Quantum Physics - Università degli Studi...

Quantum Physics

(student handout version)

Philip G. Ratcliffe

([email protected])

Dipartimento di Scienza e Alta TecnologiaUniversità degli Studi dell’Insubria in Como

via Valleggio 11, 22100 Como (CO), Italy

(last revised 7th March 2016)

Preface

The present is a written version of lecture notes for the last in a series of threecourses on Quantum Physics, presented in the second and third years of the firstdegree course in Physics and the two-year Master’s equivalent degree course start-ing in the academic year 2002/03 at Insubria University in Como. They have,however, been augmented and edited with the aim of being self-contained andhopefully therefore of more general use. These notes are thus intended primarilyfor use by students with a basic knowledge of classical electromagnetism and spe-cial relativity besides, of course, a solid grounding in the fundamentals of quantummechanics.

i

ii PREFACE

Contents

Preface i

Contents iii

III Advanced Quantum Mechanics 1

1 Introduction to Part III 31.1 Suggested supplementary reading . . . . . . . . . . . . . . . . . . . 31.2 Aims and philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Conventions and notation . . . . . . . . . . . . . . . . . . . . . . . 41.4 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Quantum Dynamics 72.1 Temporal evolution in quantum mechanics . . . . . . . . . . . . . . 7

2.1.1 The Schrödinger equation . . . . . . . . . . . . . . . . . . . 92.1.2 Energy eigenstates . . . . . . . . . . . . . . . . . . . . . . . 122.1.3 Spin precession in a magnetic field . . . . . . . . . . . . . . 13

2.2 A brief review of the “pictures” . . . . . . . . . . . . . . . . . . . . 152.2.1 The Heisenberg and Schrödinger pictures . . . . . . . . . . . 152.2.2 The Heisenberg equations of motion . . . . . . . . . . . . . . 162.2.3 Ehrenfest’s theorem . . . . . . . . . . . . . . . . . . . . . . . 182.2.4 Basis states . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.5 The interaction picture . . . . . . . . . . . . . . . . . . . . . 19

2.3 Propagators and the Feynman path integral . . . . . . . . . . . . . 222.3.1 Propagators in wave mechanics . . . . . . . . . . . . . . . . 222.3.2 Propagators as transition amplitudes . . . . . . . . . . . . . 262.3.3 Path integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 272.3.4 The Feynman path integral . . . . . . . . . . . . . . . . . . 28

2.4 Potentials and gauge transformations . . . . . . . . . . . . . . . . . 332.4.1 Constant potentials . . . . . . . . . . . . . . . . . . . . . . . 33

iii

iv CONTENTS

2.4.2 Gravity in quantum mechanics . . . . . . . . . . . . . . . . . 352.4.3 Gauge transformations in electromagnetism . . . . . . . . . 382.4.4 The Aharonov–Bohm effect . . . . . . . . . . . . . . . . . . 41

2.5 The adiabatic approximation . . . . . . . . . . . . . . . . . . . . . . 452.5.1 The adiabatic theorem . . . . . . . . . . . . . . . . . . . . . 452.5.2 The Berry phase . . . . . . . . . . . . . . . . . . . . . . . . 47

2.6 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3 Angular Momentum 513.1 Angular-momentum eigenstates and eigenvalues . . . . . . . . . . . 52

3.1.1 Commutation relations and ladder operators . . . . . . . . . 523.1.2 The eigenvalues of J

2 and Jz . . . . . . . . . . . . . . . . . 533.1.3 Matrix elements of angular-momentum operators . . . . . . 543.1.4 Rotations in quantum mechanics . . . . . . . . . . . . . . . 553.1.5 Spherical harmonics and the rotation matrices . . . . . . . . 56

3.2 Composition of angular momenta . . . . . . . . . . . . . . . . . . . 583.2.1 A simple example . . . . . . . . . . . . . . . . . . . . . . . . 583.2.2 Clebsch–Gordan coefficients . . . . . . . . . . . . . . . . . . 593.2.3 Recurrence relations for the CG coefficients . . . . . . . . . 613.2.4 The Clebsch–Gordan series . . . . . . . . . . . . . . . . . . . 66

3.3 The EPR paradox and the Bell inequalities . . . . . . . . . . . . . . 673.3.1 The Einstein–Podolsky–Rosen paradox . . . . . . . . . . . . 673.3.2 Proof of the Bell inequalities . . . . . . . . . . . . . . . . . . 71

3.4 Tensorial operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.4.1 Cartesian and irreducible tensors . . . . . . . . . . . . . . . 753.4.2 Tensor products . . . . . . . . . . . . . . . . . . . . . . . . . 783.4.3 Tensor matrix elements . . . . . . . . . . . . . . . . . . . . . 813.4.4 The Wigner–Eckart theorem . . . . . . . . . . . . . . . . . . 81

3.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4 Symmetry 874.1 Symmetries and conservation laws . . . . . . . . . . . . . . . . . . . 87

4.1.1 Symmetries in classical physics . . . . . . . . . . . . . . . . 874.1.2 Noether’s theorem . . . . . . . . . . . . . . . . . . . . . . . 884.1.3 Continuous symmetries in quantum mechanics . . . . . . . . 88

4.2 Discrete symmetries in quantum mechanics . . . . . . . . . . . . . . 894.2.1 Charge conjugation (C) . . . . . . . . . . . . . . . . . . . . . 894.2.2 Spatial inversion (P) . . . . . . . . . . . . . . . . . . . . . . 904.2.3 Time reversal (T) . . . . . . . . . . . . . . . . . . . . . . . . 97

4.3 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

CONTENTS v

5 Perturbation Theory and Scattering 1095.1 Time-dependent potentials . . . . . . . . . . . . . . . . . . . . . . . 109

5.1.1 The Rabi formula . . . . . . . . . . . . . . . . . . . . . . . . 1095.1.2 Magnetic spin resonance . . . . . . . . . . . . . . . . . . . . 1115.1.3 The maser . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.2 Time-dependent perturbation theory . . . . . . . . . . . . . . . . . 1145.2.1 The Dyson series in the interaction picture . . . . . . . . . . 1145.2.2 Transition amplitudes . . . . . . . . . . . . . . . . . . . . . 1155.2.3 Constant perturbations . . . . . . . . . . . . . . . . . . . . . 1175.2.4 Fermi’s golden rule . . . . . . . . . . . . . . . . . . . . . . . 1195.2.5 Harmonic perturbations . . . . . . . . . . . . . . . . . . . . 1215.2.6 Interaction of EM radiation and matter . . . . . . . . . . . . 1225.2.7 Energy shift and decay width . . . . . . . . . . . . . . . . . 127

5.3 Scattering theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1315.3.1 The Lippmann–Schwinger equation . . . . . . . . . . . . . . 1325.3.2 The Born approximation . . . . . . . . . . . . . . . . . . . . 1385.3.3 Higher-order Born approximations . . . . . . . . . . . . . . . 1425.3.4 The optical theorem . . . . . . . . . . . . . . . . . . . . . . 1435.3.5 The eikonal approximation . . . . . . . . . . . . . . . . . . . 1455.3.6 The free particle . . . . . . . . . . . . . . . . . . . . . . . . 1515.3.7 Partial waves . . . . . . . . . . . . . . . . . . . . . . . . . . 1555.3.8 The scattering of identical particles . . . . . . . . . . . . . . 175

5.4 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

6 Relativistic Quantum Mechanics 1796.1 The Klein–Gordon equation . . . . . . . . . . . . . . . . . . . . . . 179

6.1.1 Derivation of the Klein–Gordon equation . . . . . . . . . . . 1796.1.2 Interpretation of the Klein–Gordon equation . . . . . . . . . 180

6.2 The Dirac equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1826.2.1 A modern derivation . . . . . . . . . . . . . . . . . . . . . . 1826.2.2 The historical derivation . . . . . . . . . . . . . . . . . . . . 1836.2.3 Probability current in the Dirac equation . . . . . . . . . . . 1866.2.4 Interpretation of the Dirac equation . . . . . . . . . . . . . . 1876.2.5 The γ-matrix algebra . . . . . . . . . . . . . . . . . . . . . . 1906.2.6 Lorentz transformations . . . . . . . . . . . . . . . . . . . . 1936.2.7 Discrete transformations (CPT) . . . . . . . . . . . . . . . . 1996.2.8 Spinor bilinears . . . . . . . . . . . . . . . . . . . . . . . . . 2016.2.9 General free-particle solutions . . . . . . . . . . . . . . . . . 2036.2.10 Wave-packets . . . . . . . . . . . . . . . . . . . . . . . . . . 2086.2.11 The Klein paradox . . . . . . . . . . . . . . . . . . . . . . . 2126.2.12 The negative-energy solutions . . . . . . . . . . . . . . . . . 214

vi CONTENTS

6.3 Relativistic scattering theory . . . . . . . . . . . . . . . . . . . . . . 2176.3.1 The Mott cross-section . . . . . . . . . . . . . . . . . . . . . 217

6.4 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

A Answers to Selected Exercises 223A.1 Exercises to Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . 223A.2 Exercises to Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . 230

B Supplementary Topics 233B.1 The Hellmann–Feynman theorem . . . . . . . . . . . . . . . . . . . 233B.2 The path integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235B.3 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

C Glossary of Acronyms 239

Part III

Advanced Quantum Mechanics

1

Chapter 1

Introduction to Part III

1.1 Suggested supplementary reading

A (very short) list of suggested supplementary reading material follows. The bookby Sakurai covers much of the material I shall present (with the notable exceptionof the relativistic formulation). Certain topics are, however, dealt with in moredetail in the other texts.

The classic text by Bjorken and Drell will be most useful for the developmentof relativistic quantum mechanics. The volume by Griffiths contains useful discus-sions of, among other topics, the Berry phase, scattering theory and Bell’s theorem.The book by Bailin and Love provides a very clear introduction to the Feynmanpath-integral formalism, while the remaining two volumes (Itzykson and Zuber;Schiff) are intended as complementary reading in general.

Reading list

Bailin, D. and Love, A. (1993), Introduction to Gauge Field Theory (Inst. of Phys.),revised edition.

Bjorken, J.D. and Drell, S.D. (1964), Relativistic Quantum Mechanics (McGraw–Hill).

Griffiths, D.J. (2005), Introduzione alla Meccanica Quantistica (Casa Editrice Am-brosiana).

Itzykson, C. and Zuber, J.-B. (1980), Quantum Field Theory (McGraw–Hill).

Sakurai, J.J. (1985), Modern Quantum Mechanics (Benjamin Cummings).

3

4 CHAPTER 1. INTRODUCTION TO PART III

Schiff, L.I. (1968), Quantum Mechanics (McGraw–Hill), 3rd. edition.

1.2 Aims and philosophy

The main aim of this third and final part of the course on quantum mechanics is tocomplete the studies carried out in the first two parts and provide the student witha more-or-less complete working knowledge of the subject (that is, for problemsolving) and the necessary preparation to take on the more advanced topic ofquantum field theory.

One of the most powerful tools, which has now become indispensable to almostevery theoretical physicist dealing with the quantum world, is the path-integralformalism developed by Feynman (1950); see also Feynman and Hibbs (1965).While it is true that this approach is far more powerful than necessary to solvemost problems in ordinary quantum mechanics, when applied to the quantisationof field theory, the simplifications and gain in insight with respect to the canonicalapproach are priceless. In view of this and considering the importance of theconcept, it seems appropriate to introduce the formalism here.

The other two main “themes” to this course will be the implications and im-portance of symmetries (both discrete and continuous) in quantum mechanicsgenerally and the relativistic formulation of quantum mechanics, as developed byDirac (1928).

1.3 Conventions and notation

It is common practice to indicate operators via a circumflex, e.g. H . In the fol-lowing chapters, however, to avoid unnecessary cluttering of formulæ, unless thereis a real ambiguity, we shall avoid such notation and rely on the context as in-dicator of the nature (operator or variable) of the various symbols used. Indeed,the circumflex will usually indicate a unit vector: thus, x≡x/|x|. Note also thatvectors in three-dimensional space will be set in bold typeface x while four-vectors(introduced in Chap. 6) will be set in normal typeface—there should be no causefor confusion however. In writing scalar products, in either three- or four-space, itis convenient to adopt the ? summation convention for repeated indices: thus, forexample, x·y≡xiyi.

The bra–ket notation, introduced in ? by ?, is now universally adopted as aconvenient and powerful short-hand. It will thus be the main language used herefrom the very start of the course. Students unfamiliar with the usage would bewell advised to consult the less advanced texts, in preparation.

1.4. BIBLIOGRAPHY 5

1.4 Bibliography

Dirac, P.A.M. (1928), Proc. Royal Soc. (London) A117, 610.

Feynman, R.P. (1950), Phys. Rev. 80, 440.

Feynman, R.P. and Hibbs, A.R. (1965), Quantum Mechanics and Path Integrals(McGraw–Hill).

6 CHAPTER 1. INTRODUCTION TO PART III

Chapter 2

Quantum Dynamics

2.1 Temporal evolution in quantum mechanics

In the first part of the quantum mechanics course mainly static systems wereexamined. That is, systems, although possibly involving motion, that did not altertheir characteristics with time—the motion being typically periodic or uniform,as in the case of an electron bound inside an atom. An important aspect ofthe quantum description of physics, introduced in the second part, is evolutionunder the influence of forces that actually alter the nature of systems. An obviousexample might be the emission of a quantum of radiation by an atom in an excitedstate, which then returns to its ground or lowest-energy state. Or, more simply, thescattering of an electron by an external electromagnetic field. We shall re-examinethis subject here using an alternative and more powerful approach.

Before attacking the problem of temporal evolution it is important to recall theparticular nature, in quantum mechanics, of the variable called time. Indeed, weshould speak of time as a parameter—it is not something we can alter at will nor,indeed, measure in the same sense that we measure a distance or a momentum. Inquantum mechanics there is no operator associated with time. It should be thoughtof as merely a label identifying, distinguishing and ordering our photographs ofreality. Anticipating the approach of quantum field theory, let us simply remarkthat the natural symmetry between space and time will be restored there, not byelevating time to the standing of a variable (or observable), but by reducing (ordegrading, as it were) position to the same rôle of mere parameter.∗ Such a moveis already nascent in the general description of quantum physics through the useof wave-functions ψ(x,t), which indicate the probability of finding a particle ata certain position x at a given instant t, but do not themselves actually measure

∗ Although we might perhaps remark that the introduction of the four-vector xµ in relativisticquantum mechanics naturally leads to a time operator.

7

8 CHAPTER 2. QUANTUM DYNAMICS

position; this only becomes necessary in relation to experimental observation.∗

The problem we are posed is the description of a given state, |α〉 say, at sometime t0 and its subsequent temporal evolution to a later instant t. We may expressthis compactly as:

|α,t0;t〉 (t > t0), (2.1.1)

which, in words, is “the state at time t that was |α〉 at an earlier time t0.” Thestandard procedure, already met, is that of introducing the so-called temporal-evolution operator U(t,t0). We shall thus write

|α,t0;t〉 = U(t, t0)|α,t0〉, (2.1.2)

where, temporarily, we indicate explicitly that |α〉 is the state at time t0.Clearly, there are natural restrictions to be placed on the operator U. First of

all, the requirement that probability be conserved in quantum mechanics (statescannot simply appear from or disappear into the vacuum) implies that the normbe preserved. In other words, U must be unitary. To see this, consider a genericstate expanded over a basis set of eigenstates |a〉 of some operator A:

|α,t0;t〉 =∑

a

ca(t)|a〉. (2.1.3)

Assuming the eigenstates to be properly normalised, the generalised Fourier coef-ficients ca(t) simply represent the probability amplitude for obtaining result a fora measurement A performed at time t; in other words, |ca(t)|2 is the probabilityof obtaining result a. We must therefore have

∑

a

c2a(t) = 1 (∀ t), (2.1.4)

or, equivalently,〈α,t0;t|α,t0;t〉 = 〈α,t0|α,t0〉. (2.1.5)

Then, since〈α,t0;t|α,t0;t〉 = 〈α,t0|U†(t,t0)U(t,t0)|α,t0〉 (2.1.6)

must hold for any state |α,t0〉, we require

U†(t, t0)U(t, t0) = 1, (2.1.7)

where 1 is just the identity operator. This fundamental condition is known asunitarity and is intimately related to the probabilistic interpretation of quantum∗ Note, however, that even experimental measurement always requires statistical or probabilisticinterpretation and analysis.

2.1. TEMPORAL EVOLUTION IN QUANTUM MECHANICS 9

mechanics.A somewhat trivial, but necessary, requirement is the property of composition:

U(t, t0) = U(t, t′)U(t′, t0) (t0 < t′ < t). (2.1.8)

With this property in hand, it makes sense to work with infinitesimal temporaltranslations:

|α,t0;t0+δt〉 = U(t0 + δt, t0)|α,t0〉. (2.1.9)

Finally, continuity of the solutions requires that

U(t0, t0) = 1. (2.1.10)

It is not difficult to see then that the required operator must take the (infinitesimal)form

U(t0 + δt, t0) = 1− iΩ(t0)δt, (2.1.11)

where the unitarity condition (2.1.7) requires that Ω be an Hermitian operator(Ω†=Ω). It is immediately obvious that Ω has dimensions of frequency. Recallnow that in classical mechanics the Hamiltonian H is the generator of temporaltranslations and that if our system is to be invariant under time-reversal, H toomust be Hermitian. A natural identification is therefore

Ω = 1~H (2.1.12)

and we shall thus takeU(t0 + δt, t0) = 1− i

~H δt. (2.1.13)

If the Ehrenfest theorem (see Sec. 2.2.3) is to hold and thus guarantee the correctclassical limit, the constant of proportionality ~ introduced must clearly be thesame as that used in the relationship involving spatial translations or, equivalently,in the de Broglie relationship. As we shall now see, the hypothesis of a universalconstant ~ is sufficient to derive the Schrödinger equation.

2.1.1 The Schrödinger equation

As noted, the preceding assumptions are sufficient to derive the differential equa-tions governing the temporal evolution of physical systems. Let us begin with theevolution operator itself: we have

U(t+ δt, t0) = U(t+ δt, t)U(t, t0) =(1− i

~Hδt

)U(t, t0). (2.1.14)


The difference in U between t and t+δt is thus

δU(t, t0) ≡ U(t+ δt, t0)− U(t, t0) = − i~HδtU(t, t0), (2.1.15)

from which we immediately obtain the desired differential, or Schrödinger, equationfor U:

i~d

dtU(t, t0) = H U(t, t0). (2.1.16)

Once we know how U evolves, the evolution of the states is also determined:

i~d

dt

[U(t, t0)|α,t0〉

]=

[i~

d

dtU(t, t0)

]|α,t0〉 = H U(t, t0)|α,t0〉, (2.1.17)

since |α,t0〉 is independent of t. We finally rewrite this as

i~d

dt|α,t0;t〉 = H|α,t0;t〉. (2.1.18)

It is, however, clear from the foregoing that once we have U, we may abandonthe Schrödinger equation as superfluous; all information regarding the evolutionof the states is unambiguously contained in U. It is important to underline thesimplification this engenders: we need only obtain the solution of Eq. (2.1.16) andapply it to any given starting ket. We shall need to consider three different possiblecases of increasing complexity:

1. H is time independent,

2. H is time dependent, with [H(t),H(t′)]=0 (∀ t and t′),

3. H is time dependent, with [H(t),H(t′)] 6=0 (for t′ 6= t).

Case 1: In this case H does not vary with time, in other words, the source ofthe force fields is constant (e.g. a spinning, charged particle in a constant magneticfield). The solution of Eq. (2.1.16) is then simply∗

U(t, t0) = exp[− i

~H·(t− t0)

]. (2.1.19)

To verify that this really is a solution, it is sufficient to expand† in powers ofH and differentiate. Alternatively, one can divide the full interval [t0,t] into N

∗ The dot placed between H and (t− t0) is used here merely to clarify that (t− t0) multiplies Hand is not its argument.

† Indeed, a function of an operator such as H is to be understood precisely in terms of itspower-series expansion.


subintervals of equal length δt=(t− t0)/N . One then has

U(t, t0) = U(t, t− δt) U(t− δt, t− 2δt) · · ·U(t0 + δt, t0)

=[1− i

~H·(t− t0)/N

]N. (2.1.20)

Taking the limit N→∞ leads to Eq. (2.1.19).

Exercise 2.1.1. Check result Eq. (2.1.19) by expansion in powers of H.

Case 2: If H depends on t but commutes with itself at different times, then theformal solution may be written as

U(t, t0) = exp

[− i

~

∫ t

t0

dt′H(t′)

]. (2.1.21)

This can be proved in a similar manner to the previous case. An example mightbe a spinning, charged particle in a magnetic field of constant direction but time-varying intensity.

Exercise 2.1.2. Prove result (2.1.21).

Case 3: This is the most general case: H may depend on t and does not neces-sarily commute with itself at different times. The example might now become aspinning, charged particle in a time-dependent magnetic field of varying direction(recall that the operators sx and sy, e.g. do not commute). The solution is a littlemore difficult here. At least formally, we may integrate Eq. (2.1.16) to obtain

U(t, t0) = 1− i

~

∫ t

t0

dt′H(t′)U(t′, t0), (2.1.22)

where the 1 appears owing to the boundary condition U(t0,t0)=1. This is clearlynot much use since U still appears on the right-hand side. We may, however,substitute the entire right-hand side, evaluated at t= t′, for U in the integrand:

U(t, t0) = 1− i

~

∫ t

t0

dt′H(t′) +

(i

~

)2∫ t

t0

dt′H(t′)

∫ t′

t0

dt′′H(t′′)U(t′′, t0). (2.1.23)

The process of substituting with Eq. (2.1.22) for U in the last integrand may beiterated to obtain the (formal) infinite power-series (in H), or Neumann series,


solution

U(t, t0) = 1+

∞∑

n=1

(− i

~

)n∫ t

t0

dt1

∫ t1

t0

dt2 · · ·∫ tn−1

t0

dtnH(t1)H(t2) · · ·H(tn).

(2.1.24)This is known as the Dyson series (although the original treatment by Dyson wasactually performed within the framework of quantum field theory).

Without proof, we express the Dyson series in a more symmetric fashion:

U(t, t0) = 1+

∞∑

n=1

1

n!

(− i

~

)n∫ t

t0

dt1

∫ t

t0

dt2 · · ·∫ t

t0

dtn T(H(t1)H(t2) · · ·H(tn)

).

(2.1.25)where the operator T stands for time ordering. It is defined by

T(O(t) O(t′)

)≡O(t) O(t′) for t > t′,

O(t′) O(t) for t < t′.(2.1.26)

This all permits a rather compact final expression:

U(t, t0) = T

exp

[− i

~

∫ t

t0

dt′H(t′)

]. (2.1.27)

Exercise 2.1.3. Derive Eq. (2.1.25) explicitly, in particular the factor 1/n!, start-ing from the less symmetric Eq. (2.1.24).Hint: consider the effect of reordering the integrals—the indices 1,... ,n are merelylabels that may be freely reordered; it may help to start by considering the two-dimensional case, where there are only two possible orderings.

Clearly, we may make no statements a priori as to the convergence propertiesof this or indeed the previous series solutions. Note that it is now trivial to obtainthe formal solutions for the previous two simpler cases from this expression. Sincea general solution in anything but the first case is however not accessible, for thetime being H will be assumed time independent. The more complicated case oftime dependence will be dealt with later.

2.1.2 Energy eigenstates

In order to evaluate the temporal evolution of a generic initial state |α〉, we mayclearly start with the evolution of some complete set of basis states. The mostconvenient will naturally be the set of energy eigenstates:

H|a〉 = Ea|a〉. (2.1.28)


We may then exploit the operator completeness relation∑

a |a〉〈a|=1 to expandthe temporal-evolution operator (for H time independent), thus

exp[− i

~H t]=∑

a

exp[− i

~H t]|a〉〈a|

=∑

a

exp[− i

~Ea t

]|a〉〈a|, (2.1.29)

where (and henceforth), for simplicity, we set t0=0 and suppress it.At this point we may directly obtain the solution to any problem in which the

expansion of the initial state over the basis |a〉 is known. We have

|α;t〉 = exp[− i

~H t]|α〉

=∑

a

exp[− i

~Ea t

]|a〉〈a|α〉 =

∑

a

ca exp[− i

~Ea t

]|a〉, (2.1.30)

where the C-number constants ca≡〈a|α〉 are none other than the Fourier coeffi-cients of the initial state. Note that the weight, |ca|2, of each component remainsconstant and that only the relative phases between components actually vary withtime.

A trivial but important case is that of a pure energy eigenstate:

|α〉 = |a〉. (2.1.31)

The corresponding state at time t is then just


~Ea t

]|a〉. (2.1.32)

Now, if the basis states |a〉 also correspond to eigenstates of some operator A thatcommutes with H (a necessary condition for having common eigenstates), thenclearly this observable (and therefore too any observable compatible with H) is aconstant of the motion.

2.1.3 Spin precession in a magnetic field

Let us examine a simple but very important example of non-trivial temporal evol-ution. Consider a system consisting of a spin-1/2 particle (e.g. an electron) situatedin a constant magnetic field B. The interaction is then via the magnetic momentof the electron,

µ = γ S, (2.1.33)


where γ= emec

is known as the gyromagnetic ratio (recall that e<0 for the electron).The interaction Hamiltonian is

Hint = −µ·B = −γS·B. (2.1.34)

For simplicity we shall take the magnetic field to be aligned along the z-axis,B=(0,0,B), and so we have

Hint = −γBSz. (2.1.35)

Since H and Sz trivially commute, we may define simultaneous eigenstates of bothenergy and spin-projection along the z-axis. For an electron at rest, we thereforefind the following energy eigenvalues,

E± = ∓ 12~γB for Sz = ±1

2~. (2.1.36)

By defining

ω ≡ |e|Bmec

, (2.1.37)

(note the modulus of the charge) the natural frequency corresponding to the energydifference between the two states, we may simply write

Hint = ωSz . (2.1.38)

The evolution operator thus becomes

U(t) = e−i~ωSzt, (2.1.39)

which, acting on a generic state |α〉 given at time t=0 by

|α〉 = c+|+〉+ c−|−〉, (2.1.40)

leads to

|α;t〉 = c+ e−i2ωt |+〉+ c− e+

i2ωt |−〉. (2.1.41)

We see that the populations |c±|2 of the ± states are unaffected and thus remainconstant although their relative phase ωt varies in time.

Indeed, if we consider an initial state polarised parallel to the z-axis, so thatc+=1 and c−=0, then we see that it is perfectly stationary. If, however, weconsider an eigenstate of, say, Sx (with c+= c−= 1√

2), then the situation changes:

let us calculate the probability that the system is in an eigenstate of Sx with

2.2. A BRIEF REVIEW OF THE “PICTURES” 15

eigenvalue ± 12~ at some later time t. We have

∣∣∣〈Sx±|α;t〉∣∣∣2

=∣∣∣[

1√2〈+| ± 1√

2〈−|] [

1√2e−

i2ωt |+〉+ 1√

2e+

i2ωt |−〉

]∣∣∣2

=∣∣∣12 e

− i2ωt ±1

2e+

i2ωt∣∣∣2

=

cos2 ωt

2= 1

2(1 + cosωt) for Sx = +1

2~,

sin2 ωt2

= 12(1− cosωt) for Sx = −1

2~.

(2.1.42)

The spin projection thus flips back and forth with frequency ω. The expectationvalues for the projections along the x-, y- and z-axes are given by

〈Sx〉 =(+~

2

)cos2

(12ωt)+(−~

2

)sin2

(12ωt)

= ~

2cos(ωt)

(2.1.43a)

and 〈Sy〉 = ~

2sin(ωt), (2.1.43b)

while 〈Sz〉 = 0. (2.1.43c)

The spin thus precesses about the z-axis. This phenomenon is known as Larmorprecession and the frequency ω= |γ|B is the Larmor frequency.

Exercise 2.1.4. Derive analogous expressions for the expectation values 〈Sx〉,〈Sy〉, and 〈Sz〉 in the most general case; that is, for any pair of coefficients c±.

Hint: given the standard normalisation of the wave-function, |c+|2+ |c−|2=1, itis convenient to parametrise the coefficients as c+=cos(α/2) and c−=sin(α/2).What is the physical significance of the angle α? Show finally that a relative phasebetween the coefficients c± simply leads to an offset in the time variable t.

Exercise 2.1.5. Show that inclusion of kinetic or other spin-independent energy(e.g. the case of an electron in motion) does not alter the above results.

2.2 A brief review of the “pictures”

2.2.1 The Heisenberg and Schrödinger pictures: states andoperators

The description at which we arrived in the previous section is known as the Sch-rödinger picture.∗ In this view of quantum dynamics the temporal-evolution op-erator is applied to the state describing the given initial conditions of a system.∗ The term “picture” is used as distinct from “representation”, which has a different meaning.


Recall, however, that we are ultimately interested in physical observables, whichare essentially expectation values of operators, and it is the temporal evolution ofthese quantities that has real physical significance.

Let us therefore consider a generalised expectation value at the instant t=0:

〈β|O |α〉. (2.2.1)

At some later time t we should then have

〈β;t|O |α;t〉 = 〈β|U†(t) O U(t)|α〉. (2.2.2)

The form of the right-hand side now suggests an alternative view: we may leavethe state vectors as they are initially and consider instead operators that evolvewith time:

OH(t) = U†(t) OS U(t), (2.2.3)

where the suffix H explicitly indicates that the operator is now expressed in theHeisenberg picture, whereas the suffix S indicates the original Schrödinger picture.∗

Analogously, we may introduce the corresponding states:

|α;t〉H = |α〉S ≡ |α,t〉S∣∣t=0

, (2.2.4)

which are then explicitly time-independent.In the Heisenberg picture all time dependence thus passes over to the operators

and it is the states that remain invariant. This clearly does not change the outcomeof any measurement; any expectation value is, by construction, numerically thesame in the two pictures. The advantage, however, is that the Heisenberg pictureallows a closer connection with classical mechanics in that we can now associateobservables (such as position, momentum etc.) with their corresponding operatorsmore directly and indeed it is these that vary with time. More importantly though,it simplifies evaluation of the dynamics; once we have identified the relevant op-erators and defined them in the Heisenberg picture, then we have already solvedthe dynamics for any and all states we might wish to construct.

2.2.2 The Heisenberg equations of motion

To complete the description in the Heisenberg picture, we shall now derive theequations of motion. Since the time dependence has been transferred from thestates to the operators, we clearly seek an equation governing the temporal evol-ution of the latter (and not the former, as in the Schrödinger picture). The time∗ We shall often omit the suffix S as implicitly understood.


derivative of Eq. (2.2.3) leads to

dOH

dt=

dU†(t)

dtOS U(t) + U

†(t)dOS

dtU(t) + U

†(t) OS

dU(t)

dt. (2.2.5)

Assuming, for simplicity, that the operator O does not depend explicitly on time(i.e. the second term above vanishes) and using Eq. (2.1.16), we obtain

i~dOH

dt=[OH, HH

], (2.2.6)

where we have introducedHH ≡ U

† HS U . (2.2.7)

Note, however that, by construction, in the case of HS time independent (or thatat least commutes at different times), U and HS commute and we thus have simply

HH = HS ≡ H. (2.2.8)

The operator equations of motion in the Heisenberg picture then take on the form

dOH

dt=

1

i~

[OH, H

]. (2.2.9)

By analogy with the classical Poisson brackets for a function of the generalisedposition and momentum variables

dO

dt=O,H

Poisson

, (2.2.10)

one is thus tempted (as was Dirac) to formulate a procedure for quantisation thatsimply replaces the Poisson brackets with a commutator (divided by i~) and allphysical observables by their corresponding quantum operators. Since, however,not all quantum observables have a classical correspondent (spin is a good ex-ample), it would seem more logical to consider the inverse procedure as the morephysically significant. Indeed, it is true to say that while classical mechanics maybe derived from quantum mechanics, the converse is not true. In other words, theclassical world is a particular limiting case of quantum physics. This is a pointof fundamental importance; there is no unambiguous passage from classical toquantum mechanics, the correct formulation can only be determined by compar-ison with experimental observation. In contrast, a complete quantum formulationimplicitly contains the laws of classical physics.


2.2.3 Ehrenfest’s theorem

The Heisenberg picture together with the standard commutation relations

[x, f(p)

]= i~

∂f

∂pand

[p, g(x)

]= − i~

∂g

∂x, (2.2.11)

where f and g are functions of p and x respectively (and which should admit powerseries expansions in their arguments), allows an elegant derivation of Ehrenfest’stheorem. For simplicity, we shall work in one dimension—the extension to threebeing trivial.

Exercise 2.2.1. Prove the commutation relations exhibited in Eq. (2.2.11).

Consider a system governed by the following (one-dimensional) Hamiltonian:

H =p2

2m+ V (x), (2.2.12)

which describes a particle of mass m subjected to an x-dependant potential V . Inthe Heisenberg picture we then have

dx

dt=

1

i~

[x,H

]=

p

m. (2.2.13)

Taking the time derivative once again leads to

d2x

dt2=

1

i~

[dx

dt, H

]=

1

i~

[p

m,H

]= − 1

m

dV

dx. (2.2.14)

Rearranging this last equation slightly and taking the expectation value between(time-independent) states, we finally obtain Ehrenfest’s theorem:

md2〈x〉dt2

=d〈p〉dt

= −〈∇V (x)〉. (2.2.15)

Note that, as stated earlier, it is vital for the correctness of this theorem (i.e. thatit give the correct classical limit) that the factor ~ appearing in Eq. (2.2.11)—spatial translations—be the same as that of Eq. (2.1.13)—temporal translations.Moreover, since we are working in the Heisenberg picture, the equation makessense even without taking expectation values; i.e. we may write

md2x

dt2=

dp

dt= −dV (x)

dx, (2.2.16)


whereas the same is not true in the Schrödinger picture.

