimed to obtain the correctform for the high temperature, low frequency spectrum. His important paper was onlytwo pages long.

As the author of The Theory of Sound, he was expert in the permitted modes ofoscillations of waves in a box. This greatly simplified the calculation of the equilibriumspectrum. The calculation in the paper is an example of working out the allowablemodes of oscillation of waves in a box. The wave equation is

∇2ψ =

1

c2∂2ψ

∂t2

To fit the waves into a cubical box of side L, we seek solutions of the form

ψ = C e−iωt sinlπx

Lsin

mπy

Lsin

nπz

L

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The Rayleigh-Jeans Law (2)

We find a dispersion relation of theform

ω2

c2=π2

L2(l2 +m2 + n2) =

π2p2

L2

where p2 = l2 +m2 + n2.We find the number of modesbetween p and p+ dp by the usualprocedure of drawing an octant ofsphere and asking how manymodes lie in the spherical shell ofthickness dp at radius p.

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The Rayleigh-Jeans Law (3)

If p is large, the number of modes is

n(p) dp = 184πp

2 dp

Expressing this result in terms of ω instead of p,

p =Lω

πcdp =

Ldω

πc

and hence

n(ω) dω =L3ω2 dω

2π2c3

For electromagnetic waves, there are two independent polarisations per mode and sowe need to multiply this result by two.

Now, Rayleigh made the key assumption. According to the Maxwell-Boltzmanndoctrine, we give kT of energy to each independent mode and hence we find theenergy density of radiation u(ν) dν in the box

u(ω) dω =ω2 dω

π3c3kT

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The Rayleigh-Jeans Law (4)

Rewriting the expression in terms of frequencies,

u(ν) =8πν2

c3kT

This is familiar. It is Planck’s result with E = kT .

• Notice that, in Rayleigh’s approach, he deals directly with the waves themselves,rather than the oscillators.

• Notice the use of the law of equipartition of energy. In the paper, Rayleigh wasaware of

‘. . . the difficulties which attend the Boltzmann-Maxwell doctrine of thepartition of energy.’

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The Rayleigh-Jeans Law (5)

• The expression diverges at high frequencies, the famous ultraviolet catastrophe,but Rayleigh proposed that the result ‘may apply to the graver modes’.

• Then, right out of the blue, in the fifth paragraph we read

‘If we introduce the exponential factor, the complete expression is’

in our notation

u(ν) =8πν2

c3kT e−βν/T

Obviously, Rayleigh included this expression so that the spectrum converged athigh frequencies and Wien’s exponential cut-off provided a good fit at highfrequencies.

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Comparison with Experiment

On October 25 1900, Rubens and Kurlbaum compared their measurements of thespectrum of black body radiation with five different predictions: (i) Planck’s formula, (ii)Wien’s law, (iii) Rayleigh’s law, and two other formulae.

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Comparison with Experiment

Planck’s result provided by far the best fit to the experiments. Rayleigh’s completeexpression was a poor fit to the data. Rayleigh was upset about this, since he regardedhis key result to be the expression for the spectrum at low frequencies, u(ν) ∝ T . Hepointed out

‘This is what I intended to emphasise. Very shortly afterwards the anticipationabove expressed was confirmed by the imporant researches of Rubens andKurlbaum who operated with exceptionally long waves.’

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Planck’s Theory of Black-body Radiation

In his scientific autobiography, Planck wrote:

‘On the very day when I formulated this law, I began to devote myself to thetask of investing it with a true physical meaning. This quest automatically ledme to study the interrelation of entropy and probability – in other words, topursue the line of thought inaugurated by Boltzmann.’

Planck recognised that the way forward consisted of adopting a point of view which hehad rejected in essentially all his previous work.

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What Planck Did

We showed how Planck derived an expression for the second derivative of the entropyof an oscillator with respect to energy

∂2S

∂E2= −

a

E(b+ E)

This can be integrated twice to find the entropy as a function of energy

S = −a

[

E

blnE

b−

(

1 +E

b

)

ln

(

1 +E

b

)]

This was the expression which Planck attempted to explain through Boltzmann’sstatistical procedures, but he was no expert in these techniques. In fact, he did notfollow the Boltzmann prescription at all.

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Planck’s analysis

Planck’s analysis begins like a classical piece of statistical mechanics. He assumedthere was a fixed total energy E = EN to be divided among the N oscillators, where Eis the mean energy of the oscillator. As in the standard Boltzmann procedure, heintroduced the microscopic energy elements ǫ. Therefore, he knew exactly how manyenergy elements P he had to distribute over the oscillators P = EN/ǫ.