2.2.4 Basis states

From the foregoing, one might be led (erroneously) into assuming that the basisstates should also be time-independent; let us look more carefully. Take a set ofbasis states that are eigenstates of some observable A, say:

A|a〉 = a|a〉. (2.2.17)

This equation is still written, however, in the traditional Schrödinger picture.Multiplying by U

† from the left overall and inserting UU† into the left-hand side,

we obtainU

†A UU† |a〉 = AH U

† |a〉 = aU† |a〉. (2.2.18)

We are thus forced to consider the eigenstates |a,t〉H constructed via

|a,t〉H ≡ U† |a〉. (2.2.19)

Note that, not only are these time dependent, but that, owing to the presence ofU

† rather than U, their phase rotation is in the opposite direction to that of theSchrödinger-picture states.∗

2.2.5 The interaction picture

In the preceding sections it was implicit that the Hamiltonian H had straightfor-ward and/or known solutions. In all but the most trivial cases this is generallynot true. It is usually possible, however, to separate the Hamiltonian into a piecewith known solution and a remaining small perturbation:

H = H0 + V (t), (2.2.20)

where we have also allowed for a time dependence in the perturbation V (t), but ex-plicitly excluded time variation in H0. We define the solutions of the unperturbedsystem as

H0|n〉 = En|n〉. (2.2.21)

Given that we know the unperturbed states, how best might we represent thetemporal evolution of the perturbed states? If we wish to maintain the closenessof the Heisenberg picture to classical mechanics, then it would seem natural to try∗ It may be helpful to compare this situation with that of ordinary spatial rotations and theireffect on the reference-frame axis vectors.


the following definitions for the generic states and operators:

|α;t〉I = U†I |α;t〉S, (2.2.22a)

AI = U†I AS UI, (2.2.22b)

where by UI we now mean

UI = exp[− i

~H0 t

]; (2.2.23)

that is, the temporal-evolution operator associated with the unperturbed Hamil-tonian; in the interaction picture it is often called the Dyson operator.

We can now derive the differential equation governing the temporal evolutionof these new interaction-picture states:

i~d

dt|α;t〉I = i~

d

dt

(U

†I(t)|α;t〉S

)

= −H0 U†I(t)|α;t〉S + U

†I(t)

(H0 + V

)|α;t〉S

= U†I(t) V |α;t〉S. (2.2.24)

It is then sufficient to insert UIU†I (between V and the ket in the last line above)

to obtain

i~d

dt|α;t〉I = VI |α;t〉I. (2.2.25)

It is also not difficult to derive the corresponding equation for the operators:

dAI

dt=

1

i~[AI, H0]. (2.2.26)

Note again that (since H0 is assumed constant) there is no need for a suffix S or Ion H0 to distinguish between the Schrödinger and interaction pictures.

Exercise 2.2.2. Prove result (2.2.26).

We see then that in the interaction picture the states evolve via a Schrödinger-like equation with H replaced by VI while the operators follow a Heisenberg-likeevolution driven by H0. A comparison of the three pictures is given in Table 2.1.The table highlights the intermediate nature of the interaction picture betweenthe previous two.

To conclude this discussion, let us now examine the evolution equation for theFourier coefficients in the interaction picture. The basis kets are the unperturbed


Table 2.1: A comparison of the state and operator temporal evolution in the threepictures described in the text.

picture Schrödinger Interaction Heisenberg

states evolve with H evolve with VI staticoperators static evolve with H0 evolve with H

eigenstates |n〉 and so we write

|α;t〉I =∑

n

cn(t)|n〉. (2.2.27)

We now project out the coefficients

cn(t) = 〈n|α;t〉I (2.2.28)

and take the time derivative using Eq. (2.2.25)

i~d

dtcn(t) = 〈n|VI|α;t〉I. (2.2.29)

Defining the matrix elements Vnm by

〈n|VI(t)|m〉 = 〈n|U†I(t)V (t)UI(t)|m〉 =: Vnm(t) e

i~(En−Em)t (2.2.30)

and inserting a complete set∑

m |m〉〈m|, we obtain

i~d

dtcn(t) =

∑

m

Vnm(t) eiωnmt cm(t), (2.2.31)

whereωnm :=

En − Em

~. (2.2.32)

We thus have a series of coupled first-order differential equations for the Fouriercoefficients. Its solution would provide us with the probabilities |cn(t)|2 of findingthe system in states |n〉 at time t. We shall return to this problem later.


2.3 Propagators and the Feynman path integral

2.3.1 Propagators in wave mechanics

As we have seen, the problem of temporal evolution (for H time independent) canbe attacked by first finding a complete basis of energy eigenstates, over which anygiven state may then be expanded. The basis states will normally be labelled byfinding a set of compatible operators or observables that completely and unam-biguously distinguish all possible distinct configurations.

Let us take a step back to the formulation in wave mechanics; we start byrecalling the expansion (2.1.30):


~H t]|α〉

=∑

a

exp[− i

~Ea t

]|a〉〈a|α〉 =

∑

a

ca exp[− i

~Ea t

]|a〉, (2.1.30)

We now move over to the coordinate-space representation via projection with po-sition eigenstates:

〈x|α;t〉 =∑

a

ca exp[− i

~Ea t

]〈x|a〉, (2.3.1a)

which, in older notation, may be rewritten as

ψ(x, t) =∑

a

ca exp[− i

~Ea t

]ua(x), (2.3.1b)

with the obvious definitions for ψ, ca and ua.Now, since one may write

〈a|α〉 =

∫dx 〈a|x〉〈x|α〉 (2.3.2a)

andca =

∫dxu∗a(x)ψ(x, 0), (2.3.2b)

the final wave-function can be cast in a more suggestive form:

ψ(x, t) =

∫dx′K(x, t; x′, 0)ψ(x′, 0), (2.3.3)

2.3. PROPAGATORS AND THE FEYNMAN PATH INTEGRAL 23

where the kernel of the integral operator is

K(x, t; x′, 0) =∑

a

〈x|a〉〈a|x′〉 exp[− i

~Ea t

]. (2.3.4)

In wave mechanics this object is known as the propagator (in mathematics it issimply Green’s function for the given differential or wave equation). The importantproperty is that it only depends on the potential of the problem and not on theinitial conditions, that is, on ψ(x,0). Once the eigenstates are known, the evolutionof any given wave-function is thus completely determined. Indeed, note that inthis sense quantum mechanics is still a deterministic theory.

The propagator also has other important properties. First of all, from thestandard definition of the completeness relation, one (trivially) has

limt→0

K(x, t; x′, 0) =∑

a

〈x|a〉〈a|x′〉 = 〈x|x′〉 ≡ δ(x− x′). (2.3.5)

Secondly, given that ua(x)= 〈x|a〉 obeys the Schrödinger wave equation in x, so toodoes K(x,t;x′,0). The propagator may thus be interpreted as the wave-function attime t for a particle that was initially localised precisely at the point x=x′. Thisinterpretation may be somewhat clearer if we rewrite the propagator as

K(x, t; x′, 0) = 〈x|U(t)|x′〉, (2.3.6)

where the action of U(t) is to generate the state at time t corresponding to a statethat was localised at x′ at t=0. Therefore, to solve the problem for a state thatis initially spatially distributed according to some wave-function ψ(x′), we needmerely to convolute K with ψ. This is in complete analogy with, e.g., the standardprocedure for solving the electrostatic potential generated by a charge distributionand is none other than Huygens’ principle.

As a (somewhat trivial) example, let us consider a free particle. The energyeigenstates are then just the eigenstates of the momentum operator p. TakingH=p2/2m and substituting the sum in Eq. (2.3.4) with an integral over momenta,we obtain

K(x, t; x′, 0) =1

2π~

∫ +∞

−∞dp′ exp

[ip′(x− x′)

~− ip′2t

2m~

]. (2.3.7)

To evaluate this integral, we need to regularise the highly oscillatory integrand atinfinity. This may be conveniently performed by attributing the particle with acomplex mass m+ iε (with ε>0 for t>0).∗ We may then use Cauchy’s theorem∗ As we shall see later, this is equivalent to assuming a finite lifetime (indeed the imaginary part


to shift and rotate∗ the path of integration in the complex p′ plane to run alonga path at 45 to the original real axis (see Fig. 2.1), this is generally known as a

Imz′′

Rez′′

Figure 2.1: A Wick rotation in the complex z′′ plane; here the rotated integration ofEq. (2.3.8c) lies along the path at 45.

Wick rotation (Wick, 1954).In other words, suppose we wish to evaluate the integral

I =

∫ +∞

−∞dz e− ia(z

2+2bz), (2.3.8a)

where a and b are real and a > 0. We then first rewrite this as

= e iab2∫ +∞

−∞dz e− ia(z+b)

2

, (2.3.8b)

after which we may shift to a new variable z′ = z+b:

= e iab2∫ +∞

−∞dz′ e− iaz

′2. (2.3.8c)

We finally change variable to z′′=e iπ/4z′ (this is equivalent to rotating the path ofintegration as shown in Fig. 2.1). To use Cauchy’s theorem, the contour must beclosed and we must ensure that there are no singularities enclosed by the contourdefined by the new and old paths. To guarantee that the integral vanishes alongthe arcs in Fig. 2.1 for |z′′|→∞, we also temporarily shift a→a− iε with ε>0

of the mass is none other than the decay rate or inverse lifetime).∗ Note that the sign of the iε term determines the sense of the rotation.


(as already noted, this corresponds to ascribing the particle a complex mass):

I = e− iπ/4 e iab2∫ +∞

−∞dz′′ e−az

′′2. (2.3.8d)

The desired integral is now trivial and the result for the propagator is

K(x, t; x′, 0) =

√m

2π i~ texp

[im(x− x′)2

2~t

]. (2.3.9)

Exercise 2.3.1. Complete the proof of result (2.3.9).

Exercise 2.3.2. Derive the form of the propagator in the case of a massive chargedparticle (e.g. the electron) in a constant and uniform electric field.

Exercise 2.3.3. Use the general form of the propagator derived above to study theharmonic oscillator (difficult but interesting).

There are various spatial and temporal integrals of interest that may be derivedfor K(x,t;x′,0). Consider first setting x=x′ and integrating over all x:

G(t) ≡∫ +∞

−∞dxK(x, t; x, 0)

=

∫ +∞

−∞dx∑

a

∣∣〈x|a〉∣∣2 exp

(− i

~Ea t

)

=∑

a

exp(− i

~Ea t

). (2.3.10)

The result is none other than the trace of the of the temporal-evolution operatorin the coordinate-space representation. Seen as a sum over all states, it is veryreminiscent of the partition function encountered in statistical mechanics. Indeed,a technique used to study quantum mechanics at non-zero temperature exploitsprecisely this similarity by defining an imaginary time:∗

β ≡ i~t. (2.3.11)

In this case G(t) becomes precisely the partition function:

Z ≡∑

a

exp(−βEa) , (2.3.12)

∗ Note that time, of course, plays no role in equilibrium systems, where, by definition, there isno temporal evolution.


where β is then just the standard inverse temperature 1/kT .Consider next the Fourier transform of G(t):

G(E) ≡ − i~

∫ ∞

0

dt ei~Et G(t)

= − i~

∫ ∞

0

dt∑

a

ei~Et e−

i~Eat . (2.3.13)

Again, it is useful to regulate the integrand via a shift E→E+ iε. Once theintegral has been performed, we may let ε→0 and thus obtain

G(E) =∑

a

1

E −Ea

. (2.3.14)

In other words, the energy spectrum is completely specified by the simple poles ofG(E) in the complex energy plane. Thus, to evaluate the spectrum of a physicalsystem, it is sufficient to study the analyticity properties of G(E).

2.3.2 Propagators as transition amplitudes

From Eq. (2.3.6), we can immediately derive another suggestive form for thepropagator:

K(x, t; x′, t′) = 〈x;t|x′;t′〉, (2.3.15)

where, for clarity, we have reinstated the time variable in the “earlier” ket and, westress, the notation |x;t〉 clearly indicates an eigenstate of the position operatorin the Heisenberg picture (recall the opposite phase motion). The meaning thenof this expression should be obvious: it represents the transition amplitude for aparticle that was located at the point x′ at time t′ and that should then find itselfat the point x at a later time t, hence also the term propagator.

As we have always insisted, the temporal evolution of a system may be de-scribed in terms of a unitary operator U(t,t′), where again we reinstate the “earlier”time t′. The action of this operator is fully determined by its action on a set ofbasis states, e.g. the complete set of position eigenstates. This in turn is completelyencoded in the propagator, as written above.

The completeness property of position eigenstates,∫

d3x |x;t〉〈x;t| = 1, (2.3.16)

allows us to consider the evolution from ti to tf , say, divided into two pieces: first


from ti to some intermediate t′ and then from t′ to tf . Thus,

〈xf ;tf |xi;ti〉 =

∫d3x′ 〈xf ;tf |x′;t′〉〈x′;t′|xi;ti〉 (tf > t′ > ti). (2.3.17)

2.3.3 Path integrals

Thus, a possible (seemingly uninteresting) exercise is to divide the time intervalbetween the initial and final instants ti and tf into n equal subintervals:

tf ≥ tn−1 ≥ tn−2 ≥ · · · ≥ t2 ≥ t1 ≥ ti, (2.3.18)

with, say, tm− tm−1= δt for m=1,... ,n (t0≡ ti and tn≡ tf). Then, by virtue ofthe completeness relation (2.3.16), we can rewrite the transition amplitude in thefollowing form:

K(xf, tf; xi, ti) =

∫dxn−1

∫dxn−2 · · ·

∫dx1

× 〈xf;tf|xn−1;tn−1〉〈xn−1;tn−1|xn−2;tn−2〉 · · · 〈x1;t1|xi;ti〉. (2.3.19)

The physical meaning is perhaps best rendered pictorially, as in Fig. 2.2. The

ti t1 t2...

. . .

tn−2 tn−1 tft

x

Figure 2.2: A pictorial representation of the Feynman path integral.

probability amplitude for propagation from point xi at time ti to point xf at time tfmay be calculated as the sum over all paths with all possible intermediate positions.

Let us briefly comment on the concept of paths in classical mechanics beforecontinuing with their significance in quantum physics. The classical Lagrangianfor a particle of mass m, subject to an external potential V (x) is

Lcl(x, x) = 12mx2 − V (x). (2.3.20)


If we now fix the boundary conditions in the usual manner and constrain the endpoints to be (xi,ti) and (xf,tf), there is usually just one path (or at least only a finitenumber) corresponding to the classically allowed motion of the particle. Consider,for example, the simple case of a particle moving in the Earth’s gravitational field,V (x)=mgx, with the following boundary conditions:

(ti, xi) = (0, h) and (tf, xf) = (√

2h/g, 0). (2.3.21)

The solution, or classical path, can then be nothing other than

x = h− 12gt2. (2.3.22)

The general treatment follows Hamilton’s principle: the classical path is that forwhich the classical action is stationary; in other words, for which

δ

∫ tf

ti

dt Lcl(x, x) = 0. (2.3.23)

The solution of this condition leads to Lagrange’s equation of motion.

2.3.4 The Feynman path integral

Moving over to quantum mechanics, the path is no longer unique. Indeed, as wehave just seen, all paths become possible. This leads to the formulation in terms ofpath integrals developed by Feynman (apparently triggered by comments of Diracin his book). Since it turns out that one still uses the classical action, let us define

Sm ≡∫ tm

tm−1

dt Lcl(x, x). (2.3.24)

While there is no explicit reference here to a particular path, it is clear that, sincewe need to know x and x to evaluate Lcl, some choice will have to be made.

Now, the comment by Dirac that struck Feynman suggested an equivalencebetween

exp

i~Sm

and 〈xm,tm|xm−1,tm−1〉. (2.3.25)

Following the product expression in Eq. (2.3.19) for the transition amplitude, weare thus led to multiply such expressions along the path:

n∏

m=1

exp

i~Sm

= exp

i~

n∑

m=1

Sm

= exp

i~S[xf, tf, xi, ti

], (2.3.26)


where now

S[xf, tf, xi, ti

]≡∫ tf

ti

dt Lcl(x, x). (2.3.27)

Taking into account the fact that in Eq. (2.3.19) there is also effectively a sum (orintegral) over all paths, we now write schematically

〈xf,tf|xi,ti〉 ∼∑

paths

exp

i~S[xf, tf, xi, ti

]. (2.3.28)

Note the appearance of ~ in the denominator of the exponent. The sense inwhich the classical limit may be obtained by allowing ~→0 should now be clear:for ~ very small, the phases in the above expressions vary very rapidly when theaction varies and thus the contributions from different paths tend to cancel oneanother. Only those paths in regions where the action is constant can contributeconstructively and this is just the classical condition that the action be extremalor stationary. In other words, assuming the above expression to be correct (with~ 6=0), only quantum paths close to the classical path can contribute appreciablyto the transition amplitude (see Fig. 2.3).

x

txi,ti

xf ,tf

Figure 2.3: Possible quantum paths close to the classical path, which provide the majorcontributions for ~ very small.

Consider now the path for which the action is stationary:

δS[xf, tf, xi, ti

]= 0. (2.3.29)

According to Hamilton’s principle, this is just the classical path mentioned above.In other words, the classical path is precisely the path around which the variation of


the action is vanishingly small and thus, as noted, fixes the region of quantum pathsthat will actually contribute to the Feynman integral. It is, of course, comfortingthat the correct classical limit is obtained from the quantum description in thelimit ~→0.

Feynman’s conjecture is then that the infinitesimal transition amplitude is pro-portional to the path integral given and that the constant of proportionality shouldnot depend on the dynamics (that is, on the potential V ) but merely on the sizeof the interval itself. We thus write

〈xm,tm|xm−1,tm−1〉 = w(δt) exp[i~Sm

], (2.3.30)

where we have introduced an, as yet, unknown proportionality or weight factorw(δt), which should only depend on δt. It is easy to see, e.g. from Eq. (2.3.3),that the dimensionality of w is [length]−D in D-dimensional coordinate space. Forsimplicity, we shall remain in one space dimension.

We are thus required to evaluate the exponent for δt→0. First of all, for aninfinitesimal path, to order δt, we may take a straight line between the initial andfinal points; i.e. the velocity and position are determined by finite differences. Themean position and velocity during the interval are thus sufficient:

x ∼ xm + xm−1

2and ¯x ∼ xm − xm−1

δt. (2.3.31)

Note that, to this order, for x we might equally even use simply xm or xm−1. Theinfinitesimal contribution to the action is then

Sm =

∫ tm

tm−1

dt

[mx2

2− V (x)

]

= δt

m

2

[xm − xm−1

δt

]2− V

(xm + xm−1

2

). (2.3.32)

In the case of a free particle (V =0) this leads to

〈xm,tm|xm−1,tm−1〉 = w(δt) exp

[im (xm − xm−1)

2

2~δt

], (2.3.33)

which should be compared to Eq. (2.3.9). We see then that the functional form isindeed identical and we may now in fact deduce the weight function by using

∫ +∞

−∞dξ exp

[imξ2

2~δt

]=

√2π i~ δt

m, (2.3.34)


or, equivalently,

limδt→0

√m

2π i~ δtexp

[imξ2

2~δt

]= δ(ξ). (2.3.35)

The condition of orthonormality obeyed by the eigenstates at equal times in theHeisenberg picture,

〈xm,t|xm−1,t〉 = δ(xm − xm−1), (2.3.36)

then leads to

w(δt) =

√m

2π i~ δt. (2.3.37)

The final expression for the infinitesimal transition amplitude is thus

〈xm,tm|xm−1,tm−1〉 =

√m

2π i~ δtexp[i~Sm

]. (2.3.38)

And the full transition amplitude now takes on the form

〈xf,tf|xi,ti〉 = limn→∞

( m

2π i~ δt

)n/2 ∫dxn−1

∫dxn−2 · · ·

∫dx1

n∏

m=1

exp[i~Sm

],

(2.3.39)where, recall, ti = t0 and tf= tn, with tf− ti =nδt fixed as n→∞. Although welldefined and very suggestive, such an expression, as written, is cumbersome. It istherefore useful to introduce the so-called functional or path integral :

∫Dx(t) ≡ lim

n→∞[w(δt)]n

∫dxn−1

∫dxn−2 · · ·

∫dx1 , (2.3.40)

where the weight function is just that given in Eq. (2.3.37). In the limit n→∞the path integral becomes infinite-dimensional, over all paths or functions x(t)running between (xi,ti) and (xf,tf). In this notation, the transition amplitude isthen

〈xf,tf|xi,ti〉 =

∫Dx(t) exp

[i

~

∫ tf

ti

dt Lcl(x, x)

]. (2.3.41)

Such an expression is generally known as a Feynman path integral. It is an al-ternative formulation of quantum mechanics. To be precise, it is based solely onthe concepts of superposition and composition of amplitudes, but, as we shallnow demonstrate, is entirely equivalent to the Schrödinger formulation. A furtherimportant aspect, leading to invaluable simplification, is the complete absence ofoperators and the accompanying complications of ordering, commutators etc.

Exercise 2.3.4. Rederive result (2.3.9) for the free-particle propagator using the


path-integral formalism.

Exercise 2.3.5. Using the path-integral formalism, rederive the form of the propag-ator found in Ex. 2.3.2 in the case of a charged massive particle in a constant anduniform electric field.

We close this section by showing explicitly that the transition amplitude definedby the Feynman path integral does indeed obey the time-dependent Schrödingerwave equation in the variables xf and tf (for xi and ti fixed). Let us split theamplitude into just two regions, [ti,t] and [t,tf], with t= tf−δt:

〈xf,tf|xi,ti〉 =

∫ +∞

−∞dx′ 〈xf,tf|x′,t〉〈x′,t|xi,ti〉

=

√m

2π i~ δt

∫ +∞

−∞dx′ exp

[im

2~

(xf − x′)2

δt−

iV(12(xf + x′)

)δt

~

]〈x′,t|xi,ti〉.

(2.3.42)

Since we are assuming xf−x′ infinitesimal, it is convenient to shift the integrationvariable to ξ≡xf−x′ and rewrite xf =x and tf = t+δt:

〈x,t+δt|xi,ti〉 =

√m

2π i~ δt

∫ +∞

−∞dξ exp

[im

2~

ξ2

δt− iV (x− 1

2ξ) δt

~

]〈x−ξ,t|xi,ti〉.

(2.3.43)From Eq. (2.3.35) we see that the dominant region for δt→0 is ξ∼0. That

being the case, we may now expand 〈x−ξ,t|xi,ti〉 on the right-hand side aroundthe point x (i.e. as a Taylor series in ξ). Note that expansions in δt and ξ2 areequivalent since the integral over ξ adds a power of

√δt for every power of ξ

present; the term in V is already O(δt). It is thus also necessary to expand theleft-hand side as a power series in δt. Only keeping terms up to and includingO(δt), we have

〈x,t|xi,ti〉+ δt∂

∂t〈x,t|xi,ti〉

=

√m

2π i~ δt

∫ +∞

−∞dξ exp

(im

2~

ξ2

δt

)[1− iV (x) δt

~+ · · ·

]

×[〈x,t|xi,ti〉+

ξ2

2

∂2

∂x2〈x,t|xi,ti〉+ · · ·

], (2.3.44)

where terms linear in ξ have already been dropped as they vanish on integrationfrom −∞ to +∞. The leading 〈x,t|xi,ti〉 terms cancel between the two sides and,

2.4. POTENTIALS AND GAUGE TRANSFORMATIONS 33

using ∫ +∞

−∞dξ ξ2 exp

(im

2~

ξ2

δt

)=

√2π

(i~δt

m

)3/2

, (2.3.45)

in the limit δt→0 we are left with

δt∂

∂t〈x,t|xi,ti〉 =

√m

2π i~ δt

√2π

(i~δt

m

)3/21

2

∂2

∂x2〈x,t|xi,ti〉

− i

~δt V (x)〈x,t|xi,ti〉. (2.3.46)

Finally, rearranging and gathering factors (δt drops out), we obtain the Sch-rödinger equation for 〈x,t|xi,ti〉:

i~∂

∂t〈x,t|xi,ti〉 =

[− ~

2

2m

∂2

∂x2+ V (x)

]〈x,t|xi,ti〉. (2.3.47)

We may thus conclude that the transition amplitude constructed following Feyn-man’s formulation is the same as that of the Schrödinger formulation in wavemechanics. Unfortunately, the integrals to be evaluated even for the simplestproblem in ordinary quantum mechanics are quite complicated and thus there islittle to be gained over the usual formulation. In the case of quantum field theory(or second quantisation), however, the Feynman approach is decidedly easier andmore intuitive. The parallel with the partition function, already mentioned, alsomakes this approach particularly suited to certain problems in statistical quantummechanics.

Later on we shall give two (important) examples, in which the notions of pathand phase variation along the path are central issues: the Aharonov–Bohm effect(1959), Sec. 2.4.4, and the Berry phase (1984), Sec. 2.5.2.

2.4 Potentials and gauge transformations

2.4.1 Constant potentials

In order to understand such phenomena as the Aharonov–Bohm effect and thegeneral question of phase variation, we first need to understand the concepts ofpotentials and gauge transformations, which form the topic of this section.

In classical mechanics, where we may set the zero of our overall energy scalearbitrarily, the absolute value of the total energy (i.e., Etot ≡Ekin+V ) is irrelev-ant to the dynamics of a system since it is only variations through forces, e.g.F =−∇V (x), that are important. Thus, in particular, we may always shift the


potential V by a constant amount. Let us now examine how this carries over toquantum mechanics. We define the following two states:

|α;t〉|α′;t〉

corresponding to the potentials

V (x, t)

V (x, t) + V0,(2.4.1)

where V0 is some constant shift in the potential. That is, the states are identicalat some initial time t=0 and are then allowed to evolve according to the differentpotentials. Recalling the definition of the evolution operator U(t), we immediatelyhave (in an obvious notation)

U′(t) = e−

i~V0 tU(t). (2.4.2)

And for the states we thus obtain

|α′;t〉 = e−i~V0 t |α;t〉. (2.4.3)

Therefore, the only difference at time t is a phase, which, by construction, isposition independent. For stationary states of energy E, this is tantamount tomaking the overall shift E→E+V0.

Note that in physical observables, such as the expectation values for position,momentum or polarisation measurements, such a phase disappears, as it should.Indeed, what we are studying is just the most trivial example of what is generallyknown as a global gauge transformation: i.e. a constant (in space) phase shift. Inthe language of the Schrödinger equation, we write

ψ′(x, t) = e−i~V0 t ψ(x, t). (2.4.4)

Now, suppose that V varies with time (but that it commutes with itself atdifferent times), then from our previous studies of U(t), we know that

|α′;t〉 = exp

[− i

~

∫ t

0

dt′ V (t′)

]|α;t〉. (2.4.5)

To see now that such potential differences can lead to observable phase shifts,consider the apparatus shown in Fig. 2.4. The two metal cylinders are maintainedat a constant potential difference. One can imagine allowing particle wave-packets(particle beams) to enter the apparatus while the potential is disconnected, it isthen switched on while the packets (particles) are inside the cylinders and switchedoff again before they emerge. From classical electromagnetic, we know that noforces act on the particles during the entire passage through the experimental ap-paratus. However, from quantum mechanics, we also know that on emerging from


Source Detector

Figure 2.4: A schematic view of the measurement of quantum interference due to apotential difference between two different quantum paths.

the two cylinders there will be a phase difference δφ between the correspondingwave-functions given by

δφ =1

~

∫ t

0

dt′ δV (t′). (2.4.6)

This will clearly lead to observable interference effects. Thus, although no forcesact, and therefore classically there can be no detectable effects, in quantum mech-anics we find that the standard phenomenon of interference leads to measurableconsequences. While not easy to analyse, the limit ~→0 simply provokes infinitelyrapid phase oscillations, which then cancel, leading to the usual classical limit. In-deed, such an effect is purely quantum mechanical; there can be no equivalent inclassical mechanics.

2.4.2 Gravity in quantum mechanics

Although there is no quantum field theory of gravity (owing essentially to its highlynon-linear nature), there are measurable effects of gravity through ordinary non-relativistic quantum mechanics. Note then that we are not concerned here withgeneral relativity, but gravity simply as the source of a force field and hence apotential.

Consider the equation of motion for a body falling freely in a gravitationalfield:

mz = −mg. (2.4.7)

Note first that the mass m of the particle cancels left and right and thus plays norôle. This is, however, a consequence of the equivalence between inertial (LHS) and


gravitational (RHS) mass, which leads to the statement that in classical mechanicsgravity is a purely geometrical theory. In quantum mechanics we shall see thatthe situation is surprisingly very different. By examining the wave equation in thepresence of a gravitational potential,

[− ~

2

2m∇

2 +mVgrav

]ψ = i~

∂ψ

∂t, (2.4.8)

we immediately see that the mass does not factor out and, indeed, that it alwaysoccurs in the combination m/~. Here quantum mechanics is thus intimately tied tothe mass. In other words, if there is some measurable effect in which the quantity~ appears, then (if gravity is at work) we should also expect the mass to appearand vice versa.

This can be verified explicitly using the Feynman definition of the transitionamplitude (for tm− tm−1= δt→0):

〈xm,tm|xm−1,tm−1〉 =

√m

2π i~ δtexp

[i

~

∫ tm

tm−1

dt(12mx

2 −mgz)]

(2.4.9)

Here too we see the appearance of the same combination m/~. Note the contrastwith the classical Hamiltonian approach:

δ

∫ tf

ti

dt(12mx

2 −mgz)= 0. (2.4.10)

However, starting from the Schrödinger equation and deriving the Ehrenfest the-orem for this case we find that the mass still disappears from the problem.

d2

dt2〈x〉 = −g 〈z〉. (2.4.11)

Indeed, even the frequency shifts observed for photons in a gravitational field donot depend on any quantum-mechanical effect, merely on the equivalence principle.A non-trivial quantum effect is thus required to reveal the presence of the mass.We shall now give an example.

Let us start by examining the relative strength of the gravitational interaction.Consider the Coulomb force responsible for binding an electron in the lowest stateof the hydrogen atom and compare it to the gravitational force between the sameproton and electron. The equivalent Bohr radius for gravitational binding may be


obtained by simple substitution:

a0 =~2

e2me

→ ~2

GNm2emp

, (2.4.12)

where the electric charge squared e2 is replaced with GNmemp (GN being New-ton’s gravitational constant). Evaluating the expression, one finds a gravitationalBohr radius of order 1029m, which is somewhat larger than the size of the knownuniverse. In other words, gravity does not normally play any discernible rôle inelementary particle interactions.∗

Weak as it may be, it turns out that gravity can induce a sufficient phase dif-ference between paths of a neutron beam that interference patterns be visible. Aninterferometry experiment can be performed using neutrons for which the traject-ories may be deflected via suitable crystals (Bragg diffraction): two possible pathsare created having different heights, with a phase difference thus induced by thepotential difference. In Fig. 2.5 we show schematically how this is achieved. The

A B

C D

a

b

Figure 2.5: A schematic view of an experiment to measure gravitationally inducedquantum interference effects.

two paths ABD and ACD will have different final phases if there is a gravitationalpotential difference; this is easily arranged by having one path, ACD say, higherthan the other. On recombining there will thus be an interference effect analogousto that of the Michelson–Morley experiment. Similarly to that experiment, theinterference “fringes” may be made evident by rotating the apparatus.∗ However, the beam-energy measurements in the LEP ring at CERN were sufficiently accurateto detect the tidal effect of the moon, which caused minute variations in the circumference ofthe ring.


Taking the sides AB and CD to have length a while BD and AC have length b,if the apparatus is tilted so that the sides BD and AC are inclined by an angle θwith respect to the horizontal, then the potential difference will be mgbsinθ. Forneutrons of velocity vn the time passed in the potential difference region is a/vnand thus the final phase difference will be

δφ =mng b sin θ a

~ vn, (2.4.13)

where mn is the neutron mass. The final intensity (∝ cos2δφ) will then (ideally)take the following form:

I(θ) = I0 cos2

(mng ab sin θ

~ vn

). (2.4.14)

For an estimate of the size of such an effect in practice, let us take a∼ b∼3 cm,mn∼1.7×10−27 kg, g∼10m sec−2 and the de Broglie wavelength of the neutronsused λn∼1Å (in order that Bragg diffraction may be exploited to alter their path).The number of complete phase oscillations encountered on rotating the apparatusfrom θ=0 to π/4 is then (recall that ~≃1.05×10−34 J s)

δφ

2π=

mng b a

2π ~ vn=

m2ng b a

h pn=

m2ng b a λn

h2

=(1.7× 10−27)2 × 10× (3× 10−2)× (3× 10−2)× 10−10

(6.6× 10−34)2

∼ O(10) phase oscillations. (2.4.15)

This macroscopic effect has been observed experimentally (Colella et al., 1975).Once again, the presence of ~ in the formula indicates that it is a quantum mech-anical phenomenon; for ~→0 the phase variations become infinitely rapid and theeffect vanishes.

2.4.3 Gauge transformations in electromagnetism

Let us now examine the case of electromagnetic potentials. Electric and magneticfields that are constant in time can be expressed in terms of scalar, φ(x), andvector, A(x), potentials as:

E = −∇φ and B = ∇∧A. (2.4.16)


The classical Hamiltonian for a particle of mass m and electric charge q in anelectromagnetic field is

H =1

2m

(p− q

cA)2

+ qφ. (2.4.17)

In quantum mechanics the potentials naturally become operators, which arejust functions of the position operator x, referred to the particle itself. Now, forB 6=0, the operators p and A(x) do not commute and we must take care to expandthis as (

p− q

cA)2

= p2 − q

c(p·A+A·p) + q2

c2A

2. (2.4.18)

The Hamiltonian is then manifestly Hermitian. The dynamics may be studied, asearlier, in the Heisenberg picture:

dxidt

=1

i~[xi, H ] =

1

m

(pi −

q

cAi

). (2.4.19)

It is thus natural to define the kinematic momentum π (cf. the canonical conjugatemomentum p)

π ≡ mdx

dt= p− q

cA. (2.4.20)

Note that, for B 6=0, different components of π do not commute, indeed

[πi, πj] =i~q

cεijkBk. (2.4.21)

We may thus rewrite the Hamiltonian as

H =1

2mπ

2 + qφ (2.4.22)

and thus obtain the quantum expression for the Lorentz force:

d2x

dt2=

1

m

dπ

dt=

q

m

[E +

1

2c

(dx

dt ∧B −B∧dx

dt

)]. (2.4.23)

It is left as an exercise to show that the canonical derivation of the continuityequation leads to identification of the current (for the particle) as

j(x) =~

mIm

[ψ∗(∇− iq

~cA

)ψ

](2.4.24)


and the spatial integral is therefore∫

d3xj(x) =

1

m

⟨p− q

cA⟩

=1

m〈π〉. (2.4.25)

Exercise 2.4.1. Prove results (2.4.24) and (2.4.25) starting from the Schrödingerequation.

We can now look at the effect of transformations on the electromagnetic po-tential. The simplest such gauge transformation is

φ(x) → φ(x) + λ, A(x) → A(x), (2.4.26)

with λ constant. This is precisely the type of transformation discussed earlier andclearly has no real physical consequences. If, instead, we transform the vector partthus

φ(x) → φ(x), A(x) → A(x) +∇Λ(x), (2.4.27)

with Λ a (real scalar) function of x (a local gauge transformation), then the situ-ation becomes highly non-trivial. Note that in both cases the physical fields E

and B are unaltered. The most general gauge transformation is

φ(x, t) → φ(x, t)− 1

c

∂Λ(x, t)

∂t, A(x, t) → A(x, t) +∇Λ(x, t). (2.4.28)

Where now the complete time-dependent definition of the fields must be used:

E(x, t) = −∇φ(x, t) − 1

c

∂A(x, t)

∂tand B(x, t) = ∇∧A(x, t). (2.4.29)

For the moment we shall only consider the time-independent case and take thegauge transformation to be that shown in Eq. (2.4.27).

In classical physics one talks of gauge transformations as symmetries sincethey have no effect on the observables. And thus the trajectory of a particleis independent of the particular choice of gauge. Note that it is π that definesthe trajectory of a particle and that is therefore a gauge-invariant quantity; thecanonical conjugate momentum p changes, to compensate changes in A.

Exercise 2.4.2. Show that the following two configurations lead to the same mag-netic field and find the gauge transformation that takes one into the other:

Ax = −12By, Ay = 1

2Bx, Az = 0; (2.4.30a)


Ax = −By, Ay = 0, Az = 0. (2.4.30b)

According to the standard quantisation procedure, we should require that thequantum-mechanical equivalents behave in the same manner since, e.g., 〈x〉 and〈π〉 must not change. Let us examine the behaviour of a generic state |α〉 definedin the presence of a field A and its transformation to |α′〉 for A

′=A+∇Λ. Werequire that

〈α|x|α〉 = 〈α′|x|α′〉 (2.4.31a)and

〈α|π|α〉 = 〈α′|π′|α′〉, (2.4.31b)

where π′=π− q

c∇Λ. Besides which, of course, the norm must be preserved:

〈α|α〉 = 〈α′|α′〉. (2.4.32)

What then is the operator that transforms |α〉 into |α′〉? Let us write

|α′〉 = U |α〉. (2.4.33)

The above gauge-invariance requirements clearly imply

U†xU = x (2.4.34a)

andU†π

′U = π. (2.4.34b)

One can easily show, by substitution, that the following unitary operator satisfiesthese criteria:

U(x) = exp

[iq

~cΛ(x)

]. (2.4.35)

Note that this is again simply a position-dependent phase shift.

Exercise 2.4.3. Starting instead from the Schrödinger equation show that gaugeinvariance holds. In other words, show that in the presence of electromagneticfields, it is invariant for the transformation defined in Eq. (2.4.27) and

ψ′(x, t) = U(x)ψ(x, t). (2.4.36)

2.4.4 The Aharonov–Bohm effect

The natural question is now as to whether the type of phase shift described abovecan, in fact, lead to physically detectable consequences. Consider first of all the


r1 r2

B

Figure 2.6: The region inside the inner cylinder is linked with a non-zero magnetic fluxwhile the region between the two cylinders, in which the particle is confined, is flux free.

cylindrical setup shown in Fig. 2.6, in which a charged particle is trapped. Sincethe walls of the enclosing cylinders are assumed rigid, we require the wave-functionto vanish there (i.e. for r= r1,2). Such confinement leads, as always, to quantisedenergy levels; the energy eigenvalues being determined precisely by the boundaryconditions. Since there is no magnetic field inside the region in which the particleis localised so that the Lorentz force must be zero there, one might imagine thatthe energy levels should be independent of B. This is not correct, as we shall nowsee.

Although the magnetic field is, by hypothesis, zero in the cavity where theparticle wave-function is non-zero (and vice versa), the vector potential A is not.Using Stokes’ theorem, it is straightforward to derive the form of A that willgenerate a magnetic field B=B z for r≤ r1 and zero outside:

A =

(B r

2

)φ for r ≤ r1,

(B r212 r

)φ for r ≥ r1.

(2.4.37)

where φ is the unit vector in the azimuthal direction. Recall that, while theSchrödinger equation in the absence of a magnetic field contains simply ∇, in thepresence of a magnetic field we must make the substitution ∇→∇−( iq/~c)A.In our case, in cylindrical polar coordinates, this is simply

∂

∂φ→ ∂

∂φ−(iq

~c

)B r212. (2.4.38)


The extra piece in the resulting Schrödinger equation clearly alters the spectrumof the system. This is in essence the basis of the so-called Aharonov–Bohm effect(1959), which we shall now describe.

Exercise 2.4.4. Verify and calculate the variation in the energy spectrum for thetransformation just described.

Consider now the experimental setup depicted in Fig. 2.7: a particle beam is

Source Detector

B⊙

Figure 2.7: A schematic view of an experimental setup to study the Aharonov–Bohmeffect. The circle (B) represents a cylinder containing the magnetic flux. Outside thecylinder the magnetic field vanishes.

directed in such a way as to find a cylinder similar to the inner cylinder of the setupdescribed above in its path. It thus has two possibilities: either it passes to theright or to the left. The two such possible paths entirely enclose the magnetic fluxcontained inside the cylinder although, as before, there is no B field and thereforeno Lorentz force outside. What we should, of course, expect to find is a phasedifference between the two paths, which, with suitable experimental parameters,will lead to an observable interference effect.

To examine this situation, it is instructive to adopt the Feynman path-integralapproach, precisely because it has to do with phase variation along the differentallowed quantum paths. We first of all need to construct the (classical) Lagrangianin the presence of a vector potential A:

Lcl0 ≡ 1

2mx

2 → Lcl0 +

q

cx·A(x)− qφ(x). (2.4.39)

In this case, though, φ(x)=0.

Exercise 2.4.5. Verify, via the variational principle, that the gauge-field Lag-rangian in Eq. (2.4.39) leads to the correct classical equations of motion.

We are now ready to attack the path-integral formulation of this problem.Consider just one subinterval, from (xm−1,tm−1) to (xm,tm) say:

Sm[x, x] = S(0)m [x, x] +

q

c

∫ tm

tm−1

dtdx

dt·A, (2.4.40)


where S(0)m is simply the classical action for the interval in the absence of the B

field. The integral in the last term may be rewritten directly as

q

c

∫ xm

xm−1

ds ·A, (2.4.41)

where ds is the line element along the trajectory (or path) for the m-th interval.In the complete product of subinterval contributions we thus have

∏

m

ei~S(0)m → exp

[iq

~c

∫ xn

x0

ds ·A]∏

m

ei~S(0)m . (2.4.42)

Now, from classical electromagnetic theory, we know that the integral∮ds ·A

around a closed path simply measures the total flux ΦB linked by the path:∮

ds ·A = ΦB. (2.4.43)

Thus, without any further detailed knowledge, we immediately see that the pathintegrals to the left and right, say, of the cylinder differ in phase by an amount

δφ =q

~cΦB. (2.4.44)

Therefore, if we vary the magnetic flux linked through the cylinder, although thereis no change in the Lorentz force acting on the particle on its route from sourceto detector, we shall observe interference variation with a period given by thefundamental unit of magnetic flux:

2π~c

|e| = 4.135× 10−7 Gauss cm2. (2.4.45)

The effect, once again, is clearly a purely quantum phenomenon since classicallythere can be no difference due to the paths taken by the particle.

A natural question then raised by Aharonov and Bohm (1959) was the fol-lowing: given two different vector potentials A and A

′, representing the samemagnetic field but differing by a gauge transformation, will the above result stillhold? In other words, is the phase difference gauge independent? Now, a genericgauge transformation is

A′ = A−∇Λ (2.4.46)

and by Stokes’ theorem the integral of ∇Λ is always zero around any closed path.Therefore, A and A

′ lead to the same phase difference and thus identical quantummechanical interference effects. Both classically and quantum mechanically, it is

2.5. THE ADIABATIC APPROXIMATION 45

only ∇∧A that is relevant, thus any function A with the correct curl will give thecorrect physical result.

Experimental observation of this effect is quite difficult. The phase oscillationspredicted by Aharonov and Bohm were actually observed, however, by Webb et al.(1985) in ordinary (i.e. not superconducting) metallic rings at very low temper-ature. This also demonstrates incidentally that electrons can maintain quantum-mechanical phase coherence even in ordinary materials.

2.5 The adiabatic approximation

The previous case is a particular example of so-called geometric phases generatedin quantum mechanics, which were first generically identified by Berry (1984) (seealso earlier work by Pancharatnam, 1956). In order to derive the Berry phase, werequire the adiabatic approximation, to which we now turn.∗

2.5.1 The adiabatic theorem

An adiabatic process is one in which a system is subject to very slow externalchanges; that is, changes on a time scale much longer that any characteristictime of the system itself. In quantum mechanics this immediately translates intorestrictions on the energy scale of the perturbations, which should thus be muchsmaller than any transition energy; so that transitions become, in fact, impossible.

Consider then a system described by a time-dependent Hamiltonian:

H(t)|n;t〉 = En(t)|n;t〉, (2.5.1)

where we emphasise the consequent time dependence of the energy eigenvaluesand also ignore spatial dependence (which is irrelevant for this discussion). Theeigenstates |n;t〉 form an orthonormal basis and the general solution of the time-dependent Schrödinger equation may thus be represented as

|α;t〉 =∑

n

e− iφn(t) cn(t) |n;t〉, (2.5.2)

where the cn(t) are time-dependent C-number coefficients and the phases aredefined to be

φn(t) :=1

~

∫ t

0

dt′En(t′), that is, φn(t) =

1

~En(t). (2.5.3)

∗ Note, however, that applicability of the adiabatic approximation is not strictly necessary forappearance of Berry-type phases.


Substituting this form into the Schrödinger equation for |α;t〉, we have

i~∑

n

e− iφn(t)

[cn(t)|n;t〉+ cn(t)

∂

∂t|n;t〉 − icn(t)φn(t)|n;t〉

]

=∑

n

cn(t) e− iφn(t) H|n;t〉. (2.5.4)

By virtue of (2.5.3), the last term on the left-hand side cancels the term on theright-hand side, leaving

∑

n

cn(t) e− iφn(t) |n;t〉 = −

∑

n

cn(t) e− iφn(t)

∂

∂t|n;t〉. (2.5.5)

Projecting onto 〈m;t|, we then obtain

cm(t) = −∑

n

cn(t) e− i [φn(t)−φm(t)] 〈m;t| ∂

∂t|n;t〉. (2.5.6)

We now take the time derivative of (2.5.1):

H(t)|n;t〉+H(t)∂

∂t|n;t〉 = En(t)|n;t〉+ En(t)

∂

∂t|n;t〉, (2.5.7)

which, projecting again onto 〈m;t|, leads to

〈m;t|H(t)|n;t〉+ 〈m;t|H(t) ∂∂t|n;t〉 = En(t) δmn + En(t) 〈m;t| ∂

∂t|n;t〉. (2.5.8)

Considering the case n 6=m and exploiting the hermiticity of H , we then have

〈m;t|H(t)|n;t〉 =[En(t)−Em(t)

]〈m;t| ∂

∂t|n;t〉, (2.5.9)

which may be used to substitute the bra-ket (for n 6=m) in the right-hand side of(2.5.6), from which we then finally obtain

cm(t) = cm(t)〈m;t| ∂∂t|m;t〉 −

∑

n 6=m

cn(t) e− i [φn(t)−φm(t)] 〈m;t|H(t)|n;t〉

En(t)−Em(t). (2.5.10)

This is still exact, but we may now approximate by assuming H(t) to be sufficientlysmall that all transition matrix elements 〈m;t|H(t)|n;t〉 be small compared with theenergy denominators En(t)−Em(t) and thus neglect the entire sum on the right-hand side; i.e. transitions are considered improbable. The remaining equation may


then be easily integrated to obtain the approximate solution

cm(t) ≈ cm(0) e− iγm(t), (2.5.11)

with

γm(t) = − i

∫ t

0

dt′ 〈m;t′| ∂∂t

′ |m;t′〉, (2.5.12)

where the explicit factor “ i ” in (2.5.12) ensures a real phase γm(t).

Exercise 2.5.1. Show that the geometric phase γn(t), as defined, is indeed real.Hint: consider the derivative ∂

∂t〈m;t|m;t〉.

The importance of this result is that a system initially in an eigenstate |n〉, say,when subjected to adiabatic changes, will remain (approximately) in the same statefor all later times. In other words, if the temporal variations of the Hamiltonianare sufficiently gentle, the system will not be essentially disturbed. The state attime t corresponding to an initial eigenstate |n;0〉 is then

|n;t〉 = e− i [φn(t)+γn(t)] |n;0〉. (2.5.13)

The overall phase variation is then seen to be the sum of a dynamical phase φn(t)and a geometric phase γn(t). The former is just the standard phase variationdriven by the energy of the system while the latter is precisely the phase identifiedby Berry (1984).

2.5.2 The Berry phase

Most of the phase differences examined earlier in this chapter were dynamical,being due to an energy or potential difference between two otherwise equivalentpaths. Berry’s geometric phase is somewhat more difficult to detect experimentally.Let us assume that the temporal dependence of H(t) is encoded in some parameterλ(t); so that we may write

∂

∂t|n;t〉 =

∂λ

∂t

∂

∂λ|n;λ〉 (2.5.14)

and thus

γn(t) = − i

∫ t

0

dt′∂λ

∂t′〈n;t′| ∂

∂λ|n;t′〉 = − i

∫ λf

λi

dλ 〈n;λ| ∂∂λ|n;λ〉, (2.5.15)

where λi,f are the initial and final values of the parameter.The difficulty in detecting γ is now evident: in the examples presented earlier

the conditions of the two beams coincide for both the initial and final states;


therefore, λi =λf and such a phase would necessarily vanish. The situation changes,however, if we consider not a single parameter but a space of, say, r parameters.We must then write (summation over the index a is implicit)

∂

∂t|n;t〉 =

∂λa∂t

∂

∂λa|n;λ〉 =

∂λ

∂t·∇λ|n;λ〉, (2.5.16)

where λ is a vector in some r-dimensional space and ∇λ is the correspondinggradient. The phase then becomes

γn(t) = − i

∫ λf

λi

dλ · 〈n|∇λ|n〉 (2.5.17)

and if, as is the usual case, the system returns to its original form (i.e. λi =λf) atsome time t= τ , we may write

γn(τ) = − i

∮dλ · 〈n|∇λ|n〉. (2.5.18)

Such a (closed, multidimensional) line integral (due to Berry, 1984∗) may, ingeneral, be different from zero. The important point is that, while in the one-dimensional case the system can only oscillate back and forth over precisely thesame path, in two or more dimensions topologically non-trivial closed paths mayexist that do not retrace themselves. It is also important to appreciate that theBerry phase only depends on the path followed and not on the time interval (or,equivalently, the velocity). This is in complete contrast to the dynamic phase,which depends inextricably on the time interval over which it accumulates.

Let us now present a simple but concrete example, which we shall, however,leave as an exercise. A beam of neutrons is polarised parallel to the direction B ofa uniform magnetic field B. The beam is separated into two equal parts, of whichone passes through the uniform field B while the other passes through a field ofthe same magnitude but of slowly varying direction. After a certain distance thetwo beams are recombined.

The two magnetic fields may be represented as:

B1 = B0

[sin θ |x〉+ cos θ |z〉

](2.5.19a)

andB2 = B0

[sin θ cosφ(t) |x〉+ sin θ sin φ(t) |y〉+ cos θ |z〉

], (2.5.19b)

where φ(t) varies from 0 to 2π just once during the entire journey. Note that B1

∗ Note though, as mentioned earlier, that such a type of phase had in fact already been discussedby Pancharatnam (1956).


is just B2 for φ(t)=0.

Exercise 2.5.2. Evaluate:

1. the ground-state wave-function in the basis of eigenstates of Sz for a constantfield B2 (in other words, for t fixed);

2. the phase difference between the two beams when they are recombined andthus the final intensity.

As a final note, we should remark that there are many possible generalisations:in recent studies it has even been shown that the adiabatic approximation is not,in fact, essential to the presence of a Berry phase; the same effects may be equallyproduced in rapidly evolving systems. Intuitively, this is at least made plausibleby the disappearance of the time variable in the final formulæ. There is, moreover,no specific need to consider closed loops.

With hindsight now, it also emerges that the Aharonov–Bohm effect discussedearlier is a manifestation of precisely the Berry phase; the role of the adiabaticparameter is played by the magnetic field inside the solenoid. That is, explicittime dependence is not necessary; a parameter set that varies along the path issufficient.

To close this section, we might mention a well-known example from classicalphysics: the Foucault pendulum.∗ Again, although there are no real physical forcesacting on the pendulum, it does indeed precess.

∗ For a simple explanation of the Foucault pendulum effect as the manifestation of a geometricphase, see Shapere and Wilczek (1989).


2.6 Bibliography

Aharonov, Y. and Bohm, D. (1959), Phys. Rev. 115, 485.

Berry, M.V. (1984), Proc. Royal Soc. (London) A392, 45.

Colella, R., Overhauser, A.W. and Werner, S.A. (1975), Phys. Rev. Lett. 34, 1472.

Huygens, Chr. (1690), Traité de la Lumiere (Pieter van der Aa); completed in1678.

Pancharatnam, S. (1956), Proc. Indian Acad. Sci. A44, 247.

Shapere, A.D. and Wilczek, F., eds. (1989), Geometric Phases in Physics (WorldSci.), p. 1.

Webb, R., Washburn, S., Umbach, C. and Laibowitz, R. (1985), Phys. Rev. Lett.54, 2696.

Wick, G.C. (1954), Phys. Rev. 96, 1124.

Chapter 3

Angular Momentum

In this chapter we shall concentrate on the problem of angular momentum (bothorbital and spin) and its representations. Many of the results derived in quantummechanics with regard to angular momentum are less intuitive than others—indeed, the concept of intrinsic spin is peculiar to quantum mechanics and, as weshall see later, is actually predicted by the relativistic formulation due to Dirac.Many of the techniques used here, such as the use of eigenvalue raising and lower-ing (or generically ladder) operators, are common to various aspects of quantummechanics and thus we have an interesting playground in which to study them.Moreover, angular momentum (again, both orbital and spin) plays a fundamentalrôle in atomic, nuclear and particle physics (not to mention such surprising phe-nomena as superfluidity). In particular, in atomic and nuclear physics, where oneis dealing with composite systems, it is often necessary to compose angular mo-menta of different origins to obtain the total value for a system. Finally, one shouldnot forget that angular momentum is intimately tied to the spatial symmetries ofa system or state.

By way of further motivation, it is also worth mentioning that the symmetriesevident in nuclear and particle physics between the proton and neutron (or so-called “mirror” nuclei) and, e.g. the triplet of pions π0,± lead to the concept ofan internal symmetry reflected in the definition of an (abstract) isotopic spin (orisospin). The idea is that in this new internal space the proton and neutron aresomehow “spin-up” and “spin-down” versions respectively of the same object. Wethus represent the proton as |1

2,+1

2〉 while the neutron is |1

2,−1

2〉 (in analogy with

the |s,sz〉 notation used for ordinary spin). The pion triplet and the Λ0, Σ0,± andΞ0,− baryons may be treated in a similar manner. Now, by way of example, the Λ0

decays principally into either pπ− or nπ0. One might imagine a priori that thesetwo decay channels should have equal probability. As we shall see, however, therules for the composition of angular momentum applied to isospin tell us that thepπ− channel is favoured over nπ0 in the ratio 2 : 1, which is indeed very close to

51

52 CHAPTER 3. ANGULAR MOMENTUM

the experimental value of 64% :36%.∗

3.1 Angular-momentum eigenstates and eigenval-

ues

We shall begin this section by reviewing the eigenstates and eigenvalues of theangular-momentum or spin† operator J and its projection onto some preferredaxis (which is conventionally taken to be the z-axis) Jz.

3.1.1 Commutation relations and ladder operators

The simplest commutation relation obeyed by the generic spin operator J is

[J2, Ji] = 0, (i = 1, 2, 3). (3.1.1)

Exercise 3.1.1. Prove the above commutation relation starting from the canonicalindividual component commutators [Ji,Jj ]= i~εijkJk.

Again, starting from the individual commutators, it is not difficult to showthat the so-called ladder operators defined by

J± ≡ Jx ± iJy (3.1.2)

have the effect of raising and lowering the eigenvalue of an eigenstate |j,m〉 withrespect to the operator Jz.

Exercise 3.1.2. Prove the following commutation relations:

[J+, J−] = 2~ Jz, (3.1.3a)

[Jz, J±] = ±~ J±, (3.1.3b)

[J2, J±] = 0. (3.1.3c)

Thus, if ‡

Jz|j,m〉 = m ~ |j,m〉 (3.1.4)∗ One expects small deviations from the predicted value since, e.g. the neutral- and charged-pionmasses are not identical.

† Henceforth, as is common usage for brevity, we shall often use the term spin to indicate genericangular momenta, i.e. both orbital and spin. However, when necessary to distinguish betweenorbital and spin angular momentum, we shall use L for the former and S for the latter.

‡ Note that ~ (being the product, e.g., of a length and a linear momentum) naturally has thedimensions of an angular momentum.

3.1. ANGULAR-MOMENTUM EIGENSTATES AND EIGENVALUES 53

(we shall examine the quantum number j shortly), then it follows that

Jz (J±|j,m〉) = (J± Jz ± ~J±)|j,m〉= J± (m± 1) ~ |j,m〉= (m± 1) ~ (J±|j,m〉). (3.1.5)

Therefore, the states J± |j,m〉 clearly have Jz eigenvalues m±1. Note, however,that the normalisation of these states is as yet undetermined.

3.1.2 The eigenvalues of J2 and Jz

We can use the ladder operators defined in the previous section to examine theeigenvalues of J

2 and Jz and the relation between the two. One can imagineapplying the raising operator J+ to some eigenstate of J2 and Jz n times, say. Ifthe initial eigenvalue of the state for Jz were m~, say, then the value for the finalstate should be (m+n)~. One naturally realises that there must be some intrinsiclimit to such a procedure; the natural classical bound is |Jz|≤ |J |.

To prove that the procedure is limited and that the classical bound indeedholds, consider the combination

J2 − J2

z = J2x + J2

y = 12J+, J− = 1

2J+, J†

+ (3.1.6)

and evaluate its expectation value for a generic state |j,m〉; we have

〈j,m|J2−J2z |j,m〉 = 〈j,m|1

2J+,J†

+|j,m〉= 1

2

∣∣J+|j,m〉∣∣2 + 1

2

∣∣J†+|j,m〉

∣∣2. (3.1.7)

The last line is evidently positive definite and we therefore obtain

〈j,m|J2−J2z |j,m〉 ≥ 0 (∀ j and m). (3.1.8)

Note that, although intuitively correct, one cannot assume a priori from the firstequality in Eq. (3.1.6) that the expectation value of J2

x+J2y is necessarily positive

definite (since the Ji are, in general, complex valued); rather, this is a proof ofsuch a statement.

We thus obtain the (classically expected) bound

〈j,m|J2z |j,m〉 ≤ 〈j,m|J2|j,m〉. (3.1.9)


It follows that there must exist some mmax ≥0 and mmin≤0 such that

J+|j,mmax〉 = 0 and J−|j,mmin〉 = 0, (3.1.10)

otherwise repeated application of the ladder operators would eventually lead toviolation of the bound. Then, since

J∓ J± = 12J+, J− ∓ 1

2[J+, J−]

= J2 − J2

z ∓ ~ Jz, (3.1.11)

we have, e.g.,

0 = J− J+ |j,mmax〉 =(J

2 − J2z − ~ Jz

)|j,mmax〉 (3.1.12)

and finallyJ

2|j,mmax〉 = mmax(mmax + 1) ~2|j,mmax〉. (3.1.13)

We thus see that it is natural to identify j with mmax and by repeating the ar-gument for mmin we see that the two must be identical in magnitude (this wasindeed obvious from the symmetry of the construction). Note that j itself is ac-tually only a quantum number (i.e. merely a label) and not the eigenvalue of anyoperator. We shall thus say that the eigenvalues of J2 and Jz are j(j+1)~2 andm~ respectively, obeying the constraint −j≤m≤ j.

3.1.3 Matrix elements of angular-momentum operators

It is often necessary to evaluate transition matrix elements of operators; let ustherefore now consider general, off-diagonal, matrix elements of the operators in-troduced so far. First of all, from the orthonormality of the states we immediatelyhave

〈j′,m′|J2|j,m〉 = j(j + 1) ~2 δj′j δm′m (3.1.14a)

and〈j′,m′|Jz|j,m〉 = m ~ δj′j δm′

m. (3.1.14b)

Consider now the action of J± on some generic eigenstate |j,m〉:

J±|j,m〉 ≡ c±jm|j,m±1〉. (3.1.15)

This simply defines (up to an arbitrary phase) the coefficients c±jm. Taking themodulus squared of Eq. (3.1.15), we have

|c±jm|2 =∣∣J±|j,m〉

∣∣2 = 〈j,m|J†±J±|j,m〉 = 〈j,m|J∓J±|j,m〉. (3.1.16)


Recalling Eq. (3.1.11), we thus obtain

|c±jm|2 = 〈j,m|J∓J±|j,m〉= 〈j,m|(J2−J2

z ∓~Jz)|j,m〉= ~

2 [j(j + 1)−m(m± 1)]

= ~2 (j ∓m)(j ±m+ 1). (3.1.17)

The arbitrary phase is conventionally chosen such that the c±jm be real and positive.The final expression for the action of J± on |j,m〉 is thus

J±|j,m〉 =√j(j + 1)−m(m± 1) ~ |j,m±1〉

=√

(j ∓m)(j ±m+ 1) ~ |j,m±1〉. (3.1.18)

and the generic matrix elements of J± are thus

〈j′,m′|J±|j,m〉 =√(j ∓m)(j ±m+ 1) ~ δj′j δm′

,m±1. (3.1.19)

3.1.4 Rotations in quantum mechanics

The angular-momentum operator in three dimensions is the generator of the ro-tational group SO(3) and, defining the matrix elements of a generic rotation R interms of a rotation axis n and angle φ (counterclockwise, i.e. as a right-handedscrew), we have:

D(j)

m′m(R) ≡ 〈j,m′|exp

(− i

~J ·nφ

)|j,m〉. (3.1.20)

The functions D(j)

m′m(R) are often known as Wigner functions or matrices. Note,

first of all, that it is sufficient to consider a bra and ket representing the sameeigenvalue j, since rotations cannot alter the total angular momentum and thusall off-diagonal matrix elements (j′ 6= j) vanish. This reflects the fact that J

2

commutes with the individual components of J .The (2j+1)×(2j+1) matrices D

(j)

m′m(R) form a (2j+1)-dimensional irredu-

cible representation of SO(3). Note also that they are trivially the amplitudes fortransitions from |j,m〉 to |j,m′〉 under the influence of a rotation D(R):

D(R)|j,m〉 =∑

m′

|j,m′〉〈j,m′|D(R)|j,m〉

=∑

m′

D(j)

m′m(R)|j,m′〉. (3.1.21)


Exercise 3.1.3. Show that the matrices D(j)

m′m(R) do indeed form a group; i.e. that

a well-defined closed multiplication exists, with inverses and an identity operator.

As is well known, rotations may be characterised by three Euler angles. It isconventional to take these as: a rotation through an angle γ about the z-axis,followed by a rotation through β about the y-axis and finally a rotation throughan angle α, again, about the z-axis. We may express this as follows:

D(j)

m′m(α, β, γ) = 〈j,m′| e− i

~Jzα e−

i~Jyβ e−

i~Jzγ |j,m〉

= e− i (m′α+mγ) 〈j,m′|e− i

~Jyβ |j,m〉. (3.1.22)

A new symbol is usually reserved for the non-trivial piece or so-called reducedWigner matrix:

d(j)

m′m(β) = 〈j,m′|e− i

~Jyβ |j,m〉. (3.1.23)

Exercise 3.1.4. Construct the matrix d(j)

m′m(β) for the case j=1.

Hint: by considering (Jy)3, show that for j=1 (but only for j=1)

e−i~Jyβ = 1− i

(Jy~

)sin β − i

(Jy~

)2

(1− cos β).

Finally, express Jy in terms of J±.

3.1.5 Spherical harmonics and the rotation matrices

There is a natural and intimate connection between the rotation matrices presen-ted in the previous section and the spherical harmonics Y m

l (θ,φ). Consider therotation that produces an eigenstate of linear motion along some arbitrary dir-ection n from the starting point of an eigenstate of motion along the positivez-axis:

|n〉 = D(R)|z〉. (3.1.24)

If the vector n is characterised by the polar angles θ and φ, then the requiredrotation may be constructed as a rotation through θ about the y-axis, followed bya rotation through φ about the z-axis.∗ We shall thus need

D(R) = D(α=φ, β=θ, γ=0). (3.1.25)∗ The first of the previous rotations (i.e. about the z-axis) is clearly superfluous in this case.


As usual, we insert a complete set of states, but this time in the basis of angularmomentum or spherical harmonics:

|n〉 =∑

l′,m

′

D(R) |l′,m′〉〈l′,m′|z〉, (3.1.26)

where we have used l (rather than j) to underline that these are indeed orbitalangular-momentum eigenstates. By projecting onto 〈l,m|, which immediatelyforces l′= l, we thus obtain

〈l,m|n〉 =∑

m′

D(l)

mm′(φ, θ, 0) 〈l,m′|z〉. (3.1.27)

The projection on the left-hand side is just the spherical harmonic Y m∗l (θ,φ) while

those on the right-hand side are spherical harmonics for particular values of theirarguments:

〈l,m|z〉 = Y m ∗l (θ, φ)

∣∣θ=0

, (3.1.28)

where, of course, φ becomes irrelevant since θ=0. Moreover, for θ=0, Y ml vanishes

unless m=0 (this is obvious since px=0=py for this state). Thus,

〈l,m|z〉 = Y m ∗l (θ, φ)

∣∣θ=0

δm0

=

√2l + 1

4πPl(1) δm0 =

√2l + 1

4πδm0, (3.1.29)

where Pl(cosθ) are the Legendre polynomials. We therefore finally obtain

Y m ∗l (θ, φ) =

√2l + 1

4πD

(l)m0(φ, θ, 0), (3.1.30)

which, suitably rewritten, is just

D(l)m0(α, β, 0) =

√4π

2l + 1Y m ∗l (β, α). (3.1.31)

A particular case is m=0:

d(l)00(β) = Pl(cos β). (3.1.32)


3.2 Composition of angular momenta

There are many situations in, e.g., atomic, nuclear and particle physics, wheretwo or more angular momenta (orbital and/or spin) must be added together toproduce a final total angular momentum or, conversely e.g., a decaying object mustdivide its total angular momentum between two (or more) final-state objects. Inquantum mechanics it turns out that such processes are largely determined bythe properties of the operators studied in the previous section. We thus find, forexample, that the relative fractions of the various contributions are fixed a prioriand are independent of the dynamics.

3.2.1 A simple example

Consider the bound-state system composed of an electron–positron pair (known aspositronium) in an L=0 spatial state. The total angular momentum of the systemis formed by a sum of the two individual (one-half) spins of the electron–positronpair. The total spin operator is then constructed from the two separate operatorsthus:

S = Se ⊗ 1

p + Sp ⊗ 1

e, (3.2.1)

where the 1p,e represents the identity (or unit) operator on the particle indicated,for simplicity, however, we shall henceforth suppress this operator and/or indexwhere superfluous. The commutation relations are as usual, except for an addi-tional set of null relations between the components of one spin and the other; forexample, [Se

x,Spy ]=0. It immediately follows that the total spin S also naturally

satisfies the standard commutation relations.There are now four possible states:

|++〉, |+−〉, |−+〉, |−−〉, (3.2.2)

where |++〉 etc. stand for |+〉e⊗|+〉p etc. The first and last clearly correspond tototal spin states with j=1 (and m=±1 respectively). We may thus immediatelyidentify

|++〉 = |1,+1〉 and |−−〉 = |1,−1〉. (3.2.3)

The second and third, however, may contribute to a total spin of either j=0 or 1,withm=0. To settle the issue, it is sufficient (and easy enough in this case) to startfrom the state |−−〉, say, and apply the total-spin raising operator S+≡Se

++Sp+:

S+|1,−1〉 = [Se+ + Sp

+] |−〉e ⊗ |−〉p, (3.2.4)

3.2. COMPOSITION OF ANGULAR MOMENTA 59

for which, using Eq. (3.1.18), we have

S+|1,−1〉 =√2 |1,0〉 (3.2.5a)

and[Se

+ + Sp+] |−〉e ⊗ |−〉p = |+〉e ⊗ |−〉p + |−〉e ⊗ |+〉p. (3.2.5b)

We are thus led to the identification

|1,0〉 = 1√2

[|+〉e ⊗ |−〉p + |−〉e ⊗ |+〉p

]≡ 1√

2

[|+−〉+ |−+〉

](3.2.6a)

and, by orthogonality, we must then also have

|0,0〉 = 1√2

[|+〉e ⊗ |−〉p − |−〉e ⊗ |+〉p

]≡ 1√

2

[|+−〉 − |−+〉

]. (3.2.6b)

Of course, the relative phase between |0,0〉 and |1,0〉 is completely arbitrary. Thefactors ±1/

√2 appearing in front of each term in the right-hand side above are

examples of so-called Clebsch–Gordan coefficients; they are none other than thematrix elements involved in transforming between the bases me,mp and s,m.We note finally that the spin-one triplet and spin-zero singlet states of positroniumare known as ortho- and para-positronium respectively; they have distinctly dif-ferent decay modes.

3.2.2 Clebsch–Gordan coefficients

Let us now consider the general case, in which two arbitrary spins J1,2 are to becomposed to form a total spin J (or, conversely, J must be decomposed into thepair J1,2). As noted in the previous example, the standard commutation relationsare augmented by null relations between the components of J1 and J2, i.e.

[J1i, J2j] = 0 ∀ i, j. (3.2.7)

The rotation operator for the combined state is then naturally

D(R) ≡ D1(R)⊗D2(R) = e−i~J1·nφ⊗ e−

i~J2·nφ, (3.2.8)

where it is crucial that n and φ are, of course, the same in the two sub-operators.Indeed, by virtue of Eq. (3.2.7), we may then write

D(R) = e−i~J ·nφ, (3.2.9)

with, as before, J = J1 + J2. (3.2.10)

In order to characterise the eigenstates, a basis set of compatible operators is


needed. One such base is naturally J21,J

22,J1z ,J2z, i.e. the direct sum of the two

separate bases. It is, however, also desirable to have a characterisation based onthe combined or total spin, thus including J

2 and Jz. The other two operatorsto complete the basis set are best chosen as J

21 and J

22 since these are defined

from the outset, but require no specific details as to the orientations of J1,2. It israther trivial to confirm that this is a complete and compatible set. Note also that,e.g. [J2,J1z] 6=0 and thus one cannot define m1,2 in this characterisation—we shallindeed see that the basis states for this choice are superpositions of those for theset J2

1,J22,J1z ,J2z. Finally, we remark that the only remaining operator J1·J2,

which might have been chosen to further characterise the states in the secondchoice, is not independent; it is moreover evidently redundant (from a multiplicitycount) and can be written as

J1·J2 = 12[J2 − J

21 − J

22]. (3.2.11)

It is thus completely determined by the quantum numbers j, j1,2.