What Planck did was simply to work out the total number of ways W in which the Pelements can be distributed over the N oscillators and then defined the entropy to be

S = C lnW.

It is left as an exercise to show that

W ≈(N + P)N+P

PPNN

(see TCP Section 10.3)

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Planck’s analysis

Therefore

S = C lnW = −C

[

E

ǫlnE

ǫ−

(

1 +E

ǫ

)

ln

(

1 +E

ǫ

)]

This is precisely Planck’s expression for the entropy of an oscillator, provided weidentify b with ǫ. We recall that b must be proportional to frequency ν and so Planckwrote the relation in the famous form which will be used forever

ǫ = hν

According to classical statistical mechanics, we should now let ǫ tend to zero, but wecannot obtain the correct expression for the entropy of the oscillator if we do this.

Furthermore, the constant C must also be a universal constant which can be foundfrom any entropy calculation. It comes as no surprise that C = k.

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Planck’s Answer

Hence he derived the final form for the Planck distribution.

u(ν) =8πhν3

c31

ehν/kT − 1

• Planck had certainly not followed Boltzmann’s procedures. As Einstein pointed out,

‘The manner in which Mr. Planck uses Boltzmann’s equation is ratherstrange to me in that a probability of a state W is introduced without aphysical definition of this quantity. If one accepts this, then Boltzmann’sequation simply has no physical meaning’

• There is a logical inconsistency in the argument. The oscillators can only take finiteenergies E = rǫ and yet a classical formula has been used to work out theradiation rate of the oscillator.

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The Invention of Quantisation

It is fair to say that no one quite understood what Planck had done. Planck spent manyyears trying to reconcile his results with classical theory, but failed. Very few scientiststook it at all seriously and Planck’s paper was essentially the only paper on the subjecttill 1905.

In fact, Rosenfeld and Klein have suggested that Planck actually worked backwardsand derived the expression for the probability from his expression for the entropy of anoscillator.

But, despite these basic problems, the concept of quantisation had appeared for thefirst time and it was not going to go away.

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From Planck’s Autobiography

Let us read what Planck said about his attitude to his analysis.

‘My futile attempts to fit the elementary quantum of action somehow into theclassical theory continued for a number of years and they cost me a great dealof effort. Many of my colleagues saw in this something bordering on tragedy.But I feel differently about it. For the thorough enlightenment I thus receivedwas all the more valuable. I now knew for a fact that the elementary quantumof action played a far more significant part in physics than I had originally beeninclined to suspect and this recognition made me see clearly the need for theintroduction of totally new methods of analysis and reasoning in the treatmentof atomic problems.’

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Why Did Planck Take the Analysis so Seriously?

Part of the reason was that, by introducing the new fundamental constant h, he coulddefine a set of fundamental units determined by the gravitational constant G, the speedof light c and Planck’s constant h.

Planck Units

Unit Defining expression SI Value

Time tPl = (Gh/c5)1/2 10−43 sLength lPl = (Gh/c3)1/2 4 × 10−35 mMass-energy mPl = (hc/G)1/2 5.4 × 18−8 kg ≡ 3 × 1019 GeV

These are the Planck units which are often used in modern cosmology for studies ofthe very early Universe.

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Einstein in 1905In 1905, Albert Einstein was 26 and wasemployed as ‘technical expert third class’ at theSwiss patent office in Bern. In that year, hepublished three papers, which are among thegreatest classics of all physics. Any one of themwould have ensured that his name remained apermanent fixture in scientific history. Thesepapers are:

1. ‘On the theory of Brownian motion’2. ‘On the electrodynamics of moving bodies’3. ‘On a heuristic viewpoint concerning the

production and transformation of light’.

All three appeared in Volume 17 of the Annalender Physik of 1905.

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‘On a Heuristic Viewpoint Concerningthe Production and Transformation of Light’

Einstein (1905)

‘There is a profound formal difference between the theoretical ideas which physicistshave formed concerning gases and other ponderable bodies and Maxwell’s theory ofelectromagnetic processes in so-called empty space. Thus, while we consider the stateof a body to be completely defined by the positions and velocities of a very large butfinite number of atoms and electrons, we use continuous three-dimensional functions todetermine the electromagnetic state existing within some region, so that a finite numberof dimensions is not sufficient to determine the electromagnetic state of the regioncompletely . . . ’

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Einstein (1905) – continued

‘The undulatory theory of light, which operates with continuous three-dimensionalfunctions, applies extremely well to the explanation of purely optical phenomena andwill probably never be replaced by any other theory. However, it should be kept in mindthat optical observations refer to values averaged over time and not to instantaneousvalues. Despite the complete experimental verification of the theory of diffraction,reflection, refraction, dispersion and so on, it is conceivable that a theory of lightoperating with continuous three-dimensional functions will lead to conflicts withexperience if it is applied to the phenomena of light generation and conversion.’