Exercise 3.2.1. Show that the set of operators J21,J

22,J

2,Jz is indeed compat-ible, i.e. that the operators do commute with one another. Verify explicitly too thatthe overall multiplicities are indeed identical and that it is therefore also complete.

The task we now wish to perform is to construct the transformation from onebasis set J2

1,J22,J1z,J2z to the other J2

1,J22,J

2,Jz:

|j1,j2;j,m〉 =∑

m1,m2

|j1,j2;m1,m2〉〈j1,j2;m1,m2|j1,j2;j,m〉. (3.2.12)

The matrix elements 〈j1,j2;m1,m2|j1,j2;j,m〉 are then just the Clebsch–Gordancoefficients we seek. The first (expected) property of such matrix elements isthat they vanish unless m=m1+m2 (i.e. the z-component of the total angularmomentum is conserved). Consider now the operator Jz−J1z−J2z: by definition,acting on any state this gives zero. Therefore,

0 = 〈j1,j2;m1,m2|(Jz−J1z−J2z)|j1,j2;j,m〉= (m−m1 −m2)〈j1,j2;m1,m2|j1,j2;j,m〉, (3.2.13)

where the last line is obtained by considering J1,2z as acting to the left.Consider next the state of maximal m (m= j= j1+j2); this is clearly the

state constructed from the two maximal m1,2 states and trivially shows that jis bounded above by j1+j2. Now, let the spin-lowering operator J− act onsuch a state; this results in a state with total-spin eigenvalue still j, but withm= j−1. The new state must clearly be some linear combination of the two states


|j1,m1;j2,m2〉|m1=j1,m2=j2−1 and |j1,m1;j2,m2〉|m1=j1−1,m2=j2. There exists just one

other independent linear combination, which may be taken as orthogonal to thisand which can then only correspond to a state with j= j1+j2−1.

Acting again with J− leads to a level with three possible states, correspondingto j= j1+j2, j1+j2−1 and j1+j2−2. It is obvious that this proliferation ofstates and reduction of j can only continue until one of the basis states coincideswith |j1,m1;j2,m2〉|m1=j1,m2=−j2

or |j1,m1;j2,m2〉|m1=−j1,m2=j2. This will happen

after n=2min(j1,j2) such steps, at which point the minimum j will be j1+j2−2min(j1,j2)= |j1−j2|. We have thus shown that

|j1 − j2| ≤ j ≤ j1 + j2. (3.2.14)

This is often known as the triangle rule.

Exercise 3.2.2. Check, by counting the tower of states so produced, that the multi-plicity of states for the two sets |j1,j2;j,m〉 and |j1,j2;m1,m2〉 is the same: namely,(2j1+1)(2j2+1).

The usual normalisation condition on the states guarantees unitarity of thematrix elements so defined and it is customary to take them as real for simplicity.Therefore, since the states are orthonormal, that is

〈j1,j2;m′1,m

′2|j1,j2;m1,m2〉 = δm1m

′1δm2m

′2, (3.2.15)

we have that the matrix-element products are also orthonormal:∑

j,m

〈j1,j2;m′1,m

′2|j1,j2;j,m〉〈j1,j2;j,m|j1,j2;m1,m2〉 = δm1m

′1δm2m

′2

(3.2.16a)

and∑

m1,m2

〈j1,j2;j′,m′|j1,j2;m1,m2〉〈j1,j2;m1,m2|j1,j2;j,m〉 = δjj′ δmm′ . (3.2.16b)

3.2.3 Recurrence relations for the CG coefficients

Once the spins j1,2 and j have been fixed, the coefficients for the different valuesof m1,2 and m=m1+m2 obey certain recurrence relations, which, among otherthings, allow their generation via simple numerical algorithms. Consider

J±|j1,j2;j,m〉 =[J1± + J2±

]∑

m′1,m

′2

|j1,j2;m′1,m

′2〉〈j1,j2;m′

1,m′2|j1,j2;j,m〉. (3.2.17)


We have already discovered the action of J± in Eqs. (3.1.18); thus, exploiting theorthonormality and projecting onto 〈j1,j2;m1,m2|, we readily obtain

√(j ∓m)(j ±m+ 1) 〈j1,j2;m1,m2|j1,j2;j,m±1〉

=√

(j1 ∓m1 + 1)(j1 ±m1) 〈j1,j2;m1∓1,m2|j1,j2;j,m〉+√

(j2 ∓m2 + 1)(j2 ±m2) 〈j1,j2;m1,m2∓1|j1,j2;j,m〉. (3.2.18)

Notice how the use of the ladder operators has altered the z-axis spin-projectionconservation condition to m1+m2=m±1. Using the above recurrence relation, itis not difficult to construct a numerical algorithm for the generation of arbitrarysets of Clebsch–Gordan coefficients. The starting point is either of the (uniquely-defined) states of maximal or minimal spin

|j1,j2;j,m〉∣∣m=±(j1+j2)

≡ |j1,j2;m1,m2〉∣∣m1=±j1,m2=±j2

:= |j1,m1〉∣∣m1=±j1

⊗ |j2,m2〉∣∣m2=±j2

, (3.2.19)

from which all other states can then be generated by repeated application of theabove relation.

The simplest example, yet one of the most important, is that of composition ofthe orbital angular momentum j1= l of a spin one-half particle with its own spinj2= s= 1/2. This example describes the angular-momentum state of an electronbound in an hydrogenoid atom or of an unpaired proton or neutron inside a nucleus(and in this last case it then describes the angular-momentum or spin state of thenucleus itself). We are thus interested in the combination

j1 = l (l ∈ N), m1 = ml (|ml| ≤ l),

j2 = s = 12, m2 = ms = ± 1

2.

(3.2.20)

There are then only two values of j to consider for any given l >0:

j = l ± 12, (3.2.21)

the case l=0 is clearly trivial and we shall not consider it further.Since the electron spin projection takes on only two values (ms=±1/2), the

recurrence relations are relatively straightforward. Applying relation (3.2.18) tothis case, we obtain (in an obvious notation)

√(j ∓m)(j ±m+ 1) 〈l, 1

2;ml,ms|l, 12 ;j,m±1〉

=√

(l ∓ml + 1)(l ±ml) 〈l, 12 ;ml∓1,ms|l, 12 ;j,m〉


+√

(12∓ms + 1)(1

2 ±ms) 〈l, 12 ;ml,ms∓1|l, 12;j,m〉. (3.2.22)

Notice that the second term on the right-hand side only contributes for ms=±1/2respectively, that is, the raising (lowering) operator, applied to the electron-spinpart of the state naturally gives zero if the electron is already spin up (down). Wemay thus simplify the recurrence relation by first considering Eq. (3.2.22) only forms=+1/2 and choosing the lower sign:

√(j +m)(j −m+ 1) 〈ml,

12|j,m−1〉 =

√(l +ml + 1)(l −ml) 〈ml+1, 1

2|j,m〉,(3.2.23)

where we have now lightened the notation by suppressing reference to the l,1/2labels in the state vectors.

Let us now consider the specific case j= l+ 1/2 and, by substitution, also elim-inate explicit reference to both j and ml=m− 3/2:

√(l + 1

2+m)(l + 3

2−m) 〈m− 3

2, 12|l+ 1

2,m−1〉

=√

(l +m− 12)(l −m+ 3

2) 〈m− 1

2, 12|l+ 1

2,m〉. (3.2.24)

This may be rewritten, cancelling the common factor√l−m+ 3/2, as

〈m− 32, 12|l+ 1

2,m−1〉 =

√l +m− 1

2

l +m+ 12

〈m− 12, 12|l+ 1

2,m〉. (3.2.25)

Finally, shifting m→m+1, we obtain

〈m− 12, 12|l+ 1

2,m〉 =

√l +m+ 1

2

l +m+ 32

〈m+ 12, 12|l+ 1

2,m+1〉. (3.2.26)

By iterating the same relation, 〈m+ 12, 12|l+ 1

2,m+1〉 may in turn be expressed in

terms of 〈m+ 32, 12|l+ 1

2,m+2〉 and so on. The procedure comes to a halt at the

largest allowed value of m on the right-hand side, which is just l− 1/2. We thusarrive at a product of (mainly cancelling) factors:

〈m− 12, 12|l+ 1

2,m〉 =

√l +m+ 1

2

l +m+ 32

√l +m+ 3

2

l +m+ 52

· · ·

· · ·√

2l − 1

2l

√2l

2l + 1〈l, 1

2|l+ 1

2,l+ 1

2〉. (3.2.27)


This last matrix element involves a maximal spin-projection state on the right-hand side, which is unique in both representations. We must therefore have |l, 1

2〉=

e iφ |l+ 12,l+ 1

2〉, where φ is an arbitrary phase. With the choice φ=0, we finally

obtain the rather simple expression

〈m− 12, 12|l+ 1

2,m〉 =

√l +m+ 1

2

2l + 1. (3.2.28)

Of course, we could have considered Eq. (3.2.22) for ms=−1/2 and taken the uppersign; the result would have been precisely the same.

However, we must now also consider the other two series, in which both termson the right-hand side remain, e.g. by considering Eq. (3.2.22) for ms=±1/2 butnow choosing the upper (lower) sign:

|j= l+ 12,m〉 =

√l+m+ 1

2

2l+1|ml=m− 1

2,ms=

12〉

+X |ml=m+ 12,ms=−1

2〉 (3.2.29a)

and

|j= l− 12,m〉 = Y |ml=m− 1

2,ms=

12〉

+ Z |ml=m+ 12,ms=−1

2〉, (3.2.29b)

where the first coefficient is already fixed by Eq. (3.2.28). This leaves the coeffi-cients X, Y and Z still to be determined. The problem is simplified by noting thatfor fixed m, by virtue of the orthonormality of the pairs, the transformation fromone basis to the other (two states to two states) must effectively be a rotation-likematrix and may be taken to be real :

(cos β sin β

− sin β cos β

). (3.2.30)

The coefficient we already have is cosβ, see Eq. (3.2.29a), and therefore the re-maining sinβ is determined, up to a phase:

sin2 β = 1− l +m+ 12

2l + 1=

l −m+ 12

2l + 1. (3.2.31)

The phase may be chosen, by convention, such that the (non-vanishing) matrixelements of J− are positive and real.

We are thus naturally led to define the following two-component composite


total angular-momentum or spin functions (χ± are Pauli spinors):

Yj=l± 1

2,m

l = ±

√l ±m+ 1

2

2l + 1Y

m− 12

l (θ, φ)⊗ χ+ +

√l ∓m+ 1

2

2l + 1Y

m+ 12

l (θ, φ)⊗ χ−

=1√

2l + 1

±

√l ±m+ 1

2Y

m− 12

l (θ, φ)√l ∓m+ 1

2Y

m+ 12

l (θ, φ)

. (3.2.32)

Note that m is half -integer here.We should remark that the states so constructed are simultaneous eigenfunc-

tions of the (compatible) set of operators L2,S2,J2,Jz. It is worth pointing outthat they are also incidently eigenfunctions of L·S; this is a general observationbased on the trivial identity (see Eq. 3.2.11)

L·S = 12

(J

2 −L2 − S

2). (3.2.33)

From this relation one can immediately derive the eigenvalues of L·S:

12~2[j(j + 1)− l(l + 1)− s(s+ 1)

] ∣∣∣s=1/2j=l±1/2

. (3.2.34)

The eigenvalues of the two possible states, often called “long” and “short”, are thenthe following:

L·S |j,m〉 =

12l ~2 |j,m〉 for j = l + 1

2,

−12(l + 1)~2 |j,m〉 for j = l − 1

2.

(3.2.35)

This result is important for evaluation of the energy due to the spin–orbit interac-tion in both atomic and nuclear physics. Such an interaction is responsible for anenergy-level splitting between, for example, the p1/2

and p3/2states (the lj notation

is standard in atomic physics), which would otherwise be degenerate. In nuclearphysics the effect is found to be comparable to the splitting between shells and isthus responsible for a partial rearrangement of the energy levels. The formulæ justderived are of great help in this case since the true form of the nuclear spin–orbitinteraction is not known from first principles.


3.2.4 The Clebsch–Gordan series

One final useful expansion is the Clebsch–Gordan series for products of rotationmatrices. Consider the product

D(j1)

m1m′1(R) D

(j2)

m2m′2(R). (3.2.36)

From the foregoing and from what we know of spin states, we might expect tobe able to express such a product in terms of a sum over rotation matrices for|j1−j2|≤ j≤ j1+j2. Let us first simplify the expression:

D(j1)

m1m′1(R) D

(j2)

m2m′2(R) = 〈j1m1|D(R)|j1m′

1〉〈j2m2|D(R)|j2m′2〉

= 〈j1j2;m1m2|D(R)|j1j2;m′1m

′2〉. (3.2.37)

On the right-hand side we now insert complete sets of states, in the |j1j2;jm〉 basis,to the right and left of D(R):

D(j1)

m1m′1(R) D

(j2)

m2m′2(R)

=∑

j,m,j′,m

′

〈j1j2;m1m2|j1j2;jm〉〈j1j2;jm|D(R)|j1j2;j′m′〉〈j1j2;j′m′|j1j2;m′1m

′2〉

=∑

j,m,j′,m

′

〈j1j2;m1m2|j1j2;jm〉D(j)

mm′(R) δjj′ 〈j1j2;j

′m′|j1j2;m′1m

′2〉

=∑

j,m,m′

〈j1j2;m1m2|j1j2;jm〉D(j)

mm′(R) 〈j1j2;jm′|j1j2;m′

1m′2〉. (3.2.38)

We shall return to this relation, known as the Clebsch–Gordan series, later, in theproof of an important theorem on tensor-operator matrix elements.

The series expansion just derived can be used to obtain a further useful integralrelation between spherical harmonics. Recall first the relation between the rotationmatrices and the spherical harmonics given in Eq. (3.1.31). If we now set j1,2= l1,2and m′

1,2=0, which also implies m′=0, in Eq. (3.2.38), then after taking thecomplex conjugate we obtain

Y m1l1

(θ, φ) Y m2l2

(θ, φ) =

√(2l1 + 1)(2l2 + 1)

4π

×∑

l,m

〈l1l2;lm|l1l2;m1m2〉〈l1l2;00|l1l2;l0〉√

4π

2l + 1Y ml (θ, φ). (3.2.39)

3.3. THE EPR PARADOX AND THE BELL INEQUALITIES 67

We may now project out by multiplying with Y m′∗

l′ (θ,φ) and integrating over the

solid angle (θ and φ). Owing to the orthogonality of the spherical harmonics, thisprojects the sum onto just one particular pair (l′,m′). We thus obtain (droppingthe primes on l and m)

∫dΩY m∗

l (θ, φ) Ym1

l1(θ, φ) Y

m2

l2(θ, φ)

=

√(2l1 + 1)(2l2 + 1)

4π(2l + 1)〈l1l2;lm|l1l2;m1m2〉〈l1l2;00|l1l2;l0〉. (3.2.40)

The formula so derived is useful for the evaluation of many integrals occurring inthe description of atomic and nuclear spectroscopy.

3.3 The EPR paradox and the Bell inequalities

3.3.1 The Einstein–Podolsky–Rosen paradox

As we have seen earlier, if two spin-1/2 states are combined to form a spin-0 orspin-1 composite system then their spin projections are completely correlated:

|1,0〉 = 1√2

[|+−〉+ |−+〉

](3.3.1a)

and|0,0〉 = 1√

2

[|+−〉 − |−+〉

]. (3.3.1b)

Consider, in particular, the latter case: i.e. suppose we know that an electron pairis in an overall spin-zero state. We then measure the spin alignment (±) of just oneof the electrons with respect to some chosen spin-quantisation axis. Now, althoughwe shall clearly obtain one of two possibilities (either parallel or antiparallel), oncewe have the result, we also know the alignment of the other without measuring it.The fact that such a correlation may persist up to macroscopic distances (provid-ing the system remains undisturbed) can clearly create problems with regard tocausality. This is the basis of the so-called Einstein–Podolsky–Rosen (EPR) para-dox (1935). One of the simplest physically realisable examples is the dominantdecay of the neutral pion (spin-zero), which proceeds via the production of twophotons. A similar case is the decay of the singlet state of positronium, again intoa pair of photons. Although photons are actually spin-one objects, gauge invari-ance demands that they have only two spin-quantisation states (e.g. positive andnegative helicity) and thus, in the present context, behave in the same manner aselectrons.

It is, however, not only the difficulty with causality that renders such con-


siderations interesting. We are not limited to measuring along one axis alone andindeed two observers measuring the two different emerging objects have the libertyto choose different axes. Note, of course, that neither can meaningfully measurethe spin projections of a single particle along two different axes, and certainly notsimultaneously. This is where all simple classical analogies with coloured balls(one white and one black, say) in a bag break down: in the classical world thetwo observers are limited to performing the same measurement (e.g. distinguishingbetween black and white) whereas quantum mechanics allows for them to performmutually incompatible operations separately but, as we shall see, not independ-ently. Considerations of this type led Bell (1964) to formulate a set of inequalitiesthat constitute the basis for a fundamental and non-trivial test of the validity ofquantum mechanics vis à vis possible, more deterministic, alternative theories.

Let us begin the discussion by recalling that the basis eigenstates for spin-1/2systems are related by the following transformation:

|x±〉 = 1√2

[|z+〉 ± |z−〉

](3.3.2a)

and its inverse|z±〉 = 1√

2

[|x+〉 ± |x−〉

], (3.3.2b)

where, in an obvious notation, we shall henceforth indicate spin projections ± withrespect to the chosen spin-quantisation axis (x and z above). The spin-singlet stateof two electrons may then be written as either

|0,0〉 = 1√2

[|z+;z−〉 − |z−;z+〉

], (3.3.3a)

or= 1√

2

[|x−;x+〉 − |x+;x−〉

]. (3.3.3b)

Note the (conventional) difference in overall sign of the second pair of states.Consider now the possible experimental scenarios: for definiteness and without

loss of generality, we may assume one observer, say B, decides to always measurethe x component. If the other, A, makes no measurement then, naturally, B mayobtain either of the two possible results (±) with equal probability. However, if Ahas already performed a measurement of the same x component on his electron,then B can now only obtain one result: namely, the opposite of A. Finally, shouldA decide to measure the z projection, then again B may obtain either of the twopossible results with equal probability. We “explain” this seemingly paradoxicalsituation by saying that the two electrons are in a coherent and entangled stateand that they do indeed therefore “know” each other’s polarisation.

There remains, however, the problem of reconciling the fact that photon B“knows” that the x- or z-spin of photon A has been measured, even though theymay be separated by light years of space and too little time has passed betweenmeasurements for any information to have reached it according to the laws of


special relativity. There are essentially two choices: one can

1. accept the postulates of quantum mechanics as laws of Nature, despite theapparently difficult coexistence with special relativity, or

2. postulate that quantum mechanics is not complete, i.e. that there is moreinformation available for the description of the two-particle system at thetime it is created, carried away by both photons, that is not known or isnot available to the observers and for which quantum mechanics does notproperly account.

In 1935 Einstein, Podolsky and Rosen were thus led to postulate the existenceof so-called hidden variables, some hitherto unknown and unknowable propertiesof the system that should account for the apparent contradiction (for a discus-sion, see also Bohm, 1951, p. 611 and 1952). Their claim was that the theory ofquantum mechanics is incomplete: it does not fully describe physical reality. Sys-tem B knows all about system A before the observer performs any measurementand thus somehow avoids the limitations imposed by the non-commutativity ofstandard quantum-mechanical observables. Furthermore, the claim is, of course,that the hidden variables must be local. Thus, in particular, no instantaneousaction-at-a-distance is necessary in such a picture. In contrast, Bohr (1935), afounder of quantum mechanics, held the opposite view and even defended a strictinterpretation (known as the Copenhagen interpretation) of quantum mechanics.The central point being that one must abandon the concept of local reality.

In 1964 Bell proposed a mechanism to test for the existence of such hiddenvariables, developing an inequality principle as the basis for such a test. He showedthat if certain inequalities were ever invalidated, then this would rule out theexistence of a local theory able to describe the above spin experiments.

Using the example of two electrons configured in a singlet state, consider thefollowing: after separation, if there does indeed exist a local reality governed bysome set of hidden variables, then each electron must possess a well-defined spinprojection simultaneously for each of the three spatial axes x, y and z, where eachprojection can, as usual, have one of two values: say up (+) and down (−). Theexperiment now consists of measuring the projection along one axis for one electronand along another axis for the other. Let us examine the statistical distributionsof such measurements (imagine performing the measurements on a large sampleof pairs of electrons). Denote the number of electrons having both (x+) and (y−)with N(x+,y−) and similarly define N(x+,y+), N(y−,z+) etc. We may alsouse N(x+,y−,z+) as the number of electrons with (x+), (y−) and (z+), and soon. Note that this is only possible in a theory that admits local reality; indeed, itis required by local reality. It is easy to see that

N(x+,y−) = N(x+,y−, z+) +N(x+,y−, z−), (3.3.4)


since, by definition, all N(x+,y−,z+) and N(x+,y−,z−) electrons are includedin the count N(x+,y−) and all else is excluded. Again, we stress that one canonly make such a claim if the measurements are connected to some real, albeithidden or unknown, property of the electrons.

Consider measurements in just two directions, x and z say. The table for allpossible results is simple, see Table 3.1. Since there are clearly equal numbers,

Table 3.1: The mutually exclusive sets of spin measurements along two different direc-tions for the spin-singlet states of two spin-1/2 objects.

Population Particle A Particle B

M1 (x+,z+) (x−,z−)

M2 (x+,z−) (x−,z+)

M3 (x−,z+) (x+,z−)

M4 (x−,z−) (x+,z+)

25%, in each set, the predictions of such a description coincide with those ofquantum mechanics. Note, for example, that should particle A be measured inthe x direction and produce a result +, then the system would belong to eitherset M1 or M2 with equal probability and thus particle B could still be measuredto have, say, spin + in the z direction in 50% of the cases.

Now, let n(x+,y+) indicate the number of measurements of pairs of electronsin which the first electron returns (x+) and the second electron (y+); and use asimilar notation for the other possible results. This is all that is necessary sincethis is all that can be measured; x and y cannot both be measured simultan-eously for the same electron. Bell then demonstrated that in such an experiment,if Eq. (3.3.4) holds (indicating properties associated with local reality—choice 2above), then the following inequality must also hold:

n(x+,y+) ≤ n(x+, z+) + n(z+,y+). (3.3.5)

Further relations can be written down by simply taking appropriate permutationsof the labels x, y and z and the two spin signs. This is the statement of Bell’sinequality principle (1964), and it is proved to be true if there exist real (perhapshidden) variables to account for properties such as spin and therefore their meas-urement. However, in quantum mechanics it is possible to find configurations inwhich it is violated.


3.3.2 Proof of the Bell inequalities

Let us now prove Bell’s inequalities. First, we may list all possible sets of meas-urement results using three different and as yet unspecified directions (a, b andc, say) as in Table 3.2. Examining the table, one sees that the population for

Table 3.2: The mutually exclusive sets of spin measurements along three different andunspecified directions (a, b and c) for the spin-singlet states of two spin-1/2 objects.

Population Particle A Particle B

N1 (a+,b+,c+) (a−,b−,c−)

N2 (a+,b+,c−) (a−,b−,c+)

N3 (a+,b−,c+) (a−,b+,c−)

N4 (a+,b−,c−) (a−,b+,c+)

N5 (a−,b+,c+) (a+,b−,c−)

N6 (a−,b+,c−) (a+,b−,c+)

N7 (a−,b−,c+) (a+,b+,c−)

N8 (a−,b−,c−) (a+,b+,c+)

which particle A has (a+) and particle B has (b+) is N3+N4. A trivial inequalityfollows from the fact that the distributions Ni are positive:

N3 +N4 ≤ (N2 +N4) + (N3 +N7), (3.3.6)

which may then be rewritten in the earlier notation as

n(a+, b+) ≤ n(a+, c+) + n(c+, b+). (3.3.7)

This is just one example of Bell’s inequalities; many others may be constructed ina similar manner.

Now, in quantum mechanics we cannot talk of sets such as N1, N2 etc. sincewe cannot simultaneously determine the projection of the spin along two differentaxes. We are not, however, precluded from determining the populations suchas n(a+,b+) since such measurements may be made. In fact, this provides uswith a nice exercise for applying the rotation relations discussed in Sec. 3.1.4.We shall consider measurements performed in the same plane (although this isnot necessary); we thus need rotations for spin-1/2 states about the z-axis. Therelevant operator is then

Rz(φ) = exp(− i

~Sz φ

)→ exp

(− i

2σz φ

), (3.3.8)


where we have now moved over to the two-dimensional spinor representation viathe Pauli matrices. The right-hand side may be simplified by expanding the ex-ponential as a power series and noting that σ2

z =1:

Rz(φ) = cos φ21− i sin φ

2σz. (3.3.9)

Without loss of generality, we may identify the a-axis with the x-axis and thusthe a-axis spin-projection eigenstates are

|a,±〉 = 1√2

[|z+〉 ± |z−〉

]. (3.3.10)

For spins along the b-axis at some angle φab with respect to the a-axis we thenhave

|b±〉 = cos φab

2|a±〉+ i sin φab

2|a∓〉 (3.3.11)

and, bearing in mind that a given state for B implies the opposite for A, we have(in an obvious notation)

|b±〉B = cos φab

2|a±〉B + i sin φab

2|a∓〉B

= cos φab

2|a∓〉A + i sin φab

2|a±〉A. (3.3.12)

And thus, defining probabilities for the combined A–B measurements,

p(a+, b+) ≡ n(a+, b+)∑8i=1Ni

, (3.3.13)

we readily findp(a+, b+) =

∣∣ 〈b+|a+〉B A

∣∣2 = 12sin2 φab

2, (3.3.14)

where the factor 1/2 arises as the probability that A obtains (a+), while sin2 φab

2is

the probability that B then obtains (b+). Note that, as to be expected, the latterprobability vanishes as φab→0.

Using these values we can now determine the two sides of Bell’s inequality(3.3.7), but now according to quantum mechanics. In terms of the probabilities pwe simply have

sin2 φab

2≤ sin2 φac

2+ sin2 φbc

2. (3.3.15)

Now, choose c as the bisector of the angle between a and b, so that φac=φ=φbc

and φab=2φ. The inequality then becomes

sin2 φ ≤ 2 sin2 φ2, (3.3.16a)

3.4. TENSORIAL OPERATORS 73

orcos2 φ

2≤ 1

2. (3.3.16b)

We thus finally obtain that, provided quantum mechanics holds as postulated, Bell’sinequalities should be violated in such an experiment whenever 0<φ<π/2.

At the time Bell’s result first became known, the experimental data recordswere reviewed to see if any known results provided evidence against locality; nonedid. An effort thus began to develop tests of Bell’s inequalities. A series of exper-iments was conducted by Aspect et al. (1981), ending with one in which the po-lariser angles were altered while the photons were ‘in flight’. Within experimentalerrors, they confirmed the quantum-mechanical description and excluded any pos-sible theory based on hidden variables. At the time this was widely regardedas being a reasonably conclusive experimental confirmation of the predictions ofquantum mechanics.

Three years later, however, Franson (1985) published a paper claiming that thetiming constraints in this experiment were not adequate to confirm that localitywas violated. Aspect et al. measured the time delays between detections of photonpairs. The critical time delay is between the instant a polariser angle is altered andthe instant this may affect detection rates of the photon pairs. Aspect et al. estim-ated this time based on the photon speed and the distance between the polarisersand detectors. Quantum mechanics does not permit knowledge or assumptionsas to where a particle is between detections. We cannot know when a particletraverses a polariser unless we actually detect the particle at the polariser.

Experimental tests of Bell’s inequalities continue but none has yet fully ad-dressed the issue raised by Franson. Moreover, there is an issue of detector effi-ciency: by postulating new laws of physics one can obtain the expected correlationswithout non-local effects unless the detectors are close to 90% efficient. Recent ef-forts have also examined particle–antiparticle correlations (rather than spin–spin)in high-energy particle experiments at one of the so-called B-factories, where vi-olation of Bell’s inequalities has been observed in excess of the 3σ level (Go, 2004;Go et al., 2007); for other recent observations, see also Gröblacher et al. (2007). Asimilar test may also be performed at the so-called φ-factories (see, for example,Ghirardi et al., 1992). It turns out, however, that the multiple nature of the pos-sible final states invalidates many of these types of experiments as a true test ofBell’s inequalities (see, e.g. Bertlmann et al., 2004).

3.4 Tensorial operators

The time has now come to examine the transformation properties of the vari-ous observables (or operators), such as x, p, S, L etc., under rotations. They


correspond to vector quantities in classical mechanics, but we must define theirproperties in quantum mechanics more carefully. In classical mechanics a vectorobeys the following transformation law for rotations:∗

Vi → V ′i ≡ RijVj, (3.4.1)

where Rij is a standard rotation matrix. Following Ehrenfest’s theorem, it isnatural to require, that the expectation value of a vectorial operator in quantummechanics should transform according to the classical rules. Since the action ofthe transformation on a generic state is

|α〉 → |α′〉 ≡ D(R)|α〉, (3.4.2)

we thus require that

〈α|Vi|α〉 → 〈α′|Vi|α′〉 ≡ 〈α|D†(R)ViD(R)|α〉and also that

〈α′|Vi|α′〉 = Rij〈α|Vj|α〉= 〈α|RijVj|α〉. (3.4.3)

If this is to be true for any state |α〉, we must then have

D†(R) Vi D(R) = RijVj. (3.4.4)

Consider now the case of a rotation through an infinitesimal angle ǫ about somearbitrary axis n:

D(R) = 1− i~ǫJ ·n. (3.4.5)

Eq. (3.4.4) then becomes, to order ǫ,

Vi − i~ǫ [Vi,J ·n] = Rij(n, ǫ) Vj. (3.4.6)

For simplicity, let us take n along the z-axis. To order ǫ, the rotation matrix R isthen given by

R(n, ǫ) =

(1 −ǫ 0ǫ 1 00 0 1

). (3.4.7)

At this point it is a simple exercise to show that Eq. (3.4.6) leads to the followingcommutation relations:

[Vi, Jj] = i~ εijk Vk. (3.4.8)

Exercise 3.4.1. Derive the commutation relations (3.4.8) from Eq. (3.4.6).

∗ As always, we adopt the Einstein summation convention for repeated indices.


This set of relations also completely determines the action of finite rotationson the vector Vi or indeed on any vector. The standard formula

exp(i~Jj φ

)Vi exp

(− i

~Jj φ

)(3.4.9)

may be evaluated simply by repeated application of the commutation relations(3.4.8). In this connection, it is useful to recall the Baker–Campbell–Hausdorffseries (Campbell, 1897; Baker, 1902; Hausdorff, 1906):

eA B e−A = B+[A,B]+ 12![A, [A,B]]+· · ·+ 1

n![A, [A, . . . [A,B] . . . ]]+· · · (3.4.10)

Exercise 3.4.2. Derive the Baker–Campbell–Hausdorff series (3.4.10).Hint: define the function F (λ)=eλABe−λA, take repeated derivatives with respectto λ and thus deduce the Taylor series expansion for F (λ); finally, set λ=1.

When [A,[A,B]]=0= [B,[A,B]],∗ the series truncates after the first term andwe can rewrite it in various useful forms:

eA eB = eB eA e[A,B], (3.4.11a)or, equivalently,

= eA+B e12[A,B] . (3.4.11b)

The first is often also called Weyl’s identity.From the above observations it should be clear that we may turn the argument

around and use the commutation relations (3.4.8) to define a vectorial operator.Recall the various similar such sets of commutation relations that have alreadybeen derived: e.g.

[xi, Lj] = i~ εijk xk, [pi, Lj ] = i~ εijk pk, . . . (3.4.12)

3.4.1 Cartesian and irreducible tensors

The Cartesian tensor is the simplest extension of a vector to an object of higherdimension or rank. It obeys the following transformation under rotations:

Tijk... → T ′ijk... ≡ Rii

′Rjj′Rkk

′ · · · Ti′j′k′.... (3.4.13)

A trivial example would be the direct product of two vectors,

Tij = AiBj. (3.4.14)

∗ This is often the case in quantum mechanics; consider, for example, the operators p and x.


Such tensors are, however, reducible, that is to say, they may be decomposedinto different pieces that effectively behave differently under rotations. Let usillustrate this using precisely the example of the product of two vectors, whichmay be rewritten (decomposed or reduced) in the following way:

AiBj = 13A·B δij +

12(AiBj −AjBi) +

[12(AiBj + AjBi)− 1

3A·B δij

]. (3.4.15)

The right-hand side contains three distinct objects. The first is clearly a scalar andis therefore invariant under rotations (recall that δij is invariant under coordinatetransformations). The second, while appearing to still have two free indices, isin fact an antisymmetric tensor, which may be expressed in terms of the vectorproduct as εijk(A∧B)k. It thus has only three independent components and there-fore actually represents a vector. Finally, the third term is a symmetric tensor,from which the δij term subtracts the trace. Again, counting the number of inde-pendent components, we find six minus the trace, or five. As a simple cross-checkwe may sum the total number of independent components counted on both sidesto find nine.

Now, it is not mere numerology or coincidence that the number of independentcomponents of each of the three terms above is just the multiplicity of the first threeorbital angular-momentum quantum numbers: l=0, 1 and 2, i.e. 2l+1=1, 3 and5 respectively. In fact, the decomposition we have just performed has reduced theCartesian tensor AiBj into the lowest three irreducible spherical tensors. Considernow the spherical harmonic Y m

l (θ,φ): since it only depends on the angles θ and φ,it may be written more simply as Y m

l (n), where n is the unit vector characterisedby θ and φ. This is precisely a spherical tensor of rank l. We may thus express ageneric tensor in terms of, say, Y m

l (A), where the vector A simply indicates somechosen direction:

T (l)m ≡ Y m

l (A). (3.4.16)

Of course, l and m should not be confused with the tensor indices. Since thevector A=(Asinθ cosφ,Asinθ sinφ,Acosθ), there is an immediate and direct cor-respondence between components: e.g. for l=1 we have∗

Y 01 (A) =

√3

4πcos θ ⇒ T

(1)0 =

√3

4πAz, (3.4.17a)

Y ±1 (A) = ∓

√3

8πsin θ e± iφ ⇒ T

(1)± = ∓

√3

8π(Ax ± iAy). (3.4.17b)

Note the presence of three independent components. The generalisation to other∗ The overall ± sign for the ± components is conventional, being dictated by the conventionalsign choice for the spherical harmonic functions themselves.


values of l is straightforward.In order, to understand how spherical tensors transform, it is necessary then

to study the transformation properties of the spherical harmonics. Now, for eigen-states corresponding to motion in a fixed direction, n, we have

|n〉 → |n′〉 ≡ D(R)|n〉. (3.4.18)

We wish to study though the behaviour of the Y ml (n). From Eq. (3.1.21), we

already know that we may write

D†(R)|l,m〉 =

∑

m′

D(l) †m

′m(R)|l,m′〉. (3.4.19)

Projecting onto 〈n|, we have

Y ml (n′) =

∑

m′

Y m′

l (n)D(l) †m

′m(R). (3.4.20)

We thus finally obtain obtain for a generic vector A

D†(R) Y m

l (A) D(R) =∑

m′

D(l) ∗mm

′(R) Ym

′

l (A). (3.4.21)

Exercise 3.4.3. Perform the complete derivation of relation (3.4.21).