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Einstein’s Perspective

The revolutionary nature of Einstein’s proposal was that, for certain purposes, it mightbe preferable to think of light as particles rather than waves. Specifically, he had inmind the problems of the spectrum of black body radiation, the photoelectric effect andfluorescence.

Recall that it was only 15 years since Hertz’s magnificent experiments had fullyvindicated Maxwell’s equations. It must have looked like an unfortunate rerun of thecontroversy concerning the corpuscular versus the wave picture of light.

Note that, unlike Planck, Einstein operated directly with the radiation itself and not withthe source of the radiation.

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Einstein and the Entropy of Black-body Radiation

Einstein began with the entropy of radiation. In outline, the computation goes as follows.First, he worked out a suitable expression for the entropy density per unit frequencyinterval φ(ν) corresponding to the energy density per unit frequency interval u(ν).

∂S

∂E=

1

T

∂φ(ν)

∂u(ν)=

1

T

Then, he considered only the Wien region of the spectrum in which all the problemswere known to arise. From Wien’s formula

u(ν) =8πα

c3ν3 e−βν/T ,

he could find the expression for φ(ν) from

1

T=

1

βνln

8παν3

c3u(ν)=∂φ(ν)

∂u(ν)

This can be integrated to find φ(ν).

40

The Change in Entropy of Black-body Radiation

Einstein now considered the entropy of the radiation in the spectral range ν to ν + ∆ν

within volume V . Its energy is E = V u(ν)∆ν.

S = V φ(ν)∆ν = −E

βν

(

lnEc3

8παν3V∆ν− 1

)

Now, suppose we expand the volume from V0 to V . Then, the change of entropy is

S − S0 =E

βνln

(

V

V0

)

But, this expression looks familiar. It is one of the classical problems of Boltzmann’sstatistical mechanics. What is the change in entropy if we expand a gas of N particlesfrom volume V0 to V at constant energy?

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The Entropy of Particles in a Box

We use Boltzmann’s relation in the form

S − S0 = k lnW

W0

where W is the probability that the N particles are in volume V and W0 that they are inV0. Obviously, W/W0 = (V/V0)

N , since the probability for one particle is V/V0. Thus,

S − S0 = kN lnV

V0

But this form is identical to the formula he had derived from the Wien formula. Hence,N = E/kβν. He immediately concluded

Monochromatic radiation of low density (within the limits of validity of Wien’sradiation formula) behaves thermodynamically as though it consisted of anumber of independent energy quanta of magnitude kβν.

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Einstein (1905)

From Planck’s analysis, he knew that β = h/k, and hence the quantum of radiation is

ǫ = hν.

Finally, he worked out the mean energy of the quanta in the Wien spectral energydistribution. It is straightforward to show that E = 3kT .

So far, Einstein had written that the radiation behaved as though it consisted ofindependent photons. He then goes on to write

‘. . . the next obvious step is to investigate whether the laws of emission andtransformation of light are also of such a nature that they can be interpreted orexplained by considering light to consist of such energy quanta.’

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Einstein’s Three Pieces of Evidence

• Stokes’ rule – the frequency of photoluminescent emission is always less than thefrequency of the incident light.

• The photoionisation of gases. The energy of quanta must be greater than theionisation potential of gas to obtain ionisation.

• The photoelectric effect. Lenard had shown that the energies of the photoelectronswere independent of the intensity of the radiation. Einstein reasoned that a certainenergy W , the work function, is needed to remove an electron from the surface andso, by conservation of energy, the kinetic energy of an electron ejected by aquantum of energy hν is

Emax = hν −W

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The Photoelectric Effect

If a reverse voltage V is applied to the tube so that no current flows, then eV = Emax.Hence,

V =h

eν −

W

e

In Einstein’s words,

‘If the formula derived is correct, then V must be a straight line function of thefrequency of the incident light, when plotted in Cartesian coordinates, whoseslope is independent of the nature of the substance investigated.’

The slope was simply h/e. This was an amazing prediction because nothing wasknown about the frequency dependence of the photoelectric effect. The relation wasonly found after ten years of difficult experimentation by Millikan in 1916.

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