In terms of the generic spherical tensor of rank l, we then have

D†(R) T (l)

m D(R) =

l∑

m′=−l

D(l) ∗mm

′(R) T(l)

m′ . (3.4.22)

Equivalently, we may write

D(R) T (l)m D

†(R) =

l∑

m′=−l

D(l)

m′m(R) T

(l)

m′ . (3.4.23)

Note that these expressions actually hold independently of the fact that T (l)m may

be expressed explicitly in terms of Y ml (A) for some vector A. Indeed, the product

considered earlier leads to a tensor of rank 2 that, depending on two vectors, doesnot have a simple expression in terms of any Y m

l .The best way to define the spherical tensors is to start from an infinitesimal


rotation:

(1+ i

~J ·n ǫ

)T (l)m

(1− i

~J ·n ǫ

)=

l∑

m′=−l

T(l)

m′ 〈lm′|

(1+ i

~J ·nǫ

)|lm〉, (3.4.24a)

or, equivalently, [J ·n, T (l)

m

]=

l∑

m′=−l

T(l)

m′ 〈lm′|J ·n|lm〉. (3.4.24b)

To see explicit expressions, we may choose n along the z-axis and then readilyobtain

[Jz, T

(l)m

]= m ~ T (l)

m (3.4.25a)and [

J±, T(l)m

]=√

(l ∓m)(l ±m+ 1) ~ T(l)m±1. (3.4.25b)

These commutation relations may serve equally as the definition of a sphericaltensor.

Exercise 3.4.4. Perform the complete derivation of the commutation relationsgiven in (3.4.25).

3.4.2 Tensor products

Earlier in this section we gave an example of the decomposition of a simple tensorproduct in terms of Cartesian tensors. The same exercise may be performed usingthe spherical tensors we have just defined. Thus,

T(0)0 = − 1√

3A·B = 1√

3(A+1B−1 + A−1B+1 − A0B0),

T (1)m = − i√

2(A∧B)m (m = −1, 0,+1),

T(2)±2 = A±1B±1,

T(2)±1 = 1√

2(A±1B0 + A0B±1),

T(2)0 = 1√

6(A+1B−1 + 2A0B0 + A−1B+1), (3.4.26)

where, recall, we conventionally define A±1≡ 1√2(∓Ax− iAy) and A0≡Az. We can

now proceed with the proof of an important theorem on tensor products

Theorem 3.4.1. Let U (l1)m1

and V (l2)m2

be irreducible spherical tensors of rank l1,2respectively, then

T (l)m =

∑

m1,m2

〈l1l2;m1m2|l1l2;lm〉U (l1)m1

V (l2)m2

(3.4.27)


is an irreducible spherical tensor of rank l (|l1− l2|≤ l≤ l1+ l2).

In other words, the coefficients on the right-hand side of Eq. (3.4.26) are justthe matrix elements 〈l1l2;m1m2|l1l2;lm〉, or Clebsch–Gordan coefficients.

Proof. 1. To prove the theorem it is merely necessary to demonstrate that thetensor so constructed transforms according to the rules already laid down, e.g.Eq. (3.4.23) or (3.4.25). We thus start from (dropping the labels l1,2 for clarity, asusual)

D†(R) T (l)

m D(R) =∑

m1,m2

〈m1m2|lm〉 D†(R)U (l1)m1

D(R) D†(R) V (l2)m2

D(R)

=∑

m1,m2

〈m1m2|lm〉∑

m′1,m

′2

U(l1)

m′1D

(l1)

m′1m1

(R−1) V(l2)

m′2

D(l2)

m′2m2

(R−1),

which, using the Clebsch–Gordan series (3.2.38), may be rewritten as

=∑

l′,m1,m2

〈m1m2|lm〉∑

m′1,m

′2,m

′,m

′′

〈m′1m

′2|l′m′〉〈l′m′′|m1m2〉

×D(l′)

m′m

′′(R−1)U

(l1)

m′1V

(l2)

m′2, (3.4.28)

where, recall, all states |lm〉, |m1m2〉 etc. are implicitly understood to be construc-ted from a composition of angular momenta l1,2 and thus |l1− l2|≤ l≤ l1+ l2. Fromthe orthogonality of the Clebsch–Gordan coefficients we then have

D†(R) T (l)

m D(R) =∑

l′,m

′1,m

′2,m

′′,m

′

δll′ δmm′′ 〈m′

1m′2|l′m′〉 D(l

′)

m′m

′′(R−1)U

(l1)

m′1V

(l2)

m′2

=∑

m′

∑

m′1,m

′2

〈m′1m

′2|lm′〉U (l1)

m′1V

(l2)

m′2

D

(l)

m′m(R−1)

=∑

m′

T(l)

m′ D

(l)

m′m(R−1) =

∑

m′

D(l)∗m

′m(R) T

(l)

m′ , (3.4.29)

which is just the required relation.

Proof. 2. An alternative derivation of the theorem is furnished via the commuta-tion relations (3.4.25). First of all, we note that a commutator involving a productof operators may be simplified via the following trivial identity:

[A,BC] = [A,B]C +B [A,C]. (3.4.30)


The commutator on the left-hand side of (3.4.25a) thus leads to

[Jz, T(l)m ] =

∑

m1,m2

〈l1l2;m1m2|l1l2;lm〉 [Jz, U (l1)m1

V (l2)m2

]

=∑

m1,m2

〈l1l2;m1m2|l1l2;lm〉[Jz, U

(l1)m1

]V (l2)m2

+ U (l1)m1

[Jz, V(l2)m2

]

=∑

m1,m2

〈l1l2;m1m2|l1l2;lm〉 (m1~+m2~)U(l1)m1

V (l2)m2

= m~ T (l)m (3.4.31)

and the commutator in (3.4.25b) gives

[J±, T(l)m ] =

∑

m1,m2

〈l1l2;m1m2|l1l2;lm〉 [J±, U (l1)m1

V (l2)m2

]

=∑

m1,m2

〈l1l2;m1m2|l1l2;lm〉[J±, U

(l1)m1

]V (l2)m2

+ U (l1)m1

[J±, V(l2)m2

]

=∑

m1,m2

〈l1l2;m1m2|l1l2;lm〉

×√

(l ∓m1)(l ±m1 + 1) ~U(l1)m1±1 V

(l2)m2

+√

(l ∓m2)(l ±m2 + 1) ~U (l1)m1

V(l2)m2±1

.

Shifting the indices on the two terms as m1,2 →m1,2∓1, the two series may berewritten to obtain

=∑

m1,m2

~U (l1)m1

V (l2)m2

×√

(l ∓m1 + 1)(l ±m1) 〈l1l2;m1∓1m2|l1l2;lm〉

+√(l ∓m2 + 1)(l ±m2) 〈l1l2;m1m2∓1|l1l2;lm〉

.

The term in braces is just the right-hand side of Eq. (3.2.18) and thus we finallyobtain

=√

(l ∓m)(l ±m+ 1) ~ T(l)m±1, (3.4.32)

which is again the desired result.

This theorem then demonstrates how to construct tensors of higher rank by


multiplication of lower-rank tensors. The derivation renders the connection withangular-momentum composition quite transparent—indeed, precisely the sameClebsch–Gordan coefficients are involved.

3.4.3 Tensor matrix elements

As already seen in earlier courses,∗ problems of emission and absorption in atomicand nuclear physics often require evaluation of matrix elements that involve rathercomplicated integrals of wave-functions and tensorial operators. In many casessuch calculations may be simplified by consideration of the geometrical propertiesof the objects concerned. In particular, one can easily derive many useful so-calledselection rules. Indeed, we may immediately derive a relatively simple selectionrule for the magnetic quantum number m:

〈α2;j2m2|T (l)m |α1;j1m1〉 = 0 for m2 6= m1 +m, (3.4.33)

where α represents some set of quantum numbers that may be defined for thesystem and thus used to label the states. The derivation is straightforward: usingcommutation relation (3.4.25a) we have

0 = 〈α2;j2m2|[Jz,T

(l)m ]−m~T (l)

m

|α1;j1m1〉

=(m2 −m1)~−m~

〈α2;j2m2|T (l)

m |α1;j1m1〉, (3.4.34)

which proves the selection rule.

3.4.4 The Wigner–Eckart theorem

We shall now close this chapter on angular momentum by proving a very importanttheorem in quantum mechanics, which finds useful applications in e.g. atomic andnuclear physics.

Theorem 3.4.2. The Wigner–Eckart theorem states the following relation formatrix elements of tensorial operators between angular-momentum eigenstates:

〈α2;j2m2|T (l)m |α1;j1m1〉 = 〈j1l;m1m|j1l;j2m2〉

〈α2;j2‖T (l)m ‖α1;j1〉√

2j1 + 1, (3.4.35)

where, again, α1,2 represent possible sets of other quantum numbers characterisingthe system.

∗ See, for example, Quantum mechanics II.


The second bra-ket on the right-hand side, known as the reduced or double-barmatrix element, is defined by this equation. A surprising but crucial observationis that the reduced matrix element does not depend on m1,2. The presence ofthe factor 1/

√2j1+1 is conventional and simply defines the normalisation of the

reduced matrix element.∗

Let us take a moment to examine the physical significance of Eq. (3.4.35). Theright-hand side consists of two factors, the first of which is simply the Clebsch–Gordan coefficient for the composition of angular momenta l and j1 to obtain j2.Moreover, this first factor does not depend on the dynamics (represented by α andT (l)m ), but only on the orientation of the initial and final states with respect to thez-axis. The dynamics is, in fact, contained entirely within the second factor, which,in contrast does not depend on the orientations with respect to the z-axis. Thus,the generic matrix element 〈α2;j2m2|T (l)

m |α1;j1m1〉 need not be re-evaluated foreach combination of m and m1,2; one is sufficient since the others follow from thestandard Clebsch–Gordan relations. In other words, the selection rules for matrixelements of tensorial operators can be deduced from those for angular-momentumcomposition—perhaps, not surprisingly. Indeed, in reading such relations (and thefollowing derivation), it may help the reader to think of T (l)

m as equivalent to a state|lm〉 to be composed with the ket to the right to produce (or project onto) the brato the left. There are two simple and immediate sum rules that follow from thetheorem: namely, that the total angular-momentum eigenvalues l, j2 and j1 mustobey the triangle rule (3.2.14) and the so-called magnetic eigenvalues must satisfym2=m1+m.

We may now proceed with a proof of the theorem.

Proof. From Eq. (3.4.25b) we have

〈α2;j2m2|[J±,T (l)m ]|α1;j1m1〉 = ~

√(l ∓m)(l ±m+ 1) 〈α2;j2m2|T (l)

m±1|α1;j1m1〉.(3.4.36)

From Eq. (3.1.18) we then have

√(j2 ±m2)(j2 ∓m2 + 1) 〈α2;j2m2∓1|T (l)

m |α1;j1m1〉=√

(j1 ∓m1)(j1 ±m1 + 1) 〈α2;j2m2|T (l)m |α1;j1m1±1〉

+√

(l ∓m)(l ±m+ 1) 〈α2;j2m2|T (l)m±1|α1;j1m1〉. (3.4.37)

Now, this recurrence relation has exactly the same form as that of Eq. (3.2.18).∗ Some authors adopt rather different normalisations and some also include a conventional phasefactor (−1)j+j1−j2 .


We thus have two sets of linear equations with identical coefficients, i.e.∑

j

aij xj = 0 and∑

j

aij yj = 0, (3.4.38)

which implies yi= cxi, with the same constant c for all i. Thus, noting e.g. that inEq. (3.2.18) 〈j1,j2;m1,m2∓1|j1,j2;j,m〉 and 〈α2;j2m2|T (l)

m±1|α1;j1m1〉 have the samecoefficient, it follows that

〈α2;j2m2|T (l)m±1|α1;j1m1〉 = c 〈j1,j2;m1,m2∓1|j1,j2;l,m〉, (3.4.39)

with j→ l and where c, the reduced matrix element, is a common factor, whichdoes not depend on any of m1,2 or m. This completes the proof.

Let us now provide an important example of the Wigner–Eckart theorem. Con-sider the case of a vector operator V , with spherical components Vm=0,±1: therelevant selection rules for its matrix elements are

∆j ≡ j2 − j1 =

0 (though j1 = 0 = j2 is prohibited∗)±1

(3.4.40a)

and ∆m ≡ m2 −m1 = 0,±1. (3.4.40b)

A further theorem is of particular use in this case:

Theorem 3.4.3. The projection theorem states that, for any vector operator V

the following relation (or projection) holds

〈α′;jm′|Vk|α;jm〉 =〈α′;jm′|J ·V |α;jm〉

~2 j(j + 1)

〈jm′|Jk|jm〉. (3.4.41)

Proof. Recall first that the spherical components of a vector A are defined by

A±1 ≡ 1√2(∓Ax − iAy) = ∓ 1√

2A± and A0 ≡ Az, (3.4.42)

The scalar product J ·V , in the spherical basis, is thus expressed as

J ·V = J0V0 − J+1V−1 − J−1V+1. (3.4.43)

∗ One clearly cannot make, e.g., a spin-one object by combining two spin-zero states.


The matrix element of such an operator is then

〈α′;jm′|J ·V |α;jm〉= 〈α′;jm′|J0V0−J+1V−1−J−1V+1|α;jm〉= m~〈α′;jm|V0|α;jm〉

+ 1√2~√(j +m)(j −m+ 1) 〈α′;jm−1|V−1|α;jm〉

− 1√2~√(j −m)(j +m+ 1) 〈α′;jm+1|V+1|α;jm〉. (3.4.44)

The Wigner–Eckart theorem tells us that all three matrix elements of V on theright-hand side are proportional to the same reduced element. We thus have

〈α′;jm′|J ·V |α;jm〉 = cj〈α′;j‖V ‖α;j〉, (3.4.45)

where the constants cj do not depend on α, α′ or on V itself. Moreover, sinceJ ·V is a scalar operator, neither can they depend on m. Then, since the equality(with the same constants) is also valid for, say, V =J , we may write

〈α;jm′|J2|α;jm〉 = j(j + 1) ~2

= cj〈α;j‖J‖α;j〉, (3.4.46)

where we have explicitly indicated that J does not alter the other quantum num-bers embodied in α,α′. Now, the Wigner–Eckart theorem leads directly to

〈α′;jm′|Vk|α;jm〉〈α;jm′|Jk|α;jm〉 =

〈α′;j‖V ‖α;j〉〈α;j‖J‖α;j〉 , (3.4.47)

We may substitute into the right-hand side with the expressions (3.4.45 & 46) andthus finally obtain the sought relation (3.4.41).

3.5. BIBLIOGRAPHY 85

3.5 Bibliography

Aspect, A., Grangier, Ph. and Roger, G. (1981), Phys. Rev. Lett. 47, 460; ibid.(1982) 91; Aspect, A., Dalibard, J. and Roger, G., ibid. (1982) 1804.

Baker, H.F. (1902), Proc. Lond. Math. Soc. s1-34, 347; ibid. s1-35 (1903) 333;ibid. s2-3 (1905) 24.

Bell, J.S. (1964), Physics 1, 195.

Bertlmann, R.A., Bramon, A., Garbarino, G. and Hiesmayr, B.C. (2004), Phys.Lett. A332, 355.

Bohm, D. (1951), Quantum Theory (Prentice–Hall).

Bohm, D. (1952), Phys. Rev. 85, 166; 180.

Bohr, N. (1935), Phys. Rev. 48, 696.

Campbell, J.E. (1897), Proc. Lond. Math. Soc. s1-28, 381; ibid. s1-29 (1898) 14.

Einstein, A., Podolsky, B. and Rosen, N. (1935), Phys. Rev. 47, 777.

Franson, J.D. (1985), Phys. Rev. D31, 2529.

Ghirardi, G.C., Grassi, R. and Ragazzon, R. (1992), in The DAPHNE PhysicsHandbook, eds. L. Maiani, G. Pancheri and N. Paver (Ist. Naz. Fis. Nucl.),vol. 1, p. 283.

Go, A., Belle Collab. (2004), in proc. of the Fifth Workshop on Mysteries, Puzzlesand Paradoxes in Quantum Mechanics (Gargnano, Sept. 2003), ed. P.L. Knight;J. Mod. Opt. 51, 991.

Go, A. et al., Belle Collab. (2007), Phys. Rev. Lett. 99, 131802.

Gröblacher, S. et al. (2007), Nature 446, 871.

Hausdorff, F. (1906), Ber. Verh. Sachs. Akad. Wiss. Leipzig 58, 19.

Chapter 4

Symmetry

4.1 Symmetries and conservation laws

Although symmetries play an undeniably fundamental role in physics, this fact israrely explicitly acknowledged in textbooks and usually only very specific examplesare treated. Indeed, historically, it was perhaps not until the work of EmmyNoether (1918) that the concept of symmetry and its general importance in physicswere fully recognised. Noether’s theorem clarified the intimate relation betweensymmetry and conservation laws that now underpins the development of physicsfrom classical mechanics through quantum mechanics up to quantum field theoryand even string theory.

4.1.1 Symmetries in classical physics

We open the discussion of symmetries by reflecting on their role in classical physics.In particular, we shall examine the close relationship between symmetries andconservation laws. Following the Lagrangian formulation of quantum mechanics,one takes the Lagrangian L(qi, qi) as the starting point. Now, if L is invariantunder translations qi→ qi+δqi, i.e.

∂L

∂qi= 0, (4.1.1)

then the Lagrange equations (the dot indicates a time derivative)

d

dt

(∂L

∂qi

)− ∂L

∂qi= 0 (4.1.2)

87

88 CHAPTER 4. SYMMETRY

lead directly todpidt

= 0, (4.1.3)

where, as usual, the canonical conjugate momentum is defined as

pi =∂L

∂qi. (4.1.4)

In other words, translation invariance of the Lagrangian in the variable qileads to conservation of the canonically conjugate momentum pi. The same resultis reached analogously via the Hamiltonian formulation:

∂H

∂qi= 0 (4.1.5)

also leads to dpidt

= 0. (4.1.6)

One can thus equally say that symmetry of L or H under qi→ qi+δqi leads toconservation of the corresponding canonical momentum pi.

4.1.2 Noether’s theorem

We have just exemplified what is a very general statement:

Theorem 4.1.1 (Noether’s theorem, 1918). In correspondence to any continuoussymmetry of the laws of physics, there exists a conservation law; and for everyconservation law, there exists a continuous symmetry.

4.1.3 Continuous symmetries in quantum mechanics

In quantum mechanics every generalised translation (including, e.g. rotations) isassociated with a unitary operator. Since it is held that, in general, translations(and rotations) are symmetries of the laws of physics, it is customary to referto such an operator as a symmetry operator. We may express the infinitesimalversion of such an operator S as

S = 1− i~ǫ G (4.1.7)

and we the call the Hermitian operator G the generator of the symmetry. If H isinvariant under the action of S,

S† H S = H or [S, H ] = 0 (4.1.8)

4.2. DISCRETE SYMMETRIES IN QUANTUM MECHANICS 89

and in terms of the generator G we also have

[G, H ] = 0. (4.1.9)

In the Heisenberg picture one then has that

dG

dt= 0, (4.1.10)

or, in other words, G represents a constant of the motion. This is just the quantummechanics version of Noether’s theorem.

Note further that since G commutes with H , it also commutes with the evol-ution operator U. We thus see equivalently that an initial eigenstate of G, witheigenvalue g say, remains such for all times.

Exercise 4.1.1. Prove explicitly the preceding statement.

In concluding this short section, we note that the above also applies to anycontinuous internal symmetry, such as isospin.

4.2 Discrete symmetries in quantum mechanics

Let us now turn to the question of discrete symmetries in quantum mechanics. Weshall briefly introduce two such symmetries: spatial inversion (usually denoted P

or parity) and time reversal (usually denoted T). Together with the operation ofcharge conjugation (usually denoted C), which exchanges matter and antimatter,they represent three pillars of our description of the physical laws. This last sym-metry, however, naturally requires discussion within the framework of the Diracequation or quantum field theory and thus a serious, in-depth examination must bepostponed until a later course. We shall see though that it already arises naturallyin connection with the relativistic formulation of quantum mechanics.

4.2.1 Charge conjugation (C)

Before discussing the more immediate symmetries of parity (P) and time reversal(T), let us briefly explain the meaning of charge-conjugation invariance. In particlephysics we know that to many particles, there corresponds an antiparticle, i.e.with all quantum numbers inverted. Now, while there is thus a clear distinctionbetween particle and antiparticle, were we to compare any physical process with itsantiparticle equivalent (e.g., n→pe−νe and n→ pe+νe), assuming C to be a perfectsymmetry of Nature, we should observe no difference in the dynamics (e.g., decayrates, angular distributions etc.). However, it turns out that Nature does not, in


fact, preserve C and this is intimately related to parity violation, which we shalldiscuss shortly.

Indeed, the three discrete symmetries of C, P and T are traditionally consideredtogether: while they are found not to be individually respected in Nature (but areviolated only by the so-called weak nuclear interaction), products in pairs areusually better respected and the product of all three is believed to be a true andexact symmetry of Nature—this is often referred to as the CPT theorem. Thus,while C and P are individually violated maximally by the weak interaction (onlyleft-handed particles and right-handed antiparticles take part in weak processes),the product CP is violated only at a very low level (precisely, the weak interactioninvolves left-handed particles and right-handed antiparticles, which behave in avery similar manner) and only in processes in which all three families of quarksplay a role. Since CPT is conjectured to be a symmetry of Nature, the non-conservation of CP necessarily implies that T is also violated. Indeed, it turns outthat it is precisely T that is manifestly violated at the Lagrangian level, although,for obvious reasons, one can normally only observe experimental evidence for thisvia CP-violating effects.

4.2.2 Spatial inversion (P)

Consider first the operation of spatial or parity inversion. Since P inverts the signof all spatial coordinates, it transforms right-handed into left-handed coordinatesystems and vice versa. As such, in three-dimensional space only, it may be viewedas a reflection and a rotation in the mirror plane∗ (see Fig. 4.1). However, as

x

y

z

reflection−−−−−→ x

y

z

rotation−−−−→ x

y

z

Figure 4.1: The parity transform may be represented as a reflection followed by arotation through 180 in the mirror plane.

always, it is more convenient to consider the operation as acting on the statesrather than on the spatial coordinates themselves. We thus wish to construct the∗ Care should be exercised in using this version of the transformation; when considering, e.g. areal physical situation, the rotation is an essential part of the transformation thus defined.


state P|α〉 for a generic state |α〉. A natural requirement is that the expectationvalue of the position operator should return a value of the same magnitude butopposite sign:

〈α|P†xP|α〉 = −〈α|x|α〉. (4.2.1)

For this to hold for any state |α〉, we clearly require

P†xP = −x, (4.2.2)

which is just what we should expect. Now, since P is unitary, we may rewrite thisas

xP = −Px (4.2.3)or simply

x,P = 0. (4.2.4)

Examining the action of P on an eigenstate of position, since the right-handside below must be an eigenstate with the opposite-sign eigenvalue, we expect

P |x〉 = e iδ |−x〉, (4.2.5)

The arbitrary phase δ is conventionally set to 0. It then follows trivially thatP2=1. Since P must also be Hermitian, one immediately finds that its eigenvalues

may therefore only be ±1.

Exercise 4.2.1. Show that P is Hermitian and that if P|α〉=χ|α〉 then χ=±1.

The case of momentum requires a little more work if we are to be rigorous.The operator p is the generator of translations. Thus, since a translation followedby P must be equivalent to a translation in the opposite direction acting on theP-inverted system, we should have

P(1− i

~p·dx

)=(1+ i

~p·dx

)P . (4.2.6)

In other words,

−Pp = pP (4.2.7)or simply

p,P = 0. (4.2.8)

Therefore, since we clearly cannot also have [p,P]=0, there cannot exist simul-taneous parity and momentum eigenstates.

It also immediately follows that, since L=x∧p, we must have

[L,P] = 0 (4.2.9)


and thus instead L and P are compatible.

Exercise 4.2.2. Demonstrate explicitly that indeed [L,P]=0.

Again, to be more rigorous and include the possibility of an intrinsic angular-momentum or spin operator, we should use the fact that a generic spin operatorJ is the generator of rotations. Consider the matrix representations of the twooperations in question:

P =

−1 0 0

0 −1 0

0 0 −1

and D(z, φ) =

cosφ − sin φ 0

sin φ cosφ 0

0 0 1

, (4.2.10)

where, without loss of generality, we take the rotation to be about the z axis. Thetwo matrices clearly commute. In general, we thus expect

PD = DP or [P,D] = 0. (4.2.11)

Once again, considering the infinitesimal form of a rotation, we require

P(1− i

~ǫJ ·n

)P† = 1− i

~ǫJ ·n (4.2.12)

and we thus obtain the more general condition

[J ,P] = 0. (4.2.13)

Exercise 4.2.3. Show explicitly that the operators S·x and S·p behave as pseudo-scalar operators under P while S·L and x·p behave as normal scalars.

Wave functions and parity

Using Dirac notation, the wave-function has the following representation in co-ordinate space:

ψα(x) = 〈x|α〉. (4.2.14)

The wave-function for the spatially inverted state is then naturally given by

〈x|P|α〉 = 〈−x|α〉 = ψα(−x). (4.2.15)

Now, if |α〉 represents a parity eigenstate, then, since P |α〉=±|α〉, we must have

ψα(−x) = 〈−x|α〉 = 〈x|P|α〉 = ±〈x|α〉= ±ψα(x) for

evenodd

parity, (4.2.16)


Note that not all wave-functions are naturally eigenstates of P: for example, planewaves, which are momentum eigenstates, are clearly not. However, while p andP do not commute, L and P do. We may thus rightly expect angular-momentumeigenstates to have definite parity.

Since L± and P commute, we immediately obtain that all states |l,m〉 for anygiven l must have the same parity (assuming it to be definite) for all m. To seethis, assume that the state |l,m〉 has some well-defined parity χ=±1. Therefore,

P |l,m±1〉 = PL±|l,m〉 = L± P |l,m〉 = L± χ|l,m〉 = χ|l,m±1〉. (4.2.17)

The case l=0 is trivial: the object is a scalar; it therefore has even parity and, inany case, only one spin-projection state (m=0). From the discussion on Clebsch–Gordan coefficients, we have seen that, for example, a spin-two object may beconstructed by composing two spin-one objects and so on. The parities of l-evenstates must thus all be the same (that is, even), as too those of l-odd states,although l-odd states could, in principle, have either odd or even parity. It is thensufficient to consider the case l=1, m=0: since Y 0

1 (θ,φ)=cosθ= z/r, we see thatall l-odd states have odd parity.

We can now demonstrate an important theorem for energy eigenstates:

Theorem 4.2.1. Assuming spatial inversion to be a symmetry of the Hamilton-ian, i.e. [P,H ]=0, then any non-degenerate energy eigenstate |n〉 is also a parityeigenstate.

Proof. Define the projection operators P±≡ 12(1±P); clearly [P±,H ]=0. Now,

PP± |n〉 = 12(P±P

2)|n〉,which (since P

2 = 1) = ± 12(±P+1)|n〉

= ±P± |n〉. (4.2.18)

The states P± |n〉 are thus clearly eigenstates of P. And, since [P±,H ]=0, theymust also clearly be eigenstates of energy with the same energy eigenvalues as|n〉. By hypothesis, the state |n〉 is non-degenerate and thus P± |n〉 and |n〉 mustrepresent the same state (up to an arbitrary and irrelevant phase). This completesthe proof.

Note that the hypothesis of non-degeneracy is vital since otherwise there couldbe states with same energy but opposite parities. Any linear combination of suchstates would still be an eigenstate of energy but clearly not of parity. Consider,e.g. the different orbital angular-momentum states of the hydrogen atom with thesame principal quantum number.


The double potential well

A very important example of a system for which parity is a symmetry of theHamiltonian is the double potential well (see Fig. 4.2). For simplicity, let us

V (x)

ψS(x)

ψA(x)

Figure 4.2: The double potential (square) well, with in the middle the lowest-energy(even parity) solution and at the bottom the first excited state (odd parity).

denote the symmetric ground state |S〉 and the antisymmetric first excited state|A〉. These are both simultaneous eigenstates of H and P. Note also that EA>ES.Using these two states we may construct two mixed-parity states:

|L〉 ≡ 1√2

(|S〉+ |A〉

)and |R〉 ≡ 1√

2

(|S〉 − |A〉

). (4.2.19)

Such states are clearly concentrated spatially in the left- and right-hand wellsrespectively; they are neither parity nor energy eigenstates and are not thereforestationary states. We may also express the states |S,A〉 in terms of |R,L〉:

|S〉 = 1√2

(|L〉+ |R〉

)and |A〉 = 1√

2

(|L〉 − |R〉

). (4.2.20)

Let us take the state that best represents the particle placed in the well on theright-hand side at some initial time t=0: namely, |R〉. From the standard resultson temporal evolution, at some later instant in time we have

|R,0;t〉 = 1√2

e−

i~Es t |S〉 − e−

i~EA t |A〉

= 12

e−

i~Es t(|R〉+ |L〉

)+ e−

i~EA t

(|R〉 − |L〉

)

= 12

(e−

i~Es t+e−

i~EA t

)|R〉+

(e−

i~Es t− e−

i~EA t

)|L〉

= e−i~E tcos(

12~∆E t

)|R〉+ i sin

(12~∆E t

)|L〉, (4.2.21)


where we have introduced E≡ 12(EA+ES) and ∆E≡ (EA−ES). We thus see that

the particle oscillates back and forth between the two wells with frequency ∆E/~.This is perhaps the simplest example of a spontaneously broken symmetry :

although the Hamiltonian is parity conserving, an initial condition that places theparticle somewhere definite, must choose either the right- or left-hand well andthus the physical solution to the system violates parity. Indeed, had we consideredinfinitely deep wells, all the states used above would have been degenerate andthus the choice of a ground state would have immediately broken the symmetry ofthe system. A physical example of such a phenomenon is a ferromagnetic material:although the magnetic domains have no intrinsic preferred direction, the groundstate has them aligned and must therefore choose a specific direction.

Note, however, that such spontaneous symmetry breaking, due to the necessarychoice of a ground state, does not imply any fundamental violation of parity. Onthe other hand, it has been known since the sensational experiments of Wu et al.in the 1950’s that the weak interaction does indeed know the difference betweenright- and left-handed spinning particles and thus violates parity at a fundamentallevel. The effects of such violation may be seen in various ways: for example, in theβ-decay of polarised Co60 nuclei more electrons emerge downwards than upwardswith respect to the polarisation axis. Moreover, nuclei that should be pure parityeigenstates are found not to be (although the parity-violating admixtures are verysmall indeed); similar effects have even been detected at the atomic level.

Another interesting and simple example in which there is an energy differenceis provided in atomic physics by the ammonia (NH3) molecule. As shown inFig. 4.3, the hydrogen atoms sit at the three vertices of an equilateral triangleand the nitrogen atom at the fourth vertex or apex of the pyramid thus formed.Since the molecule rotates about the symmetry axis running through the nitrogen

H

H

H

N

N ′

Figure 4.3: The ammonia molecule. The symmetry axis (and axis of rotation) indicatedruns through the two equivalent nitrogen sites N,N ′ and the centre of the equilateraltriangle defined by the hydrogen atoms H.


sites and the centre of the equilateral triangle, this axis is defined. The nitrogenatom thus has two possible, equivalent sites: either above or below the hydrogenplane (in precise analogy with the two potential wells above). The natural (dipole)oscillation frequency of the system is 23.87GHz; this is the frequency correspondingto the energy difference between the ground state (symmetric wave-function withrespect to the two nitrogen sites) and the first excited state (antisymmetric).

In the quantum field theory of particle physics the same basic principle leadsto the spectacular phenomenon of particle oscillation. There the role of the doublewell is played by two states of some neutral particle and its antiparticle (e.g. theneutral kaon or even the neutrino). The CPT theorem states that particles andtheir antiparticles must have the same mass. Now, while a particle may representa well-defined state of P it will not necessarily represent an eigenstate of C. Indeed,C transforms K0 (a down–antistrange quark pair) into K0 (antidown–strange) andvice versa. The superposition states 1√

2(|K0〉±|K0〉) are, however, eigenstates of

CP and mass, with different masses. Assigning the states |K0,K0〉 the same roleas |L,R〉 above, we thus see that a beam (produced, e.g. in a high-energy collision)initially containing only |K0〉, say, will at some later time contain an admixtureof |K0〉 (and vice versa). The fact that these states decay with lifetimes of theorder of 10−10 s complicates the experimental situation. Fortunately, however, theoscillation period is of the same order of magnitude and thus such oscillations are,in fact, experimentally visible. The measured frequency corresponds to a rest-massenergy difference between the two states of ∼3.5×10−6 eV. As an aside, we mightremark that the decays of the K0–K0 system also provide the first example of thenon-conservation of CP.

Parity selection rules

The conservation of parity (just as of, e.g., angular momentum) leads to selectionrules for transitions involving states and operators of definite parity. Let |α〉 and|β〉 be two such states with

P |α〉 = χα|α〉 and P |β〉 = χβ|β〉 (χα,β = ±1). (4.2.22)

Consider, e.g. the matrix element 〈β|x|α〉:

〈β|x|α〉 = 〈β|P−1PxP

−1P|α〉 = χα χβ〈β|(−x)|α〉 = −χα χβ〈β|x|α〉, (4.2.23)

which is only possible if either 〈β|x|α〉=0 or χαχβ =−1. This selection rule,known phenomenologically as Laporte’s rule, is well established experimentally. Aparticular case is the matrix element 〈n|x|n〉, which is proportional to the electricdipole moment (EDM) of the state. If |n〉 is non-degenerate and the Hamiltonian


preserves parity, then according to the previous argument such a matrix elementmust vanish and the corresponding state must therefore have zero EDM. Aninteresting test of parity violation is thus provided by the study of the neutronEDM, which is indeed found to be zero within present experimental accuracy.

4.2.3 Time reversal (T)

Although somewhat less easy to imagine than the previous two discrete symmet-ries, one can perhaps best picture the effects of T as the projection of a film orsequence of photographs in reverse. If we were to view, for example, the motion ofone or more billiard balls on a frictionless table in this way, it would be impossibleto discern whether the film were being projected normally or in reverse.∗ Friction,or dissipation in general, clearly violates T. However, if we examine any classicalmotion at a microscopic level, T is preserved. Formally, if x(t) is a solution of theclassical equations of motion

E =p2

2m+ V (x), (4.2.24)

where V (x) depends only on x and not on t either directly or indirectly througha dependence on p (dissipation), then so too must x(−t) be a solution. This istrivially true since, by hypothesis, time only enters H through p, which occursquadratically and is thus insensitive to the sign of t.

In quantum mechanics the situation is subtly different. Consider the usualSchrödinger equation

i~∂ψ(x, t)

∂t=

[− ~

2

2m∇

2 + V (x)

]ψ(x, t). (4.2.25)

Although a given ψ(x,t) may be a solution, ψ(x,−t) is not in general a solution,unless (of course) ∂ψ/∂t=0. The problem can be traced to the presence of asingle derivative with respect to t on the left-hand side. It is not difficult to seethat ψ∗(x,−t) is, however, a solution. This can be more clearly understood byexamining the case of energy eigenstates:

ψ(x, t) = un(x) e− i

~En t and ψ∗(x,−t) = u∗n(x) e

− i~En t . (4.2.26)

In view of this, it is natural to conjecture that the second wave-function aboveis the time-reversed partner of the first. We should thus say, in Dirac’s notation,∗ Provided, that is of course, that there are no pockets into which the balls may disappear! Infact, the concept of, e.g., particle decay does lead to a direction or arrow of time.


that 〈x|α〉∗ is simply the time-reversed wave-function corresponding to 〈x|α〉 attime t=0.

Exercise 4.2.4. Verify the above statement for the case of plane waves.

Antilinear operators

In order to discuss the symmetry operation of time reversal, it is necessary tointroduce the idea of an antilinear operator. Let us begin by examining the actionof a generic symmetry operator:

S : |α〉 → |α〉. (4.2.27)

In order to be a symmetry operator, S should not change the overall magnitude ofinternal products (probability amplitudes) nor change the relative phase betweendifferent products. The first requirement may be expressed as

∣∣〈β|α〉∣∣ =

∣∣〈β|α〉∣∣, (4.2.28)

which, combined with the second, allows two distinct possibilities (ignoring arbit-rary and irrelevant phase rotations): either

〈β|α〉 = 〈β|α〉, (4.2.29a)or

〈β|α〉 = 〈β|α〉∗ = 〈α|β〉. (4.2.29b)

The first follows naturally from the requirement of unitarity since, for a unitarytransformation U, one has

〈β|α〉 = 〈β|U†U|α〉 = 〈β|α〉. (4.2.30)

The second is, however, equally valid and defines what is known as an antiunitarytransformation, which is just a special case of an antilinear operation.

Recall that a linear operator must satisfy the following distributive rule:

OL

(c1|α〉+ c2|β〉

)= c1OL |α〉+ c2OL |β〉. (4.2.31a)

It is obvious here that, in order to satisfy the restrictions on the internal product,it is necessary to extend the rule to cover the case of an antilinear operator:

OA

(c1|α〉+ c2|β〉

)= c∗1OA |α〉+ c∗2OA |β〉. (4.2.31b)

The effect is thus to transform all C-number coefficients into their complex con-jugates. Indeed, any antilinear operator can be written as the product of some


linear operator and the complex-conjugation operator K:

OA = OLK . (4.2.32)

Note that (by convention) K sits to the right of OL. As a special case, anyantiunitary operator may be expressed as the product of a unitary operator andK. The operation of time reversal is of just such a type.

Exercise 4.2.5. Show, by considering the expansion over a basis set of states |a〉,that Eqs. (4.2.29b) and (4.2.31b) are satisfied by an operator of the form UK.

The time-reversal operator

Armed with the definition of an antiunitary operator, we are now in a position toexamine the operation of time reversal. Let us first examine the effect of such atransformation on the Schrödinger equation, which becomes

T

(i~∂ψ

∂t

)= THψ. (4.2.33)

If [T,H ]=0, this leads to

− i~∂(T ψ)

∂(−t) = i~∂(T ψ)

∂t= H(T ψ), (4.2.34)

which is, indeed, the Schrödinger equation for Tψ. Thus, if ψ is an energy eigen-state, of energy E, so too must Tψ be, with the same energy.

Let us now approach the problem in the Dirac formalism, starting from aninfinitesimal time translation applied to the state |α〉 at time t=0:

|α;δt〉 =(1− i

~Hδt

)|α〉. (4.2.35)

Consider the evolution of the time-reversed state in the case where H is invariantunder T: we should expect

T |α;−δt〉 =(1− i

~Hδt

)T |α〉. (4.2.36)

In other words,

T(1− i

~H(−δt)

)|α〉 =

(1− i

~Hδt

)T |α〉. (4.2.37)


For this to be generally true, we must have

−T iH = iH T, (4.2.38)or

T, iH = 0, (4.2.39)

where we have deliberately left in the complex factor “ i ”. Now, if we wish toimpose time-reversal invariance on H , we also require the standard commutationrule [T,H ]=0. For both to be true, the action of T on C-numbers must be that ofcomplex conjugation. One might question whether the rule for such an operatorshould not instead simply be an anticommutator. However, the anticommutationof T and H would imply, for instance,

T p2 = −p

2T and T V (x) = −V (x)T, (4.2.40)

where the second example should hold for any potential. This would, of course,be absurd since, by taking the expectation value for any state, one would findnegative kinetic and potential energies for the transformed case.∗

Let us take a moment to underline a problem with the notation: thus far wehave always read an expression such as 〈β|O|α〉 as meaning either O acts on theket to the right or on the bra to the left. However, in the case of T this is bestavoided, i.e. it should only be considered as acting to the right. This will notgenerally be an obstacle though; consider the following pair of kets:

|α〉 ≡ T |α〉 and |β〉 ≡ T |β〉. (4.2.41)

Now, define |γ〉 by|γ〉 ≡ O

† |β〉. (4.2.42)

Standard dual correspondence then leads to

〈γ| ≡ 〈β|O . (4.2.43)

Hence,

〈β|O|α〉 = 〈γ|α〉 = 〈γ|α〉∗ = 〈α|γ〉= 〈α|TO† |β〉 = 〈α|TO†

T−1T |β〉

= 〈α|TO†T−1 |β〉. (4.2.44)

In particular, for an Hermitian operator, such as corresponds to a physical observ-∗ We might remark that, as we shall see later, this becomes a delicate but very central issue inthe case of relativistic quantum theory.


able, we have〈β|O|α〉 = 〈α|TOT

−1 |β〉. (4.2.45)

Now, we shall say that an operator is T even or odd according as to whether

T OT−1 = ±O . (4.2.46)

If, further, |β〉 is the same as |α〉, then we simply have an expectation value:

〈α|O|α〉 = ±〈α|TOT−1|α〉. (4.2.47)

An example of this is given by the case of the momentum operator p, alreadydescribed.

Time reversal for spinless states: Since we clearly require that the effect oftime reversal on velocities, linear momenta and angular momenta should be toreverse their sign (since they are all linear in the time derivative), then we musthave

〈α|p|α〉 = −〈α|p|α〉. (4.2.48a)

Now, as usual, |α〉 = T |α〉 and thus

〈α|T−1pT |α〉 = −〈α|p|α〉. (4.2.48b)

For this to be true for all states |α〉, we clearly require

T−1

pT = −p, (4.2.49)or, equivalently,

T,p = 0. (4.2.50)

Applying a similar argument to x, with the obvious change of sign, we have

〈α|x|α〉 = 〈α|x|α〉 (4.2.51a)and thus

〈α|T−1xT |α〉 = 〈α|x|α〉. (4.2.51b)

For this to be true for all states |α〉, we clearly require

T−1

xT = x, (4.2.52a)or, equivalently,

[T,x] = 0. (4.2.52b)


Similarly, from the definition L :=x∧p, it is straightforward to obtain

T−1

LT = −L, (4.2.53a)or, equivalently,

T,L = 0. (4.2.53b)

We might remark at this point that the commutation relation [xi,pj ]= i~δij issimultaneously preserved if and only if T is antiunitary (since then the right-handside changes sign to compensate the change in sign of p).

Now, in the coordinate-space representation p=− i~∇ and thus the obviouschoice for T is simply T=K. Note, however, that in the momentum-space rep-resentation x= i~∂/∂p while p is a real operator. Here we should thus chooseT=UK, with U the operator that transforms p into −p. It is important to recog-nise and remember this representation dependence of transformation operators.

Time reversal for non-zero spin states: By analogy with Eq. (4.2.53b), it isnatural to expect

T,S = 0 and T,J = 0. (4.2.54)

Indeed, the commutator rules for angular momenta,

[Ji, Jj] = i~ εijkJk, (4.2.55)

by virtue of the complex factor “ i ” on the right-hand side, actually require suchan anticommutation relation. Now, we have just seen that the precise form of Tdepends on the representation adopted and thus we expect that, before proceedinghere, we should specify the representation of the spin operator. In the standardrepresentation for spin-1/2, we have the three Pauli matrices:

Sx =~

2

(0 11 0

), Sy =

~

2

(0 − ii 0

), Sz =

~

2

(1 00 −1

). (4.2.56)

For the spin-one case, we adopt the following dimension-three matrices:

Sx =~√2

0 1 01 0 10 1 0

, Sy =

~√2

0 − i 0i 0 i0 − i 0

, Sz =

~√2

1 0 00 0 00 0 −1

.

(4.2.57)In this representation Sx,z are real while Sy is purely imaginary. Writing T=UK,T,S=0 thus requires that

U, Sx = 0, [U, Sy] = 0, U, Sz = 0. (4.2.58)


Clearly then, U must be a function of Sy only. Indeed, recalling that Sy is thegenerator of rotations about the y-axis, it is obvious that the operator exp(− i

~πSy)

transforms Sx,z into −Sx,z and Sy into Sy. We thus take

T = e−i~π Sy K . (4.2.59)

For the spin-1/2 case, since σ2y =1, we may write this as

T = − iσy K . (4.2.60)

N.B. this last equality does not hold for the higher-dimensional representations.

Time reversal for multiparticle states: For composite systems, the time-reversal operator may be constructed using the spin operators for each sub-system:

T = e−i~π S

(1)y e−

i~π S

(2)y · · · e− i

~π S

(n)y K . (4.2.61)

Note that, since each exponential is purely real (Sy is purely imaginary) and[S(i)

y ,S(j)y ]=0,the ordering in the product is irrelevant in this case.

We now discuss an aspect of particles with intrinsic spin that leads to surprisingresults connected with Fermi–Dirac statistics. Consider a particle with an arbitraryspin direction n; it may be represented by applying a suitable rotation to a statepolarised along the positive z-axis:

|n+〉 ≡ D(α,β,γ)|+〉∣∣∣α=φβ=θγ=0

= e−i~Szφ e−

i~Syθ |+〉, (4.2.62)

where θ and φ are the angles associated with n. We then apply a rotation throughsome angle α, which, since the polarisation is taken in a generic direction, may bechosen without loss of generality about the z-axis:

|n+〉 → |n′+〉 = D(α,β,γ)|n+〉∣∣∣β=0γ=0

= e−i~Szα e−

i~Szφ e−

i~Syθ |+〉. (4.2.63)

The two exponentials on the left commute trivially and we are thus left with theproblem of commuting the first with the last:

e−i~Szα e−

i~Syθ |+〉 =

∑

r,s

1

r!

(− i

~αSz

)r 1

s!

(− i

~θ Sy

)s |+〉

=∑

r,s

1

r!

1

s!(−1)rs

(− i

~θ Sy

)s (− i~αSz

)r |+〉


=∑

r,s

1

r!

1

s!(−1)rs

(− i

~θ Sy

)s (− imα)r |+〉

=∑

s

1

s!

(− i

~θ Sy

)se−(−1)

simα |+〉. (4.2.64)

In order to proceed further, we may assign a value to α. Choosing α=2π rendersthe exponential in the last line above trivially unity in the case of integer spin(m∈N) and −1 in the half-integer case (m=n+ 1

2with n∈N). In either case the

terms may now be resummed to restore the original rotation and hence the initialstate. We are thus left with

D(2π, 0, 0)|n+〉 = ±|n+〉, (4.2.65)

where the sign is positive for spin-one objects (or bosons) and negative for half-integer spins (or fermions). The surprising result is then that a rotation through2π does not leave a fermionic state entirely unaltered: a non-trivial phase (−1) isintroduced. This result has many and profound consequences.

Returning now to the time-reversal operator for a multiparticle state, since K

commutes with iSy and also K2=1, we have

T2 = e−

i~2π S

(1)y e−

i~2π S

(2)y · · · e− i

~2π S

(n)y . (4.2.66)

Each of the exponentials in this expression is equivalent to a rotation through 2π.Following the above discussion, we thus have a factor +1 for each boson and −1for each fermion. Overall, T2=+1 if the number of fermions is even and −1 if itis odd. To be precise, the eigenvalue of T2 is (−1)2j for a particle of spin j.

Now, as we have already seen for time-reversal invariant systems, if |n〉 isan energy eigenstate, then so too is T |n〉, with the same energy. If there is nodegeneracy, the two states must therefore be identical, up to an irrelevant phase:

T |n〉 = e iδ |n〉. (4.2.67)Consider now T

2,T2 |n〉 = T e iδ |n〉 = e− iδ

T |n〉 = |n〉. (4.2.68)

Clearly, in the case of an odd number of particles with half-integer spin such aresult is in contradiction with the deduction that the eigenvalue of T2 is −1 foreach fermion. The hypothesis of non-degeneracy must therefore fall. Indeed, onecan easily show that |n〉 and T |n〉 are orthogonal for particles with half-integerspin.

Now, consider an atom inside a crystal with a very low level of symmetry.Since, by hypothesis, the atom is in an asymmetrical environment, one would not


expect its electronic states to be degenerate. Note, however, that interaction withthe rest of the crystal cannot violate time-reversal invariance and thus T2=(−1)2j

for each electron must still hold. Hence, if there is an odd number of electrons,despite the explicit lack of symmetry in the potential there must be (at least) atwo-fold degeneracy ; this is known as Kramers’ degeneracy (1930). The degeneracymay be lifted by the action of a magnetic field since this induces interactions of theform B·L (or p·A+A·p) and B·S, which, according to the commutation relations(4.2.53b & 54), change sign under the action of T. A magnetic field applied to acrystal with an odd number of electrons per atom will thus cause a splitting ofthe degenerate energy levels. Note that this is by no means an indication of anyfundamental violation of time-reversal invariance; the magnetic field is external tothe system and it is unaffected by the T operator applied to the crystal (which, ofcourse, is not the case for any magnetic fields inside the crystal produced by theatoms themselves).

The reality of eigenfunctions

Consider a system in which spin plays no role, then in the coordinate-space rep-resentation we have T=K. Now, if there exists some operator O that commuteswith K and that has non-degenerate eigenstates |α〉:

O |α〉 = ωα|α〉, (4.2.69)

then, by hypothesis, |α〉 and T |α〉 represent one and the same state. We canrewrite this as a wave-function equation by projecting with 〈x| to obtain

Oψα(x) = ωαψα(x) (4.2.70a)and

T ψα(x) = cαψα(x) (cα ∈ C). (4.2.70b)

Let us now write the wave-function in terms of a real amplitude A(x) and phaseangle W (x):

ψα(x) = Aα(x) eiWα(x) . (4.2.71)

Equation (4.2.70b) then becomes

Aα(x) e− iWα(x) = e iwα Aα(x) e

iWα(x), (4.2.72)

where we have also expressed the complex constant cα in terms of a constant phasewα (the magnitude is necessarily unity). We thus have

−Wα(x) + 2nπ = wα +Wα(x) (n ∈ Z). (4.2.73a)


orWα(x) = nπ − 1

2wα. (4.2.73b)

In other words, we have shown that the phase Wα(x) is actually constant. Sincea constant phase is irrelevant, this is implies that the wave-function is effectivelyreal. The foregoing argument may also be extended to the degenerate case, withimportant implications for scattering theory (a topic to which we shall returnlater).

Consider now the case in which the Hamiltonian is real and the scatteringpotential is spherically symmetric (i.e. central). We therefore have the followingset of commutation relations:

[H,L2] = 0, [K,L2] = 0, [K, H ] = 0 (4.2.74)

Moreover, the eigenstates |nlm〉 are distinguished by the energy eigenvalue En andthe orbital angular-momentum number l, leaving a (2l+1)-fold degeneracy withrespect to the magnetic quantum number m. Now, the operator associated withm (Lz) is purely imaginary:

Lz = − i~∂

∂φ. (4.2.75)

and thus it clearly does not commute with K. One cannot then make the sameargument for the reality of the wave-functions, as was made for the purely energy-eigenvalue case. Indeed, we know that the spherical harmonics are not in generalreal functions.

On the other hand, if we restrict ourselves to the case m=0 (i.e. the eigenvaluesof Lz vanish), then it follows from the relation Lz,K=0 that Kψnl0 is still aneigenfunction of Lz, with eigenvalue zero. Therefore, assuming there to be noother degeneracy, the wave-functions ψnl0 are real (in the sense that their phaseis constant in space). Now, these are precisely the eigenfunctions that appear inscattering theory and thus we have shown that the induced phase shifts δl (seelater) are real or complex according as to whether the Hamiltonian is or is nottime-reversal invariant (i.e. whether or not there is absorption). We shall examinethis case more closely in the next chapter.


4.3 Bibliography

Kramers, H.A. (1930), Kon. Ned. Akad. Wetensch. Proc. 33, 959.

Noether, E. (1918), Nachr. v. d. Ges. d. Wiss. zu Göttingen, 235; transl., TransportTheory and Statistical Mechanics 183.

Wu, C.-S., Ambler, E., Hayward, R.W., Hoppes, D.D. and Hudson, R.P. (1957),Phys. Rev. 105, 1413.

Chapter 5

Perturbation Theory and Scattering

Much of the ground-work has already been laid for the approach to scatteringtheory. In this chapter we shall first quickly review time-dependent perturbationtheory and the derivation of Fermi’s golden rule, after which we shall move on tothe general description of scattering.

We open this chapter then with an examination of simple, exactly solvablesituations involving time-dependant potentials, thus preparing the way to discusstime-dependent perturbation theory. Following this we shall delve into the all-important subject of scattering theory.

5.1 Time-dependent potentials

5.1.1 The Rabi formula

Only a few time-dependent potentials lead to an exactly solvable model and onemust usually resort to perturbation theory, or other methods of approximation, toobtain any sort of working solution. There are, nevertheless, particular systemsthat are of some practical importance and that are essentially reducible to a two-level model, which may then be treated exactly.

Such a model may be described as follows:

H0 = E1|1〉〈1|+ E2|2〉〈2| (5.1.1a)and

V (t) = V0 e+iωt |1〉〈2|+ V0 e

− iωt |2〉〈1|, (5.1.1b)

where V0 and ω may be taken real and positive, as too the difference E2−E1. TheHamiltonian is constructed so as to be explicitly Hermitian (indeed, both H0 and

109

110 CHAPTER 5. PERTURBATION THEORY AND SCATTERING

V (t) are so separately): defining Vij = 〈i|V (t)|j〉, we have

V12 = V ∗21 = V0 e

iωt (5.1.2a)and

V11 = V22 = 0. (5.1.2b)

The interaction term explicitly connects the two eigenstates of the bare or freeHamiltonian and may thus be expected to cause transitions. The system of equa-tions to be solved is then, following Eq. (2.2.31),

i

(c1(t)

c2(t)

)=

(0 β e+i(ω−ω21)t

β e− i (ω−ω21)t 0

)(c1(t)

c2(t)

), (5.1.3)

where β=V0/~ and ci(t) are the coefficients describing the state:

|α;t〉 =

2∑

i=1

ci(t)|i〉 (5.1.4)

and ω21=(E2−E1)/~.This simple system of coupled equations can be solved exactly and, with the

boundary conditions c1(0)=1 and c2(0)=0, one obtains the Rabi∗ formula† forthe occupancy of the upper level

∣∣c2(t)∣∣2 =

β2

β2 + ω2 sin2

(√β2 + ω2 t

), (5.1.5a)

from which the occupancy of the lower level may be obtained via

∣∣c1(t)∣∣2 = 1−

∣∣c2(t)∣∣2 (5.1.5b)

and where, for clarity, we have defined

ω = (ω − ω21)/2. (5.1.6)

The system thus oscillates between the two states (this is very reminiscent ofthe double-well problem) with angular frequency

√β2+ ω2. The amplitude of

these oscillations, β2/(β2+ ω2), depends on the resonance condition: it is largefor ω∼ω21 and small if they differ appreciably. When the oscillating potential∗ The 1944 Nobel Prize for Physics was awarded to Isidor Rabi “for his resonance method forrecording the magnetic properties of atomic nuclei.”

† Formulæ (5.1.5) were first derived by Rabi in connection with molecular beams.

5.1. TIME-DEPENDENT POTENTIALS 111

is perfectly resonant (i.e., ω=ω21), the amplitude is unity; that is, the systemoscillates perfectly between the two states, with frequency β/~. There is thus acontinual pumping of energy to and fro between the driving potential V (t) andthe two-level system.

Exercise 5.1.1. Derive explicitly the Rabi formula Eq. (5.1.5).

5.1.2 Magnetic spin resonance

The two-level model discussed above has many applications in physics, such asnuclear magnetic resonance (NMR), molecular beams, masers and so on. Considera spin-1/2 two-level system (e.g. an electron in a magnetic field B). We may takethe unperturbed Hamiltonian as the magnetic interaction of the electron with aconstant B field in the z-direction while we shall take the time-varying piece as aconstant-magnitude B field rotating in the x−y plane, thus:

B = B0z +B1(x cosωt+ y sinωt), (5.1.7)

with B0 and B1 constant. The form of the interaction is that given in Eq. (2.1.34).We thus have

H0 = −12~γB0

(|+〉〈+| − |−〉〈−|

)(5.1.8a)

and

V (t) = −12~γB1

[cosωt

(|+〉〈−|+ |−〉〈+|

)− i sinωt

(|+〉〈−| − |−〉〈+|

)]

= −12~γB1

[e+iωt |−〉〈+|+ e− iωt |+〉〈−|

].

(5.1.8b)

Since the charge e of the electron is negative, we have E−<E+ and therefore |−〉 isthe lower level and |+〉 the upper (corresponding respectively to the earlier levels1 and 2). The energy difference between the two leads to a characteristic (Larmor)precession frequency given by

ω21 =|e|B0

mec. (5.1.9)

In the absence of the rotating field B1 we know that the system has a naturalprecession frequency of ω21. That is, the (non-zero) expectation value of S in thex−y plane rotates while the expectation value of Sz remains fixed. It is thereforenot surprising to again find a resonance phenomenon when we turn on a fieldrotating in the x−y plane (B1). Indeed, the equations are those just derived, with


the same frequency ω and the simple substitution

β = 12γB1. (5.1.10)

Now, the interpretation of the solutions is that the system still precesses but theexpectation value of Sz is no longer constant. The values of Sz =±1

2correspond to

the two levels. We thus see that, besides precessing, the system continually flipsits vertical spin component.

Exercise 5.1.2. Show that an electron placed in a constant magnetic field B0 anda rotating magnetic field B1, as above, simply precesses about an axis parallel tothe directions of B0 with the angular frequency given in Eq. (5.1.9), i.e. calculatethe expectation values of Sx and Sy.

Arranging a rotating magnetic field is clearly quite impractical for real ap-plications. The same physical effects may be attained, however, with a field ofconstant direction (perpendicular to the main field B0) but sinusoidally oscillatingmagnitude. Such a field may be decomposed into a superposition of oppositelyrotating fields in the horizontal plane:

B1 cosωt x = 12B1

(cosωt x+ sinωt y

)+ 1

2B1

(cosωt x− sinωt y

). (5.1.11)

Here the signs of the rotations play an important role: if the clockwise rotatingcomponent satisfies ω≃ω21 (i.e. it is close to resonance), then clearly the other hasan equivalent ω≃−ω21 and is therefore far off resonance. Indeed, if we arrangefor B1≪B0, so that β≪ω21, then when one rotation is close to resonance theother will be very far off. The system will thus behave very much as if only theresonant rotating component were present, leading to the same physical behaviouras described above.

In an operational situation, by varying the frequency of the oscillating field, onecan, for example, extract information on the magnetic environment that differentnuclei experience. The basic idea is to examine the absorption spectrum as a func-tion of the frequency of the oscillating field. The observed spectrum will displayshifts from the known spectrum for the nuclei in the free condition; such differencesindicate the nature of the local magnetic fields and thus the chemical environment.This is the basis of the diagnostic technique known as nuclear magnetic resonance.

5.1.3 The maser

We shall now examine the theory and workings of a device known as the maser(microwave amplification by stimulated emission of radiation). This device was

5.1. TIME-DEPENDENT POTENTIALS 113

invented in 1953 by Gordon, Zeiger and Townes (1954)∗ and is the forerunner ofthe laser (light amplification by the stimulated emission of radiation), inventedin 1960 by Maiman (1960)† and which was first simply called an optical maser.

The basis is the ammonia molecule NH3 already discussed. We define d to bethe EDM of the molecule. The structure of NH3 leads one to expect that d shouldbe proportional to xN , the position (operator) of the nitrogen atom with respectto the centre of the hydrogen triangle. Ammonia molecules are placed inside amicrowave cavity, in which there is an oscillating electric field,

E(t) = E0 cosωt z. (5.1.12)

The interaction energy is then simply −d·E. Now, although the oscillating electricfield may have some spatial dependence, on the scale of the ammonia moleculesuch variations will be negligible since the wavelength of microwaves is somewhatgreater than the size of a single molecule. Recall now that the nitrogen atom maysit at either of two equivalent sites and, indeed, oscillates between them. Therewill, as usual, be a symmetric ground state |S〉 and an antisymmetric first excitedstate |A〉. The resonant frequency is then given by the energy difference:

ωAS =(EA −ES)

~. (5.1.13)

From the earlier discussion on matrix elements of the electric-dipole operator, weknow that the diagonal elements must vanish:

〈S|d|S〉 = 0 = 〈A|d|A〉 (5.1.14a)

while the off-diagonal transitions may be non-zero:

〈S|d|A〉 6= 0 6= 〈A|d|S〉. (5.1.14b)

We thus have a classic two-level coupled system, as before.Fundamental to the workings of the maser is the concept of population inver-

sion. In ammonia, where historically the first population inversion was attained,this is established by physical separation of molecules in the upper and lowerquantum states. Such separation is achieved by projecting a beam of ammonia∗ One half of the 1964 Nobel Prize for Physics was awarded to Charles Hard Townes and onequarter each to Nicolay Gennadiyevich Basov and Aleksandr Mikhailovich Prokhorov “for funda-mental work in the field of quantum electronics, which has led to the construction of oscillatorsand amplifiers based on the maser–laser principle.”

† It should be mentioned, however, that while Theodore Maiman is usually credited as the in-ventor of the laser, earlier work was performed by R. Gordon Gould, who also held most of thepatents for such devices.


molecules through a system of focussing electrodes, which generate a cylindricalquadrupole electrostatic field in the direction of the beam. Molecules in the lowerquantum state experience an outward force and rapidly leave the beam while thosein the upper quantum state experience a radial inward (and thus focussing) forceand so enter the cavity. This is somewhat analogous to the action of a Stern–Gerlach apparatus.

When microwave power passes through the cavity at the resonant frequency23.87GHz, amplification occurs due to the population inversion. That is, energyis extracted from the ammonia molecules as they pass, by cascade transition orstimulated emission, from the upper to the lower state; this is coherent with andthus reinforces the beam. If the output power emitted is sufficiently large, self-sustained oscillations result and the input beam is actually no longer required.When operating beyond this threshold the internal losses are compensated bya large enough gain and the device behaves as an oscillator. Below thresholdit behaves as an amplifier—precisely, via microwave amplification by stimulatedemission of radiation.

5.2 Time-dependent perturbation theory

Unfortunately, in most situations of physical interest an exact solution to thecoupled evolution equations is unattainable and we must resort to perturbativemethods. The expansion is via the Dyson series, already derived, which here willbe an expansion in powers of the (small) perturbing potential. The derivation inthis case is very similar to that already performed for the evolution operator, butproceeds from the interaction picture.

5.2.1 The Dyson series in the interaction picture

Our starting point will be the interaction-picture Schrödinger equation for thetemporal-evolution operator UI:

i~d

dtUI(t, t0) = VI(t)UI(t, t0), (5.2.1)

with the usual boundary (or initial) condition

UI(t0, t0) = 1. (5.2.2)

In the most general case VI may depend on t and does not necessarily commutewith itself at different times. Again, we may formally integrate Eq. (5.2.1) to

5.2. TIME-DEPENDENT PERTURBATION THEORY 115

obtain

UI(t, t0) = 1− i~

∫ t

t0

dt′ VI(t′) UI(t

′, t0), (5.2.3)

the 1 just reflects, as usual, the initial condition (5.2.2). Via the standard proced-ure of substituting the entire right-hand side (evaluated at t= t′) into the integrand,we obtain

UI(t, t0) = 1− i~

∫ t

t0

dt′ VI(t′) +

(i~

)2∫ t

t0

dt′ VI(t′)

∫ t′

t0

dt′′ VI(t′′) UI(t

′′, t0). (5.2.4)

And iterating the process, we obtain the (formal) infinite power-series (in VI)solution:

UI(t, t0) = 1+

∞∑

n=1

(i~

)n∫ t

t0

dt1

∫ t1

t0

dt2 · · ·∫ tn−1

t0

dtn VI(t1)VI(t2) · · ·VI(tn). (5.2.5)

This is the Dyson series, derived now in the interaction picture. As before, wemay express the Dyson series in a more symmetric manner:

UI(t, t0) = 1+

∞∑

n=1

(1n!

)(i~

)n∫ t

t0

dt1

∫ t

t0

dt2 · · ·∫ t

t0

dtn T(VI(t1) VI(t2) · · ·VI(tn)

).

(5.2.6)where, recall, T is the time-ordering operator defined in Eq. (2.1.26). Recall toothat we may make no statement a priori as to the convergence properties of suchan expansion. In principle at any rate, the terms of the series can be evaluatedorder-by-order to any desired level. Using the more compact notation alreadyintroduced, we may also formally write

UI(t, t0) = T

exp

[− i

~

∫ t

t0

dt′ VI(t′)

]. (5.2.7)

5.2.2 Transition amplitudes

Given UI(t,t0), formally we know the evolution and can therefore calculate thefuture state |α,t0;t〉 corresponding to any initial state |α,t0〉. Of course, we donot intend to do this for each and every state, but only for the basis of energyeigenstates since these are sufficient. We thus consider

|m;t〉I = UI(t)|m〉 =∑

n

|n〉〈n|UI(t)|m〉, (5.2.8)

where, for simplicity, we have set t0=0 and suppressed it.


The object 〈n|UI(t)|m〉 appearing on the right-hand side above, known as atransition amplitude, expresses the probability of passing from the state |m〉 to thestate |n〉 under the influence of the perturbation VI(t). Note, however, that it differsslightly from the transition amplitude defined earlier in the Schrödinger picture.To see this, let us compare the evolution operator in the two pictures. From thedefinition of the interaction-picture states, Eq. (2.2.22a), we have (reinstating t0temporarily for clarity)

|α,t0;t〉I = ei~H0t |α,t0;t〉S = e

i~H0t US(t, t0)|α,t0;t0〉S

= ei~H0tUS(t, t0) e

− i~H0t0 |α,t0;t0〉I. (5.2.9)

Since this holds for any state |α〉, we finally obtain (as expected)

UI(t, t0) = ei~H0tUS(t, t0) e

− i~H0t0 . (5.2.10)

In the interaction picture the transition amplitude 〈n|UI(t)|m〉 is thus

〈n|UI(t,t0)|m〉 = 〈n|e i~H0tUS(t,t0)e

− i~H0t0 |m〉

= ei~(Ent−Emt0) 〈n|US(t,t0)|m〉. (5.2.11)

We see that the difference is actually only a phase and thus the correspondingtransition probabilities remain unaltered. Note that this statement only holds forenergy eigenstates: if we were to consider transitions between states other thanthese then there would be phase mismatching between the various components.Fortunately, in problems of physical interest both the initial and final states arenaturally energy eigenstates (these being the asymptotic states of the theory). Iffor some reason this is not the case, then we have simply to expand the states inquestion over a basis of energy eigenstates.

Let us now expand the final state over the basis of eigenstates at t= t0. Notefirst of all that, since t0 could be taken as zero, the associated phase in Eq. (5.2.11)is irrelevant. Indeed, it may be eliminated explicitly by a suitable choice of thephase associated with the Schrödinger state:

|m,t0;t0〉S ≡ e−i~Emt0 |m〉, (5.2.12a)

and thus|m,t0;t0〉I ≡ |m〉. (5.2.12b)

Note that we may not, however, simultaneously do the same for the final-statephase. Since this last then evolves at a later time t to

|m,t0;t〉I = UI(t, t0)|m〉, (5.2.13a)


which expands as=∑

n

cn(t)|n〉, (5.2.13b)

we then finally havecn(t) = 〈n|UI(t)|m〉. (5.2.14)

In other words, the transition amplitude is just the corresponding coefficient inthe basis expansion of the time-t state.

Given we know the perturbative expansion of UI(t), by inserting where necessarycomplete sets 1≡

∑k |k〉〈k|, we can immediately obtain perturbative expansions

for the coefficients cn(t):

c(0)n (t) = δnm, (5.2.15a)

c(1)n (t) = − i~

∫ t

0

dt′ 〈n|VI(t′)|m〉,

= − i~

∫ t

0

dt′ e iωnmt′Vnm(t

′), (5.2.15b)

c(2)n (t) =(

i~

)2∑

k

∫ t

0

dt′∫ t

′

0

dt′′ e iωnkt′Vnk(t

′) e iωkmt′′Vkm(t

′′), (5.2.15c)

c(3)n (t) = . . . , (5.2.15d)

where, as usual, we have defined ωnm≡ 1~(En−Em). Since the zeroth-order term

is non-vanishing only for n=m (i.e. no transition), for a non-trivial transition(n 6=m), we have that the probability is just

P (m→ n) =∣∣∣c(1)n (t) + c(2)n (t) + · · ·

∣∣∣2

(n 6=m). (5.2.16)

5.2.3 Constant perturbations

Let us now study some examples. The simplest is a constant perturbation, orrather one that is zero until it is switched on at some instant, t=0 say, after whichit remains constant:

V (t) =

0 for t < 0,

V (constant) for t > 0.(5.2.17)

For an initially pure state, |m〉 say, we thus have

c(0)n (t) = c(0)n (0) = δnm, (5.2.18a)


c(1)n (t) = − i~Vnm

∫ t

0

dt′ e iωnmt′,

=Vnm

En − Em

(1− e iωnmt

), (5.2.18b)

which, squaring, gives

∣∣∣c(1)n (t)∣∣∣2

=

∣∣∣∣Vnm

En − Em

∣∣∣∣2

2(1− cosωnmt

)

= 4

∣∣∣∣Vnm

En −Em

∣∣∣∣2

sin2

[(En − Em)t

2~

]. (5.2.18c)

This is an oscillating (sin2) function of En with a large peak at En=Em and arapidly falling envelope due to the (En−Em)

2 denominator (see Fig. 5.1). In other

0

0.2

0.4

0.6

0.8

1

1.2

-20 -15 -10 -5 0 5 10 15 20

ω= En−Em

~

4

ω2 sin

2(ωt2

)

Figure 5.1: The transition coefficient (5.2.18c) for a constant perturbation as a functionof the final-state energy for t fixed (here t=1 in units of 1/ω); the outer envelope (dottedline) is just 4/ω2.

words, the transition probability is non-negligible only for states sufficiently closein energy to the initial state:

|En − Em| <∼ ~/t. (5.2.19)


This is more suggestive if we call the time interval for which the perturbation isactive ∆t, then we find that transitions are likely when the variation in energy ∆Eis such that

∆E∆t <∼ ~. (5.2.20)

This is just another face of the uncertainty principle: the time for which theperturbation has been active also determines the time available for “measuring” theenergy of the states. In other words, a temporary violation of energy conservationis “tolerable” within the limits of the uncertainty principle.

Now, there is immediately a very surprising result at very short times for thosestates that are very close in energy to the original state (i.e. ωnmt≪1). The widthof the major peak is inversely proportional to t but the height is proportional tot2 (this can be seen by taking the limit En→Em). In this case we have

∣∣∣c(1)n (t)∣∣∣2

≃∣∣Vnm

∣∣2 t2/~2. (5.2.21)

We thus see that in the early stages of a scattering (or decay) process the transitionprobability is not linear in t, but quadratic. Such an effect has no counterpart inclassical physics, where the probability per unit time for a transition is alwaysconstant and therefore the probability grows linearly with time. This leads to theso-called quantum Zeno effect: i.e. if observed continuously, an unstable system willnever decay (see, e.g., Misra et al., 1977). With modern laser-based techniques,where femtosecond timing is now possible, such effects have been studied andobserved experimentally (see, e.g., Itano et al., 1990).

Note that this is not, however, in contradiction with classical physics at longtime scales (where the Ehrenfest theorem must take over). In real physical situ-ations, for large times, the peak becomes a Dirac δ-function and one has to sumover similar configurations with energies close to the initial value. Since the widthof the peak is proportional to 1/t the area under the curve (which correspondsthen to the total transition probability) is proportional to t, as expected and aswe shall now see. Hence, moreover, the long-term transition rate, which is theinteresting physical quantity, is precisely constant.

5.2.4 Fermi’s golden rule

Indeed, we can now immediately derive Fermi’s golden rule for the transition rate.As stated above, we are interested in the total rate to all possible final states andmust therefore sum over transition probabilities to states with energies close tothe initial Em:

P (m→ all allowed) ≃∑

n,En∼Em

∣∣∣c(1)n (t)∣∣∣2

. (5.2.22)


There may, of course, be (infinitely) many such states or a continuum and it istherefore normal to define a density of states ρ(E), such that ρ(E)dE is just thenumber of states in the energy interval (E,E+dE). We then have

∑

n,En∼Em

∣∣∣c(1)n (t)∣∣∣2

→∫

dEn ρ(En)∣∣∣c(1)n (t)

∣∣∣2

= 4

∫dEn ρ(En)

∣∣∣∣Vnm

En −Em

∣∣∣∣2

sin2

[(En − Em)t

2~

]. (5.2.23)

For t large (with respect to ~/∆E), we may exploit one of the many standardrepresentations of the Dirac δ-function:

limα→∞

sin2 αξ

αξ2= πδ(ξ). (5.2.24)

Thus,

limt→∞

1

|En − Em|2sin2

[(En − Em)t

2~

]=

πt

2~δ(En −Em), (5.2.25)

from which we obtain

limt→∞

∑

n,En∼Em

∣∣∣c(1)n (t)∣∣∣2

=2π

~|Vnm|2 ρ(En) t

∣∣∣En∼Em

, (5.2.26)

where, as promised, the probability is indeed now proportional to t. Defining thetransition rate wm→n as the time derivative of the left-hand side and using nto denote the set of allowed final states, we finally derive Fermi’s golden rule:

wm→n =2π

~|Vnm|2 ρ(En)

∣∣∣En=Em

, (5.2.27)

which is constant, independent of t. The reader should remember that this for-mula was derived from the first-order perturbation theory term and that we haveneglected all higher orders. In the case of continuous eigenvalues this is known asthe Born approximation.

The procedure outlined above may now be repeated for the higher-order terms.Inclusion of the second-order perturbation leads to

wm→n =2π

~

∣∣∣∣∣Vnm +∑

k

VnkVkmEm −Ek

∣∣∣∣∣

2

ρ(En)∣∣∣En=Em

. (5.2.28)


In the language of Feynman, this expression may be interpreted as a sum overall possible paths (up to second order) for passing from the initial to final states.At leading order the passage is direct, via the action of the perturbing potentialV . At second order it becomes a two-stage process: the system first “jumps” tothe (classically energetically forbidden and therefore denoted virtual) intermediatestate |k〉 and then from there to the allowed final states. Such a virtual transitionis suppressed in proportion to the inverse of the energy discrepancy or so-calledenergy denominator (Em−Ek) (in other words, in proportion to the time that oc-cupation of such a virtual state is permitted by the uncertainty principle), which isjust the remnant of the propagator. A problem obviously arises if such intermedi-ate states have the same energy as the initial and final states since the denominatoris then infinite—we shall treat this problem in detail later.

Exercise 5.2.1. Derive explicitly the second-order expression Eq. (5.2.28) forFermi’s golden rule.

5.2.5 Harmonic perturbations

A useful case to consider is that of a potential varying sinusoidally with time:

V (t) = V0 eiω t+V †

0 e− iω t, (5.2.29)

where V0 is constant in time (but may depend on position, momenta, spin etc.).This is the same form already adopted in the two-level model.

We shall again start from a situation in which just one level (energy eigenstate)is populated and then abruptly switch on the perturbation at time t=0. FromEq. (5.2.15b), we thus have

c(1)n (t) = − i~

∫ t

0

dt′ e iωnmt′ (Vnm e iω t

′+V †

nm e− iω t′)

=1

~

[1− e i (ωnm+ω)t

ωnm + ωVnm +

1− e i (ωnm−ω)t

ωnm − ωV †nm

], (5.2.30)

where Vnm≡〈n|V0|m〉 and V †nm≡〈n|V †

0 |m〉. The only essential difference with re-spect to a constant perturbation is the substitution ωnm→ωnm±ω. In the limitt→∞, we see that the transition has non-zero probability only for those finalstates with energies En≃Em±~ω. In either case (±) only one of the two po-tential terms is relevant: for En≃Em+~ω the system absorbs energy from theperturbing potential and for En≃Em−~ω releases energy to it. In the first casethis phenomenon is simply termed absorption, but in the second stimulated emis-sion, for obvious reasons.


From Fermi’s golden rule, Eq. (5.2.27), for the transition rate we then obtain

wm→n =2π

~

∣∣∣V (†)nm

∣∣∣2

ρ(En)∣∣∣En=Em±~ω

, (5.2.31)

where the upper (lower) sign refers to the case Vnm (V †nm). Now, from the definition,

it is easy to see that∣∣V †

nm

∣∣2=∣∣Vnm

∣∣2. This equivalence between the forward andbackward processes is known as detailed balance, which is quite a general principle.It essentially reflects time-reversal invariance and implies that the forward andbackward rates are relatively proportional only to the respective densities of finalstates.

5.2.6 Interaction of EM radiation and matter

Absorption and stimulated emission

The most obvious application of an harmonic perturbation is the interaction betweena classical electromagnetic radiation field and (charged) matter. Let us first stressthat we shall not deal with a quantised radiation field as such; this would takeus into the realm of second quantisation or quantum field theory. The radiationconsidered will thus be a real electromagnetic field. We shall nevertheless be lednaturally to conjecture elementary field quanta (photons) carrying fixed amountsof energy and that interactions are described as collisions between single objects,with discrete energy exchanges (determined by the frequencies).

Leaving aside a term in |eA/c|2 (which is non-negligible only for exceedinglyintense fields), the Hamiltonian determining the motion of an electron in the pres-ence of a generic electromagnetic field is

H =p2

2me

+ eV (x)− e

mecA·p, (5.2.32)

where we have made the usual choice of the radiation gauge, ∇·A=0. To bespecific, we shall take the radiation to be in the form of a plane wave directedalong some vector q with (linear) polarisation ξ, such that q·ξ=0 and ξ

2=1; atypical choice might be q=(0,0,q) and ξ=(1,0,0). The potential may thus bewritten in the following form:

A(x, t) = 2A0ξ cos(q·x− ωt)

= A0ξ[e− i (ωt−q·x)+e+i(ωt−q·x)

], (5.2.33)

where |q|≡ω/c is just the wave-number of the incident radiation. This leads to


an interaction potential of the form

− e

mecA·p = −eA0

mecξ·p

[e− i (ωt−q·x) +e+i(ωt−q·x)

]. (5.2.34)

Comparing this to (5.2.29), we see that the first term in square brackets will beresponsible for absorption and the second, stimulated emission.

DefiningV (+)nm ≡ V †

nm and V (−)nm ≡ Vnm, (5.2.35)

absorption (+) and emission (−) rates are governed then by the interaction terms

V (±)nm = −eA0

mec〈n|e± iq·x

ξ·p|m〉 (5.2.36)

and the corresponding transition rate, to a specific state |n〉, will be

w(±)m→n =

2π

~

e2|A0|2

m2ec

2

∣∣〈n|e± iq·xξ·p|m〉

∣∣2 δ(En − Em ∓ ~ω). (5.2.37)

The δ-function density of states would normally indicate sharp spectral lines,corresponding to En=Em±~ω. There are, however, many effects that may giverise to broadening. As we shall see shortly (in Sec. 5.2.7), a finite lifetime in-troduces a non-zero width to the state (via the uncertainty principle); moreover,recoil effects in the collision, though usually small, will also contribute to changingthe final energy from its nominal value. One should also be aware that the struckelectron will typically be bound in an atom and may thus either end up in anotherbound state (discrete spectrum) or be ejected from the atom into the continuumof free states. Finally, the incident radiation itself will never be perfectly mono-chromatic and thus, in all practical situations, the δ-function will be replaced bya broader function.

We may now calculate the elementary cross-section for the absorption process:

σabs ≡ w(+)m→n

Φrad

, (5.2.38)

where Φrad stands for the incident radiation flux in terms of quanta (photons) perunit time per unit area. Since each photon carries a fixed energy ~ω, the cross-section may equally be defined as the energy absorbed per unit time divided bythe incident energy flux. From classical electromagnetic theory, the energy flux is

cρE =1

2

(E2

max

8π2 +B2

max

8π2

)=

ω2|A0|22πc

. (5.2.39)


We thus have (~ω is the energy absorbed per collision)

σabs =~ω (2π/~) (e2|A0|2/m2

ec2)∣∣〈n|e iq·xξ·p|m〉

∣∣2 δ(En −Em − ~ω)

ω2|A0|2/(2πc)

=4π2

~α

m2eω

∣∣〈n|e iq·xξ·p|m〉∣∣2 δ(En − Em − ~ω), (5.2.40)

where we have used the fine-structure constant α≡ e2/π~c)≃1/137.

The dipole approximation

In general, it is not possible to evaluate the matrix element in Eq. (5.2.40) exactly.We may, however, expand the exponential in powers of q·x:

e iq·x = 1 + iq·x+ 12!( iq·x)2 + · · · . (5.2.41)

Recalling that qc=ω and that the frequency of the radiation should be equivalentto the energy spacing between typical atomic electron levels ∆E whereas 〈x〉should be of the order of the atomic radius a, we see that the expansion parameter,namely q·x∼∆Ea/~c, is small. To a first approximation the exponential maytherefore normally replaced by unity.

Exercise 5.2.2. Show that, for a typical atomic transition, the wave-vector in-volved is, indeed, such that q·x≪1.

Applying this approximation to Eq. (5.2.40), we see that the matrix elementto be evaluated simplifies considerably:

〈n|e iq·xξ·p|m〉 → ξ·〈n|p|m〉. (5.2.42)

which can be conveniently rewritten with the aid of the commutation relation

[x, H0] =i~p

me

. (5.2.43)

We thus have

〈n|p|m〉 =me

i~〈n|[x,H0]|m〉

= imeωnm〈n|x|m〉. (5.2.44)

The final expression for the transition amplitude will thus be proportional toe〈n|x|m〉, which is just the so-called transition dipole moment dnm and hencethe expression dipole approximation.


This simple form immediately leads to an important set of selection rules.Recalling that x is a rank-1 tensorial operator, V (1)

±,0, we have that |jf−ji|=0,1and |mf−mi|=0,1. If, moreover, we take the radiation as propagating along thez-axis with polarisation vector ξ along (say) the x-axis, then, using the definitions(3.4.17), we see that V (1)

0 cannot contribute and thus only |mf−mi|=1 is allowed(with therefore no transition 0→0). Choosing ξ along the y-axis leads to preciselythe same rule, while along the z-axis (with the radiation thus propagating in thex−y plane) we only have V (1)

0 and therefore mf =mi.It is important to realise that such selection rules correspond to symmetries

of the system and thus to conservation laws. They essentially reflect the conser-vation of angular momentum and parity. Note also that one is thus naturally ledto consider the photon as a spin-1 particle although this is not in any way anexplicit ingredient of the calculation performed. It is, rather, merely implicit inthe adoption of a vector description of the gauge potential and the quantisationof the energy exchanged.

The absorption cross-section in the dipole approximation is now simply

σabs = 4π2αωnm

∣∣〈n|x|m〉∣∣2 δ(ω − ωnm). (5.2.45)

Let us assume that the initial electron is in the ground state, so that ωnm>0, thenintegration over ω gives

∫dω σabs(ω) = 4π2α

∑

n

ωnm

∣∣〈n|x|m〉∣∣2 , (5.2.46)

where the sum runs over all energetically allowed final states |n〉. In atomic physicsit is usual to define the oscillator strength:

fnm ≡ 2meωnm

~

∣∣〈n|x|m〉∣∣2 . (5.2.47)

Consider now explicitly the sum over the oscillator strengths in the cross-sectionformula:

∑

n

fnm =2me

~

∑

n

ωnm

∣∣〈n|x|m〉∣∣2

=2me

~

∑

n

ωnm〈m|x|n〉〈n|x|m〉

= − i

~

∑

n

(〈m|x|n〉〈n|px|m〉 − 〈m|px|n〉〈n|x|m〉

)


= − i

~〈m|[x,px]|m〉

= − i

~〈m| i~|m〉 = 1. (5.2.48)

This result is known as the Thomas–Reiche–Kuhn sum rule. Finally, for the in-tegrated cross-section we obtain

∫dω σabs(ω) =

4π2α~

2me

=2π2e2

mec, (5.2.49)

which is just the result obtained in classical electrodynamics. Indeed, note that~ is no longer present. We may thus consider an atom as acting effectively as aclassical antenna.

Multipole transitions

In some cases the dipole approximation is not sufficient. There are various reasonsfor which this may be so: the size of the object may be comparable to the radi-ation wavelength (q·x∼1), the selection rules previously derived may prohibit therelevant (leading-order) transition, or the dipole term may simply be absent (thisis the case, e.g. of interactions solely with a nucleus, where only positive chargesare present).

Let us examine then the next term in the expansion, iq·x, which depends onthe following matrix element

M (1)mn ≡ 〈n| iq·xξ·p|m〉. (5.2.50)

For definiteness, let us take, as usual, the radiation as propagating along the z-axiswith polarisation vector ξ along the x-axis. The matrix element is then

M (1)mn ≡ iq〈n|zpx|m〉. (5.2.51)

Now, zpx can be rewritten as follows:

2zpx = (zpx − xpz) + (zpx + xpz)

= Ly − i~me

(z[x,H0] + [z,H0]x

)

= Ly − i~me[zx,H0]. (5.2.52)

Comparison of this and the other directional configurations with the original ex-


pression leads to the following general form:

M (1)mn = 1

2

i (q∧ξ)·〈n|L|m〉 −meωnm qi ξj〈n|xixj |m〉

. (5.2.53)

The first term is proportional to the transition magnetic (dipole) moment 〈n|L|m〉while the second is proportional to the transition electric quadrupole moment〈n|3xixj−r2δij|m〉 (note that the trace term δij gives zero since q·ξ).Exercise 5.2.3. By considering their tensorial nature, determine the selectionrules for the magnetic dipole and electric quadrupole transitions just derived.

It is interesting to examine the relative strength of the different multipoletransitions. From the matrix elements just derived, we immediately see that

TE2 ∝ eme q c a, (5.2.54a)

TE4 ∝ eme q2 c a2, (5.2.54b)

TM2 ∝ e q ~, (5.2.54c)

where a is just the atomic radius (i.e. a few Å). Taking ratios, we find

TE4

TE2

≃ qa ≃ a

λand

TM2

TE2

≃ ~

amec≃ λCompton

a, (5.2.55)

where λCompton ≡h/mec=386 fm is just the Compton wavelength of the electron.(Recall that the Compton wavelength is the natural unit and scale describing thewavelength shift that occurs in Compton scattering.)

Exercise 5.2.4. Verify explicitly the ratios given in Eq. (5.2.55) and discuss theirsignificance in real physical situations.

5.2.7 Energy shift and decay width

Thus far, we have only dealt with transitions from a state |m〉 to |n〉 with n 6=mexplicitly. We shall now examine the temporal evolution of the coefficient cm(t), i.e.for n=m, or the population of the initial state. Moreover, in the cases consideredwe specifically avoided the problem of intermediate states with the same energyEm. In order to deal with this possibility, we shall now introduce the idea of aslowly varying (or adiabatic) perturbation:

V (t) = eεt V0, (5.2.56)

where ε is chosen to be small and positive while V0 is constant. We shall find thatthe presence of a non-zero ε regulates all possible divergences and permits the


calculations to be performed unambiguously. Finally, we shall find an expressionfor, say, the rate that in the limit ε→0 does not depend on ε (to be expected sinceit should not depend in any essentially way on t).

In the remote past (t→−∞) the perturbation is not yet active and it may thusbe assumed that we start with some well-defined (asymptotic) state |m〉. Let usfirst verify the previous results for transitions to |n〉, with n 6=m, using the newadiabatic method:

c(0)n (t) = 0, (5.2.57a)

c(1)n (t) = − i

~Vnm

∫ t

−∞dt′ e iωnmt

′eεt

′

= − i

~Vnm

e iωnmt+εt

iωnm + ε, (5.2.57b)

where, as usual, we have adopted the natural notation Vnm≡〈n|V0|m〉. Taking themodulus squared and a derivative with respect to time, we find that the transitionrate is given by

d

dt

∣∣∣c(1)n (t)∣∣∣2

≃∣∣Vnm

∣∣2

~2

[2ε e2εt

ω2nm + ε2

]. (5.2.58)

Notice that, for ε 6=0, this equation is well defined even for ωnm=0. We maynow take the limit ε→0 by exploiting a further standard definition of the Diracδ-function:

limε→0

ε

x2 + ε2= π δ(x). (5.2.59)

We thus recover the previous expression for Fermi’s golden rule (5.2.27).Let us now turn to our goal in this section: namely, the temporal evolution of

the coefficients for n=m:

c(0)m (t) = 1, (5.2.60a)

c(1)m (t) =(

i~

)Vmm

∫ t

−∞dt′ eεt

′=(

i~

)Vmm

eεt

ε, (5.2.60b)

c(2)m (t) =(

i~

)2∑

k

|Vkm|2∫ t

−∞dt′∫ t

′

−∞dt′′ e iωmkt

′+εt

′e iωkmt

′′+εt

′′

=(

i~

)2∑

k

|Vkm|2∫ t

−∞dt′ e iωmkt

′+εt

′ 1

iωkm + εe iωkmt

′+εt

′

=(

i~

)2∑

k

|Vkm|21

iωkm + ε

∫ t

−∞dt′ e2εt

′


=(

i~

)2∑

k

|Vkm|21

iωkm + ε

e2εt

2ε

=(

i~

)2 |Vmm|2e2εt

2ε2+(

i~

)2∑

k 6=m

|Vkm|2e2εt

2ε( iωkm + ε). (5.2.60c)

Up to and including now second order in the perturbation, we then have

cm(t) ≃ 1 +(

i~

)Vmm

eεt

ε+(

i~

)2 |Vmm|2e2εt

2ε2

+(

i~

)2∑

k 6=m

|Vkm|2e2εt

2ε( iωkm + ε). (5.2.61)

As always, we are interested in the transition rate (here the loss rate); we thusnow take the time derivative. At this point we may set ε=0 in the exponents(though not yet in the denominators) of the right-hand side:

cm(t) ≃(

i~

)Vmm +

(i~

)2 |Vmm|21

ε+(

i~

)2∑

k 6=m

|Vkm|21

iωkm + ε. (5.2.62)

We now see that the case of degeneracy can be dealt with in a straightforwardmanner by following the standard procedure of diagonalising Vkm in the degeneratesubspace to eliminate the singular terms. Here, for simplicity, we shall assume non-degeneracy (i.e. for k 6=m, Ek 6=Em).

Now, since we are neglecting all third- and higher-order terms, we may rewriteEq. (5.2.62) as

cm(t) ≃(

i~

)Vmm +

(i~

)∑

k 6=m

|Vkm|2iωkm + ε

cm(t), (5.2.63)

by noting that on the right-hand side here it is sufficient to write cm(t) to firstorder only (since the multiplying expression is already first order in V ). In otherwords,

cm(t) ∼ 1 +(

i~

)1εVmm +O(V 2). (5.2.64)

We thus see that (to this order in perturbation theory) we have a simple first-orderdifferential equation for cm(t), with a constant coefficient. To leading order in Vthe coefficient is imaginary and so, taking into account the boundary conditioncm(0)=1, we shall write

cm(t) = e−i~∆mt, (5.2.65)


where the leading-order part of ∆m will be real although the higher-order contri-butions may be complex (we shall see that, indeed, they are). We may re-expressthe above differential equation as

cm(t) =(

i~

) [∆(1)

m +∆(2)m + · · ·

]cm(t). (5.2.66)

The expansion of ∆m is understood, as always, to be in powers of V .The first-order term is then simply

∆(1)m = Vmm, (5.2.67)

which, not surprisingly, is just as obtained in time-independent perturbation the-ory. Let us now examine the second-order term:

∆(2)m = − i

~

∑

k 6=m

|Vkm|2iωkm + ε

= −1

~

∑

k 6=m

|Vkm|2ωkm − iε

. (5.2.68)

Yet another useful relation defining the Dirac δ-function is

limε→0

1

x± iε= Pr

1

x∓ iπδ(x), (5.2.69)

where Pr indicates the principal value. This identity then provides the real andimaginary parts of ∆(2)

m :

Re(∆(2)

m

)= −Pr

∑

k 6=m

|Vkm|2Ek − Em

, (5.2.70a)

Im(∆(2)

m

)= −π

∑

k 6=m

|Vkm|2 δ(Ek − Em). (5.2.70b)

The first of the two expressions above is just the second-order contributionin perturbation-theory to the shift in the energy eigenvalue for the original statewhile the second is none other than Fermi’s golden rule. Indeed, we may write

wm→n =2π

~

∑

k 6=m

|Vkm|2 δ(Ek − Em) = −2

~Im(∆(2)

m

), (5.2.71)

where wm→n indicates a sum over transitions from m to all allowed states n 6=m.This shift may be seen as the interaction-energy contribution due to the interme-

5.3. SCATTERING THEORY 131

diate states. If we now define

Γm ≡ −2 Im∆m, (5.2.72)

then, from the solution of the differential equation for cm(t), we have

∣∣cm(t)∣∣2 = e−

1~Γmt . (5.2.73)

And therefore the imaginary part of ∆m (up to the factor −2) just corresponds tothe decay rate or width (we shall explain this latter term shortly).

It is always important to verify the conservation of probability (to the relevantorder in perturbation theory): for t small we have

∣∣cm(t)∣∣2 +

∑

n 6=m

|cn(t)|2 ≃(1− 1

~Γmt

)+ wm→n t = 1, (5.2.74)

which is precisely the desired equality to second order in V .Now, why do we refer to the quantity Γ, the imaginary part of ∆m, as the

width? To understand this, let us examine the Fourier transform of cm(t):

cm(E) ≡∫

dt e−i~Et e−

i~[Em+Re∆m− i

2Γm]t

∝ 1

[E − (Em + Re∆m) +i2Γm]

. (5.2.75)

We thus find for the spectrum, the following distribution:

∣∣cm(E)∣∣2 ∝ 1

[E − (Em + Re∆m)]2 + 1

4Γ2m

. (5.2.76)

Clearly, Γm is precisely the full-width at half maximum (FWHM) of the spectrum,which now has a peak centred on the shifted energy (Em+Re∆m). Furthermore,the decay law

∣∣cm(t)∣∣∝ e−Γmt implies a mean lifetime for the state of τm=1/Γm.

Note that this is perfectly in accordance with (and indeed simply reflects) theuncertainty principle: ∆t∆E≥~. We shall examine the above form for

∣∣cm(E)∣∣2

in more detail later within the framework of the Breit–Wigner (BW) formalism.

5.3 Scattering theory

We now turn to the problem of scattering theory in non-relativistic quantum mech-anics. The following discussion is of central importance to many aspects of physics:almost every experimental investigation involves scattering at some level. Even the


very process by which these pages are being read involves light scattering on theretina and interpretation of the resulting information.

5.3.1 The Lippmann–Schwinger equation

It will be easiest to begin the examination of scattering via time-independentpotentials (the most useful and, indeed, common case). We shall thus take aperturbative approach based on a Hamiltonian typically of the form

H = H0 + V, (5.3.1)

where now, for definiteness, the unperturbed Hamiltonian H0 is just the kineticenergy of a particle, whose scattering off a constant potential V we wish to describe.The form of H0 will thus be taken simply as the usual kinetic term:

H0 =p2

2m, (5.3.2)

where p is the particle momentum and m its mass. The free-particle momentumeigenstates will be represented as |p〉. The presence of the perturbing potentialV will alter these states. As a first approach we shall just be interested in elasticscattering, in which the energy of the state is assumed to remain unaltered. Thus,if some generic unperturbed eigenstate |φ〉 is such that

H0|φ〉 = E|φ〉, (5.3.3)

then the corresponding perturbed solution |ψ〉 will be of the form

(H0 + V )|ψ〉 = E|ψ〉. (5.3.4)

In reality, in assuming some sort of simple correspondence between unperturbedand perturbed states, we are implicitly dealing with small perturbations. By smallwe mean something like 〈V 〉≪〈H0〉 for all states of interest. In particular, weshould impose the requirement that as V →0, ψ→φ in a smooth way. Note thatthe same energy E appears in both equations owing to the implicit assumptionthat V →0 asymptotically (i.e. at large distances and/or times). This is necessaryin order to invoke asymptotically free (well-defined) states.


Formally, we may then write the solution as∗

|ψ〉 = |φ〉+ 1

(E −H0)V |ψ〉. (5.3.5)

That this is, indeed, a solution can be seen by simply applying the operator E−H0

to both sides of the equation. Note that it also has the desired behaviour for V →0.Unfortunately, we have no means of dealing with the singular operator (E−H0)

−1;the standard techniques used in degenerate perturbation theory are of no use forthe case of a continuous spectrum. The trick here, already implicitly suggested bythe adiabatic treatment given earlier, is to shift the energy a little in the complexplane, thus

|ψ(±)〉 = |φ〉+ 1

E −H0 ± iεV |ψ(±)〉. (5.3.6)

This is known as the Lippmann–Schwinger equation (Lippmann and Schwinger,1950). We shall examine the relevance of the sign of the iε term shortly.

The above equation, as presented in Dirac notation, makes no reference to anyparticular representation but, in order to proceed, we must now choose and it isconvenient to move first into coordinate space. We thus project over the basis ofposition eigenstates, using 〈x|:

〈x|ψ(±)〉 = 〈x|φ〉+∫

d3x′⟨x

∣∣∣∣1

E −H0 ± iε

∣∣∣∣x′⟩〈x′|V |ψ(±)〉. (5.3.7)

The form then is that of an integral equation. For the case in which we are dealingwith (unperturbed) plane-wave states, we have

〈x|φ〉 =e

i~p·x

(2π~)3/2. (5.3.8)

In the momentum-space representation we would, for example, obtain instead

〈p|ψ(±)〉 = 〈p|φ〉+ 1

E − (p2/2m)± iε〈p|V |ψ(±)〉. (5.3.9)

We shall use this form later on.Returning to the coordinate-space representation, we must evaluate the kernel

∗ To appreciate that this is a sensible approach, it may help to consider an iterative series solutionto such an (implicitly) integral equation along the lines of the Dyson-series approach.


of the integral equation,

G±(x,x′) ≡ ~

2

2m

⟨x

∣∣∣∣1

E −H0 ± iε

∣∣∣∣x′⟩. (5.3.10a)

As we shall now see, this is just the well-known Green function (or propagator)for the problem of a free-particle wave equation, which is given by

G±(x,x′) = − 1

4π

e± ik|x−x′|

|x− x′| , (5.3.10b)

where k is just the wave-number corresponding to the energy eigenvalue E:

E =~2k2

2m. (5.3.11)

This is just a spherical wave emanating from or converging onto the point x′.

To evaluate the kernel we proceed as follows:

~2

2m

⟨x

∣∣∣∣1

E −H0 ± iε

∣∣∣∣x′⟩

=~2

2m

∫d3p′∫

d3p′′ 〈x|p′〉

⟨p′∣∣∣∣

1

E − (p′′2/2m)± iε

∣∣∣∣p′′⟩〈p′′|x′〉. (5.3.12)

The central “bra-ket” can now be evaluated directly:

⟨p′∣∣∣∣

1

E − (p′′2/2m)± iε

∣∣∣∣p′′⟩

=δ(3)(p′ − p

′′)

E − (p′2/2m)± iε. (5.3.13)

Using plane-wave representations for the other factors,

〈x|p′〉 =e

i~p′·x

(2π~)3/2

and 〈p′′|x′〉 =e−

i~p′′·x′

(2π~)3/2, (5.3.14)

we thus obtain

~2

2m

⟨x

∣∣∣∣1

E −H0 ± iε

∣∣∣∣x′⟩

=~2

2m

∫d3p′

(2π~)3e

i~p′·(x−x

′)

[E − (p′2/2m)± iε]. (5.3.15a)

It is now convenient to transform momentum variables to wave-vectors k′ :=p

′/~


and rewrite this as

=1

(2π)3

∫ ∞

0

dk′ k′2∫ +1

−1

dcos θ

∫ 2π

0

dφe ik

′|x−x′| cos θ

k2 − k′2 ± iε, (5.3.15b)

where k′ = |k′| and we have implicitly rescaled ε by an irrelevant constant factor2m/~2. The angular integrals are trivial, leaving

=1

4π2

1

i |x− x′|

∫ ∞

0

dk′ k′e+ik

′|x−x′| − e− ik

′|x−x′|

k2 − k′2 ± iε. (5.3.15c)

Noting that the integrand is even in k′ and that the second term is equivalent tothe first with k′ →−k′. It can thus be rewritten, by extending the lower limit to−∞, as

=1

4π2

1

i |x− x′|

∫ ∞

−∞dk′ k′

e ik′|x−x

′|

k2 − k′2 ± iε. (5.3.15d)

The remaining integral may now be evaluated using Cauchy’s theorem: first,rewrite the denominator as

k2 − k′2 ± iε ≃ (k − k′ ± iε)(k + k′ ± iε), (5.3.16)

where again, on the right-hand side ε has been rescaled by an irrelevant factor 2k(which, by definition, is positive—the sign is important) and terms in ε2 have beenneglected. We thus see that the integrand has two poles in the complex k′ plane:one situated above and one below the real k′ axis (which is above and which belowdepends on the sign in front of the iε regulator). It is easy to see that closingthe contour in either half plane forces the solution k′=±k for the choice ± iε. Wethus finally obtain the kernel (5.3.10b).

Given the explicit expression for G±, the Lippmann–Schwinger equation in theform of (5.3.7) then becomes

〈x|ψ(±)〉 = 〈x|φ〉 − 2m

~2

∫d3x′ e± ik|x−x

′|

4π|x− x′| 〈x

′|V |ψ(±)〉. (5.3.17)

Examining the right-hand side, we see that the two terms have a straightforwardinterpretation: the first represents the unperturbed incident wave 〈x|φ〉 while theother evidently represents the wave scattered by the potential. We shall latershow explicitly that, assuming the potential to be of finite range (so that r′ iseffectively bounded), the scattered wave behaves as e± ikr/r for r large (r→∞),


as one might expect. Therefore, the positive and negative signs correspond tooutgoing and incoming spherical waves respectively. While it is clear that undernormal conditions we shall only be interested in outgoing solutions (one cannotgenerally arrange for an incoming spherical wave), the incoming solutions willactually play a role in the general formalism.

To proceed further, we need to make some assumptions as to the behaviourof V . The most useful (and physically realistic) is that it is local, i.e. that it is afunction of position only (e.g., it does not contain derivatives) and therefore

〈x′|V |x′′〉 = V (x′) δ(3)(x′ − x′′). (5.3.18)

Using this, we obtain

〈x′|V |ψ(±)〉 =

∫d3x′′ 〈x′|V |x′′〉〈x′′|ψ(±)〉

= V (x′) 〈x′|ψ(±)〉. (5.3.19)

The integral in (5.3.17) then simplifies to

〈x|ψ(±)〉 = 〈x|φ〉 − 2m

~2

∫d3x′ e± ik|x−x

′|

4π|x− x′| V (x

′) 〈x′|ψ(±)〉. (5.3.20)

We may now attempt to understand the physical meaning of the solution. Theposition x represents the point at which the particle is observed (or detected)while x

′ is the point at which it “interacts” with the scattering potential V (x′). Ifthis last is of finite range, then we may choose x sufficiently far removed from thescattering centre such that, for all relevant x

′,

|x| ≫ |x′| or r ≫ r′ (r ≡ |x|, r′ ≡ |x′|). (5.3.21)

In this case

|x− x′| =√r2 − 2rr′ cos θ + r′2 = r

[1− 2r′

rcos θ +

r′2

r2

]1/2

,

where θ is the angle between x and x′, and therefore,

≃ r − x·x′, (5.3.22)

where x≡x/r and we neglect terms that are O(r′2). We also need to define amomentum vector in the x direction, k′≡kx. This is a natural definition since


a plane wave observed at the point x must have propagated along the directiongiven by x; moreover, by conservation of energy, its wave-vector will have modulus|k|, as given. At large distances we may use the approximation

e± ik|x−x′| ≃ e± ikr e

∓ ik′·x′. (5.3.23)

To the same approximation, i.e. neglecting O(r′2) terms, we may also write

1

|x− x′| ≃ 1

r. (5.3.24)

A useful trick to eliminate the factor ~ in many of the expressions is to usestates defined as |k〉 in place of |p〉 since p≡~k. Then, with the natural choicefor the normalisation of |k〉 as

〈k′|k〉 = δ(3)(k′ − k), (5.3.25)

we have, for the incident wave-function,

〈x|k〉 =e ik·x

(2π)3/2. (5.3.26)

In this way we have the asymptotic expression

〈x|ψ(+)〉 r→∞−−−→ 〈x|k〉 − 2m

~2

e ikr

4πr

∫d3x′ e− ik

′·x′V (x′)〈x′|ψ(+)〉

=1

(2π)3/2

[e ik·x+

e ikr

rf(k′,k)

], (5.3.27)

where we now clearly see the incident (plane) and outgoing (spherical) waves,respectively the first and second term in brackets. The latter has amplitude

f(k′,k) = −2m

~2

(2π)3

4π

∫d3x′ e

− ik′·x′

(2π)3/2V (x′)〈x′|ψ(+)〉

= −2m

~2

(2π)3

4π〈k′|V |ψ(+)〉. (5.3.28)

Following the same procedure for 〈x|ψ(−)〉, we again obtain a plane wave propagat-ing along the direction of k together with a spherical wave now propagating inwards


towards the scattering centre:

〈x|ψ(−)〉 r→∞−−−→ 1

(2π)3/2

[e ik·x+

e− ikr

rf(−k

′,k)

], (5.3.29)

where the scattering amplitude f is

f(−k′,k) = −2m

~2

(2π)3

4π〈−k

′|V |ψ(−)〉. (5.3.30)

Clearly, such a solution would represent the physically improbable situation inwhich an inwardly propagating spherical wave is so prearranged as to recombineon the scattering centre and form an outgoing plane wave. As such, of course, itis essentially the time reversal of the previous case.

We now wish to calculate the cross-section. As usual, it is defined as the ratioof the scattered over incident fluxes. We should thus calculate the scattered andincident fluxes implicit in expressions (5.3.27 & 28). We then have

dσ

dΩdΩ =

scattered flux in dΩ

incident flux

=jscatt r

2dΩ

jinc

=∣∣f(k′,k)

∣∣2 dΩ. (5.3.31)

The differential cross-section is thus quite simply

dσ

dΩ=∣∣f(k′,k)

∣∣2 . (5.3.32)

It is important to remark at this point that a complete and rigorous treatmentof the scattering of physical particles should be formulated in terms of wave-packetsof finite spatial extension. It is, however, also important to appreciate the relevantscales: in this case the size of the scattering centre (in other words, the rangeof the scattering potential) determines the natural length scale. Providing thecorresponding wave-packet is much larger than this scale, then a naïve treatmentin terms of plane waves is sufficient.

5.3.2 The Born approximation

To evaluate the cross-section given in Eq. (5.3.32) or, equivalently, f(k′,k), we stillneed to know the perturbed solution |ψ(+)〉, since it appears on the right-hand sideof Eq. (5.3.27). We note, however, that it appears already multiplied by V . In the


general spirit of the Born approximation encountered earlier, we may thus simplyreplace |ψ(+)〉 on the right-hand side with |φ〉, since they only differ by terms oforder V . In other words (on the right-hand side only),

〈x′|ψ(+)〉 ≃ 〈x′|φ〉 =e ik·x

′

(2π)3/2. (5.3.33)

With this substitution, we have

f (1)(k′,k) = −2m

~2

(2π)3

4π〈k′|V |φ〉

= − 2m

4π~2

∫d3x′ e i (k−k

′)·x′

V (x′). (5.3.34)

This then is the first-order Born approximation to the scattering amplitude. Ex-amining the precise form, we see that it is simply proportional to the Fouriertransform of the scattering potential V (x) with respect to the momentum transferq≡k−k

′.In the case of a spherically symmetric potential, f (1)(k′,k) only depends on

|k−k′|. Now, since we are considering elastic scattering, |k′|= |k|=k, thus

q = |k − k′| =

√(k − k

′)2

=√

2k2 − 2k2 cos θ = k√4 sin2 θ

2= 2k sin θ

2, (5.3.35)

where θ is the angle between k and k′. The angular integration then gives

f (1)(θ) = −1

2

2m

~2

1

iq

∫ ∞

0

dr′ r′(e iqr

′− e− iqr

′ )V (r′)

= −2m

~2q

∫ ∞

0

dr′ r′ sin qr′ V (r′). (5.3.36)

A problem now reveals itself: if we were to consider a potential behaving like1/r, such as is generated by the Coulomb interaction, the integral would clearlybe ill defined. As we shall now see, however, the Coulomb potential is, in fact, theborderline case between short- and long-range interactions. In order to appreciatethis, let us consider the Yukawa potential:

VYukawa(r) ≡ V0 e−µr

r. (5.3.37)


The range of this interaction is finite and is given by R=1/µ, where µ is a constant(with dimensions of mass or energy in natural units, where ~=1= c). Note alsothat the Yukawa potential tends smoothly to the Coulomb potential in the limitµ→0. In this case we now have

f (1)(θ) = −2m

~2q

∫ ∞

0

dr′ r′ sin qr′V0 e

−µr′

r′= −2mV0

~2

1

q2 + µ2 . (5.3.38)

The cross-section may then be written as

dσ

dΩ≃(2mV0

~2

)21

(4k2 sin2 θ

2+ µ2

)2 . (5.3.39)

The above expression is well defined even in the limit µ→0. We thus obtain thecross-section for the Coulomb interaction as just

dσ

dΩ≃(2mQ1Q2

~2

)21

16k4 sin4 θ2

=

(Q1Q2

E

)21

16 sin4 θ2

, (5.3.40)

where Q1,2 are the charges of the incident and “struck” particles and where we havemade the identification E=p2/2m=~

2k2/2m as the kinetic energy of the incidentparticle. Note that, once again, ~ has disappeared from the final expression, whichmust therefore (and, indeed, does) correspond to the classical Rutherford cross-section.

Note then that the Coulomb potential has an effectively infinite range andis thus at the limit of applicability of this approach. To fully appreciate theimportance of this, one should work with wave-packets; it then becomes clear thatthe extent of the region in which a potential with poorer convergence propertiesacts is larger than the size of any wave-packet. A rigorous treatment using wave-packets may be found, e.g., in Caldirola et al. (1982).

Going back now to expression (5.3.36) for the Born approximation to the scat-tering amplitude in the case of a central potential, assuming the approximationto be valid, we can make several important general statements concerning thescattering amplitude and the cross-section:

1. The amplitude f(θ) is, in fact, only a function of q; it depends on the kin-etic energy E=~

2k2/2m (or momentum p=~k) and the scattering angle θthrough q=2k sin θ

2.

2. In the Born approximation f(θ) is always purely real.

3. The differential cross-section does not depend on the sign of V (r).


4. For k small (whence q is also small, so that λ∼1/k≫R, where R is therange of the potential), f(θ) becomes a simple volume integral of V and istherefore independent of k and θ.

5. For q large and therefore θ 6=0, the rapidly oscillating integrand leads to f(θ)small.

As always, it is important to appreciate the domain of validity of the approxim-ation made. Implicit in the Born approximation is that 〈x|ψ(+)〉 should be similarto 〈x|φ〉 even inside the region where the potential is active, i.e. for x≤R. Inother words, the distortion of the incoming wave in the interaction region must besmall.∗ Examining the still exact Eq. (5.3.20) for small x we see that fulfilment ofthis condition requires

∣∣∣∣∣2m

4π~2

∫d3x′ e

± ikr′

r′V (x′)

∣∣∣∣∣ ≪ 1. (5.3.41)

In the explicit case of the Yukawa potential for low energy (such that k≪µ)this becomes

2m|V0|~2µ2 ≪ 1. (5.3.42)

It turns out that this is numerically very close to the condition (for an attractiveYukawa potential) that there be no bound states:

2m|V0|~2µ2 < 2.7. (5.3.43)

In other words, providing that the potential is too weak to support bound states,the approximation should be reliable. This is not surprising: at low energies theincident particle would otherwise risk being trapped by the potential well (at leastfor an appreciable period of time). In the high-energy limit the condition fornegligible distortion becomes

2m|V0|~2kµ

ln

(k

µ

)≪ 1. (5.3.44)

Again, this is not surprising: as the incident energy increases, the perturbing effectof the potential will, in proportion, clearly become less and less.∗ Note that this is not the same as saying that the scattered particle is only slightly deviatedfrom its initial trajectory.


5.3.3 Higher-order Born approximations

One can, of course, systematically improve on the first-order Born approximationby calculating higher-order terms. It is then convenient to use a more compactnotation. Let us define what we shall call a transition operator T , such that

V |ψ(+)〉 = T |φ〉. (5.3.45)

Taking |φ〉 as a momentum eigenstate |k〉 in Eq. (5.3.28), we have

f(k′,k) = −2m

~2

(2π)3

4π〈k′|T |k〉. (5.3.46)

Therefore, f(k′,k) is completely determined by T .Now, acting on the Lippmann–Schwinger equation (5.3.6) from the left with

V , we may write

T |φ〉 = V |φ〉+ V1

E −H0 + iεT |φ〉. (5.3.47)

Since the equality holds for any state |φ〉, it must also hold in operator form:

T = V + V1

E −H0 + iεT . (5.3.48)

In the same manner as for the derivation of the Dyson series, we may write aformal iterative solution:

T = V + V1

E −H0 + iεV

+ V1

E −H0 + iεV

1

E −H0 + iεV + · · · . (5.3.49)

Expanding f(k′,k) in a similar power series in V ,

f(k′,k) =∞∑

n=1

f (n)(k′,k), (5.3.50)

we have

f (1)(k′,k) = −2m

~2

(2π)3

4π〈k′|V |k〉, (5.3.51a)

f (2)(k′,k) = −2m

~2

(2π)3

4π〈k′|V 1

E−H0+ iεV |k〉, (5.3.51b)


...

This may be expressed in terms of the Green function (or propagator) introducedearlier: for example,

f (2)(k′,k) = −2m

~2

(2π)3

4π

∫d3x′∫

d3x

× 〈k′|x′〉V (x′) 〈x′| 1

E−H0+ iε|x〉 V (x) 〈x|k〉

=1

4π

∫d3x′∫

d3x e− ik

′·x′V (x′)G+(x

′,x) V (x) e ik·x

(5.3.52)

Again, the interpretation follows that of Feynman: at n-th order the scatteringinto the final state proceeds via a series of n interactions with the potential at thepoints x, x′ etc. while the wave propagates between the interactions according tothe Green functions G+(x

′,x) etc.

5.3.4 The optical theorem

Let us now examine the form of the total scattering cross-section: using (5.3.32),for σtot we find

σtot ≡∫

dΩdσ

dΩ=

∫dΩ

∣∣f(k′,k)∣∣2 .

=

[2m

~2

(2π)3

4π

]2 ∫dΩ

∣∣∣〈k′|V |ψ(+)〉∣∣∣2

=

[2m

~2

(2π)3

4π

]2~2

km

∫d3k′ δ

(E − ~

2k′2

2m

)〈ψ(+)|V †|k′〉〈k′|V |ψ(+)〉,

(5.3.53a)

where we have used the fact that ~2k′2

2m= Ek

′ = E = Ek =~2k2

2mand therefore k′ = k.

This, may then be rewritten as

=

[2m

~2

(2π)3

4π

]2~2

km

∫d3k′ 〈ψ(+)|V †δ(E−H0)|k′〉〈k′|V |ψ(+)〉 (5.3.53b)


and the integral over the complete set of states may thus be removed to leave∫

d3k′ 〈ψ(+)|V †δ(E−H0)|k′〉〈k′|V |ψ(+)〉 = 〈ψ(+)|V †δ(E−H0)V |ψ(+)〉. (5.3.54)

Recall now one of the many identities involving the Dirac δ-function: namely,

1

E −H0 ± iε= Pr

1

E −H0

∓ iπδ(E −H0). (5.3.55)

The imaginary part of this equation is just

Im1

E −H0 ± iε= ∓ πδ(E −H0) (5.3.56)

and so we can rewrite the expression for the total cross-section thus

σtot =

[2m

~2

(2π)3

4π

]2~2

kmπ

⟨ψ(+)

∣∣∣∣V† Im

(1

E −H0 + iε

)V

∣∣∣∣ψ(+)

⟩.

Then, since both H0 and V are Hermitian, this is equivalent to

=

[2m

~2

(2π)3

4π

]2~2

kmπIm

⟨ψ(+)

∣∣∣∣V† 1

E −H0 + iεV

∣∣∣∣ψ(+)

⟩.

Exploiting the Lippmann–Schwinger equation (5.3.6), we then have

=

[2m

~2

(2π)3

4π

]2~2

kmπIm⟨

ψ(+)∣∣∣V †

∣∣∣ψ(+)⟩−⟨ψ(+)

∣∣∣V †∣∣∣φ⟩

.

Since again V is Hermitian, the first term (being real) vanishes and we obtain

=

[2m

~2

(2π)3

4π

]2~2

kmπIm⟨φ∣∣∣V∣∣∣ψ(+)

⟩,

or finally

σtot =4π

kIm f(k,k) ≡ 4π

kIm f(0). (5.3.57)

That is, up to a factor 4π/k, the total cross-section is just the imaginary part ofthe forward scattering amplitude (i.e. for k′=k or θ=0).


Now, there is an apparent contradiction: the left-hand side in Eq. (5.3.57) is, bydefinition, proportional to an amplitude squared while the right-hand side is linearin the amplitude. This may be reconciled by carefully considering the physicalorigin of a scattering cross-section. In Eq. (5.3.27) we see that the general form ofthe wave-function is a sum of two terms:

ψ(+)(x) ∝ e ik·x +e ikr

rf(k′,k). (5.3.58)

Now, the flux loss in the forward direction (which is clearly proportional to thetotal cross-section) must be due to the interference between these two terms (tosee this, consider the difference in flux along the z-axis between points before andafter the scattering centre) and is thus linear in f(k′,k). Indeed, if we performthe calculation in this way, it becomes clear that the theorem is very general andholds even in the presence of inelastic scattering (or absorption). As one mightimagine, the expression optical theorem is borrowed from classical optics, wherethe phenomenon is well known (a bright central spot that appears behind a blackdisc diffracting a light source of suitable wavelength).

Exercise 5.3.1. Derive the optical theorem by considering the total cross-sectionas just considered, i.e. as given by the flux lost from the beam.

5.3.5 The eikonal approximation

We come now to the eikonal∗ approximation, which has a wide range of validityand which thus finds applications in various areas of physics. The condition to beapplied is that the scattering potential be slowly varying, i.e. it does not changeappreciable within the space of one wavelength of the incident particle. In otherwords (but very roughly), we require E≫|V |. As we shall see, in such a case theclassical notion of a trajectory and the wave-function |ψ(+)〉 may be approximatedsemiclassically with

|ψ(+)〉 ∼ ei~S(x) . (5.3.59)

Indeed, let us first rewrite the wave-function in a somewhat suggestive form:

ψ(x, t) =√ρ(x, t) exp

[i~S(x, t)

]. (5.3.60)

∗ From the German Eikonal, from the Greek ε,ικων, meaning image.


The classical limit

Recall first the definition of the current,

j(x, t) :=~

mIm(ψ∗

∇ψ)=

ρ∇S

m. (5.3.61)

The last form highlights the importance of the phase S(x,t): the spatial variationof S determines the probability flux. In the case of a plane wave,

ψ(x, t) ∝ exp[i~(p·x−Et)

], (5.3.62)

as might be expected, we simply have

∇S = p or j(x, t) = ρv. (5.3.63)

We may now examine the classical limit of wave mechanics. Upon substitutingform (5.3.59) for the wave-function, the Schrödinger equation becomes

i~

[∂√ρ

∂t+

i

~

√ρ∂S

∂t

]

= − ~2

2m

[∇

2√ρ+ 2i

~(∇

√ρ)·(∇S)− 1

~2

√ρ (∇S)2 +

i

~

√ρ∇2S

]+√ρ V.

(5.3.64)

Consider the limit of ~ very small, so that, in particular, we may assume

~|∇2S| ≪ |(∇S)2| (5.3.65)

and similarly for other quantities. Keeping only the leading terms (i.e. those oforder ~

0) above, we then have

1

2m

[∇S(x, t)

]2+ V (x) +

∂S(x, t)

∂t= 0. (5.3.66)

This is none other than the (non-linear) Hamilton–Jacobi equation of classicalmechanics, where S(x,t) is Hamilton’s principal function. Recall, in classicalmechanics

∂S(x, t)

∂t= piqi −H = L. (5.3.67)

For a stationary state, whose time dependence lies only in the phase

ψ(t) ∝ e−i~Et, (5.3.68)


such a dependence could have been derived from Hamilton’s principal function fora classical system with constant Hamiltonian, for which S is separable:

S(x, t) = W (x)− Et, (5.3.69)

where W (x) is Hamilton’s characteristic function. Indeed, a surface of constantS moves in time just as a wave front. In Hamilton–Jacobi theory the classicalmomentum is

p = ∇S = ∇W. (5.3.70)

We may thus identify ∇S/m with the velocity of the particle and, following thewavefront, trace out its classical trajectory (or ray).

The semi-classical approximation (JWKB)

We now seek an approximate stationary solution to the Schrödinger equation. Forsimplicity, we shall consider the one-dimensional case. Consider the solution tothe corresponding classical Hamilton–Jacobi equation,

S(x, t) = W (x)−Et = ±∫ x

dx′√2m[E − V (x′)]−Et. (5.3.71)

A stationary state implies ∂ρ/∂t=0 everywhere. The continuity equation, whichtakes the form

∂ρ

∂t+

1

m

∂

∂x

(ρ∂S

∂x

)= 0, (5.3.72)

then leads to

ρ∂W

∂x= ±ρ

√2m[E − V (x)] = const. (5.3.73)

and thus √ρ =

const.

[E − V (x)]1/4

∝ 1√vcl

. (5.3.74)

Putting this all together for the wave-function, we finally obtain what is known asthe Jeffreys–Wentzel–Kramers–Brioullin (JWKB) approximation:

ψ(x, t) ≃ const.

[E − V (x)]1/4

exp

[±

i

~

∫ x

dx′√

2m[E − V (x′)]− i

~Et

]. (5.3.75)

We can now, a posteriori , be more precise as to what is meant by assuming ~


small. The actual approximation made (5.3.65) becomes

~

∣∣∣∣d2W

dx2

∣∣∣∣ ≪∣∣∣∣dW

dx

∣∣∣∣2

, (5.3.76)

which, rewritten in terms of the de Broglie wavelength λ of the particle, is equi-valent to

λ =~√

2m[E − V (x)]≪ 2[E − V (x)]

dV/dx. (5.3.77)

The wavelength of the particle should therefore always be small compared to thelength scale of the variations in V (x). Or, in other words, the potential should notvary appreciably over one wavelength. In particular then, we should expect theapproximation to be valid at high energies. Note that the approximate solution(5.3.75) is also valid for the case in which [E−V (x)] becomes negative: it is just theusual exponentially decaying (or evanescent) solution one obtains in the classicallyprohibited region.

Following the semi-classical approximation for ψ(+), as in form (5.3.60), theHamilton–Jacobi Eq. (5.3.66) for S becomes

1

2m

[∇S(x, t)

]2+ V (x) = −∂S(x, t)

∂t= E =

~2k2

2m, (5.3.78)

where, of course, k is the asymptotic wave-number (i.e. sufficiently far removedfrom the scattering centre so that V →0).

While it is not possible to find the general exact solution to this equation, ifwe assume that the classical (or most probable) path approximates a straight line(along the z-axis, say), then we may proceed further. Indeed, we can now formallyintegrate (5.3.78):

S =

∫ z

dz′√~2k2 − 2mV (r), (5.3.79)

where r2= z′2+b2 and b is the usual impact parameter (see Fig. 5.2). We mayrender the integral definite by imposing the limiting condition

SV→0−−−→ ~kz, (5.3.80)

leading to

S = ~kz +

∫ z

−∞dz′

[√~2k2 − 2mV (r)− ~k

]

≃ ~kz − m

~k

∫ z

−∞dz′ V

(√b2 + z′2

), (5.3.81)


z-axis

rb b

z=0

interaction region

Figure 5.2: Scattering in the eikonal approximation of a linear classical trajectory.

where we have expanded the integrand in powers of V/E (E=~2k2/2m) and kept

only the leading term.Using this solution for S, we can now write an approximation for ψ(+), the

so-called eikonal wave-function:

ψ(+) ≃ 1

(2π)3/2

exp

i

[kz − m

~2k

∫ z

−∞dz′ V

(√b2 + z′2

)]. (5.3.82)

While this does not have the correct asymptotic form to be the complete solution(incident plane wave and outgoing spherical waves), it may nevertheless be usedto calculate an approximation to the scattering amplitude f(k′,k):

feik(k′,k) = − 1

4π

2m

~2

∫d3x′ e− ik

′·x′V(√

b2 + z′2)e ik·x

′

× exp

[− im

~2k

∫ z′

−∞dz′′ V

(√b2 + z′′2

)]. (5.3.83)

Note that this expression differs from the first-order Born approximation owing tothe presence of the final exponential factor—a phase.

To perform the integral over x′, it is convenient to transform to cylindricalpolar coordinates (d3x′= bdbdφdz′). First, we find

(k − k′)·x′ = (k − k

′)·(b+ z′z) ≃ −k′·b. (5.3.84)

To derive this, it is important to remember that b and k are, by definition, or-thogonal while, by hypothesis, k′ and k are approximately parallel. To simplifythe expressions, we may take the scattering plane as x−z. Note that the impactvector b does not, in general, lie in the same plane (but it does lie in the x−y


plane). We thus write

k′ = (k sin θ, 0, k cos θ) (5.3.85a)

andb = (b cos φb, b sinφb, 0), (5.3.85b)

where, recall, θ is just the usual (small) scattering angle. And therefore

k′·b ≃ kb θ cos φb. (5.3.86)

Substituting this into the expression for feik(k′,k), leads to

feik(k′,k) = − 1

4π

2m

~2

∫ ∞

0

b db

∫ 2π

0

dφb

∫ +∞

−∞dz′ e− ikbθ cosφb V

(√b2 + z′2

)

× exp

[− im

~2k

∫ z′

−∞dz′′ V

(√b2 + z′′2

)]. (5.3.87)

The integral over φb leads to a Bessel function:∫ 2π

0

dφb e− ikbθ cosφb = 2π J0(kbθ) (5.3.88)

and

∫ +∞

−∞dz′ V

(√b2 + z′2

)exp

[− im

~2k

∫ z′

−∞dz′′ V

(√b2 + z′′2

)]

= −~2k

imexp

[− im

~2k

∫ z′

−∞dz′′ V

(√b2 + z′′2

)] ∣∣∣∣∣

z′=+∞

z′=−∞

. (5.3.89)

The lower boundary value on the right-hand side gives zero and we thus finallyobtain the so-called eikonal approximation

feik(k′,k) = − ik

∫ ∞

0

b db J0(kbθ)[e−2i∆(b)−1

], (5.3.90a)

where

∆(b) = − m

2~2k

∫ +∞

−∞dz V

(√b2 + z2

). (5.3.90b)

The integral here is to be understood as running along the contour following astraight line parallel to the z-axis at a perpendicular distance b from the scatteringcentre (see Fig. 5.2). Note that if b is larger than the range of the potential V


then ∆(b)=0 and thus also f(k′,k)=0. Note further that what we have obtainedcorresponds to the JWKB approximation with the additional simplification thatV is neglected in the denominator of the amplitude of ψ(+), cf. (5.3.75).

5.3.6 The free particle

In discussing scattering off spherically symmetric potentials (and, in fact, moregenerally), the conservation of total angular momentum naturally leads us to con-sider spherical waves since these are precisely constructed as waves of definiteangular momentum. However, the states one normally deals with experimentally,both as incident and final outgoing (detected) particles, are plane waves of definitelinear momentum. In this section we shall examine the necessary transformationbetween the two bases. Indeed, note that for a free particle the operators k, L2

and Lz commute with the Hamiltonian, which coincides with just the kinetic en-ergy. We may thus consider the basis-state sets of either |k〉 or |E,l,m〉 equally.The transformation coefficients we seek are then simply 〈k|E,l,m〉.

First of all, we should define our normalisation convention for spherical waves;this we do in the most natural way:

〈E ′,l′,m′|E,l,m〉 = δ(E ′ − E) δl′l δm′m. (5.3.91)

A first observation we can make regarding plane waves is that, since the momentumis directed along some specific axis, (say, the z-axis) there can be no component ofthe angular momentum vector in same direction (L≡r∧k). This may be expressedformally as

Lz|kz〉 = 0. (5.3.92)

Taking, as always, the z-axis for angular-momentum quantisation, we must con-sequently have

〈kz|E,l,m〉 = 0 (for m 6= 0). (5.3.93)

The expansion of plane waves |kz〉 along the z-axis in terms of the spherical waves|E,l,m〉 will then take the form

|kz〉 =∑

l′

∫dE ′ |E ′,l′,m′〉〈E ′,l′,m′|kz〉

∣∣∣m

′=0. (5.3.94)

For the general case |k〉, we need only apply a rotation:

|k〉 = D(α=φ,β=θ,γ=0)|kz〉. (5.3.95)


Projecting onto 〈E,l,m| then leads to

〈E,l,m|k〉 =∑

l′

∫dE ′ 〈E,l,m|D(φ,θ,0)|E ′,l′,m′〉〈E ′,l′,m′|kz〉

∣∣∣m

′=0

=∑

l′

∫dE ′

D(l)m0(φ,θ,0) δ(E

′ − E) δl′l 〈E′,l′,m′|kz〉

∣∣∣m

′=0

= D(l)m0(φ,θ,0) 〈E,l,0|kz〉. (5.3.96)

Recall here the relation between the Wigner rotation matrices and the sphericalharmonics (3.1.31):

D(l)m0(α, β, 0) =

√4π

2l + 1Y m ∗l (β, α). (3.1.31)

Now, 〈E,l,0|kz〉 is only a function of E and l (although it must also “select”the correct value of k to give E as the kinetic energy for a particle of mass m).We may thus write

〈E,l,0|kz〉 = N(l, k) δ

(E − ~

2k2

2m

), (5.3.97)

where N(l,k) is completely determined by the orthonormality conditions (5.3.91)applied, e.g., to states of identical l and m=0:

〈E ′,l,0|E,l,0〉 = δ(E ′ − E) (5.3.98a)

=

∫d3k 〈E ′,l,0|k〉〈k|E,l,0〉

=

∫d3k∣∣N(l, k)

∣∣2 δ(E ′ − ~

2k2

2m

)δ

(E − ~

2k2

2m

)4π

2l + 1

∣∣∣Y 0l (k)

∣∣∣2

.

The radial k integral is performed using either of the energy δ-functions while theangular integration is given by the normalisation of the spherical harmonics. Wethus have

〈E ′,l,0|E,l,0〉 =∣∣N(l, k)

∣∣2 4π

2l + 1δ(E ′ − E)

mk

~2

∣∣∣∣~k=

√2mE

. (5.3.98b)


and, comparing Eqs. (5.3.98a & b), we finally obtain

N(l, k) =

√2l + 1

4π

~√mk

. (5.3.99)

We therefore have (taking the complex conjugate)

〈k|E,l,m〉 = N∗(l, k) δ

(E − ~

2k2

2m

)√4π

2l + 1Y ml (k)

and thus

=~√mk

δ

(E − ~

2k2

2m

)Y ml (k). (5.3.100)

This then is the coefficient in the expansion of |k〉 over a basis of spherical waves|E,l,m〉. Note, in particular, that the expansion contains an infinite number ofangular momentum contributions (i.e. l=0,1,... ,∞):

|k〉 =∑

l,m

∫dE |E,l,m〉〈E,l,m|k〉

=~√mk

∞∑

l=0

l∑

m=−l

Y m∗l (k)|E,l,m〉

∣∣∣E=~

2k2/2m

. (5.3.101)

Let us now try to understand the physical reason for the presence of an infinitenumber of waves in the representation of a plane wave (and therefore possibly ina scattering process too). For any given (finite) momentum p the (theoretically)infinite transverse spatial extension of the wave front implies that the impactparameter b can be infinitely large. The classical orbital angular momentum isL=r∧p, and we should thus expect l~∼pr∼pb. We also see, however, that for afinite-range potential the sum will be truncated naturally at l <∼ prpot/~.

So far we have worked in the momentum-space representation; now let us derivethe corresponding expressions in coordinate space. The free-particle solutions ofthe Schrödinger equation may be expressed in terms of spherical waves:

jl(kr) Yml (x) and nl(kr) Y

ml (x), (5.3.102)

where jl (nl) are the spherical Bessel (Neumann) functions, which are regular(singular) at the origin. We need therefore only consider jl for the present analysisand thus take

〈x|E,l,m〉 = cl jl(kr) Yml (x). (5.3.103)

Again, we are required to evaluate the coefficient cl. To do this we may exploit


the knowledge we have just gained in the momentum-space case:

〈x|k〉 =∑

lm

∫dE 〈x|E,l,m〉〈E,l,m|k〉

=∑

lm

∫dE cl jl(kr) Y

ml (x)

~√mk

δ

(E − ~

2k2

2m

)Y m∗l (k)

=∑

l

2l + 1

4πPl(k·x)

~√mk

cl jl(kr), (5.3.104)

where we have used the following standard identity for spherical harmonic func-tions: ∑

m

Y ml (x) Y m∗

l (k) =(2l + 1)

4πPl(k·x). (5.3.105)

Now, for a plane wave we also have (recall the identity i l≡ e i lπ/2)

〈x|k〉 =e ik·x

(2π)3/2

=1

(2π)3/2

∑

l

(2l + 1) i l jl(kr)Pl(k·x). (5.3.106)

This last equality can been proved using the following integral representing for thespherical Bessel functions:

jl(kr) =1

2i l

∫ +1

−1

dcos θ e ikr cos θ Pl(cos θ). (5.3.107)

Comparing Eqs. (5.3.104 & 106), we find simply

cl =i l

~

√2mk

π. (5.3.108)

We thus have the two projections of the angular-momentum eigenstates:

〈k|E,l,m〉 =~√mk

δ

(E − ~

2k2

2m

)Y ml (k), (5.3.109a)

〈x|E,l,m〉 =i l

~

√2mk

πjl(kr) Y

ml (x). (5.3.109b)

These are the two forms we shall exploit in the derivation of the so-called partial-wave expansion we are about to undertake.

The formulæ just derived may be used immediately in the description of asimple decay process. Consider, for example, the α-decay Ne20 ∗→ O16 + He4 . The


spins of the particles involved are unity for the decaying (excited) neon nucleus andzero for the two daughter oxygen and helium nuclei. Now, if the spin (or magneticmoment) of the neon nucleus has been aligned (e.g. with the aid of a magneticfield along the z-axis), then since the final state must have l=1 (i.e. it must be ap-wave) we know that the decay angular distribution must be given by

∣∣Y ±11 (θ,φ)

∣∣2according as to whether the spin is aligned up or down. In this manner, i.e. byexamining the decay angular distributions, it is possible to determine the spin ofa complex nucleus or an elementary particle.

5.3.7 Partial waves

We can now examine the case of scattering off a potential V , which we take to bespherically symmetric. This means that [V,L2]=0= [V,Lz] and thus the scatteringmatrix T introduced earlier, see Eq. (5.3.45), must be a scalar operator. We cantherefore apply the Wigner–Eckart theorem to obtain

〈E ′,l′,m′|T |E,l,m〉 = T l(E) δl′l δm′m. (5.3.110)

That is, being a spherical tensor of rank zero, T does not alter either l or m and,moreover, the non-zero elements do not depend on m.

The scattering amplitude is

f(k′,k) = −2m

~2

(2π)3

4π〈k′|T |k〉

= −2m

~2

(2π)3

4π

∑

lml′m

′

∫dE dE ′ 〈k′|E ′l′m′〉〈E ′,l′,m′|T |E,l,m〉〈Elm|k〉

= −2m

~2

(2π)3

4π

~2

mk

∑

lm

T l(E) Yml (k′) Y m∗

l (k)∣∣∣E= ~

2k2

2m

= −4π2

k

∑

lm

T l(E) Yml (k′) Y m∗

l (k)∣∣∣E= ~

2k2

2m

. (5.3.111)

We make the usual choice of an initial wave directed along the positive z-axis andso have

Y ml (k) =

√2l + 1

4πδm0, (5.3.112)

where, as already noted, cosθk=1, Pl(1)=1 and only the m=0 waves contribute.


Writing θkk′ = θk′ = θ, we are thus only interested in

Y 0l (k

′) =

√2l + 1

4πPl(cos θ). (5.3.113)

If we now define the partial-wave amplitude fl(k) as

fl(k) ≡ −π T l(E)

k, (5.3.114)

we may then write

f(k′,k) = fk(θ) =

∞∑

l=0

fl(k)Pl(cos θ). (5.3.115)

As discussed earlier, it only makes sense to study fl(k) at large distance, wherethe influence of the potential is negligible and the particle is effectively free. Firstof all then, we need the large-distance behaviour of the spherical Bessel functions:

jl(kr)r→∞−−−→ sin(kr − 1

2lπ)

kr. (5.3.116)

Recalling (5.3.106) and inserting all this into expression (5.3.27) for 〈x|ψ(+)〉, wehave

〈x|ψ(+)〉 r→∞−−−→ 1

(2π)3/2

[e ikz +fk(θ)

e ikr

r

]

=1

(2π)3/2

∞∑

l=0

(2l + 1)

[Pl(cos θ)

(e ikr − e− i (kr−lπ)

2ikr

)+ fl(k)Pl(cos θ)

e ikr

r

]

=1

(2π)3/2

∞∑

l=0

(2l + 1)Pl(cos θ)

[(1 + 2ikfl(k)

) e ikr

2ikr− e− i (kr−lπ)

2ikr

].

(5.3.117)

The interpretation should be clear: the unperturbed plane wave may be expressedas a sum of outgoing (e ikr/r) and incoming (e− i (kr−lπ)/r) spherical waves (contain-ing an infinite number of partial waves). The presence of a scattering potentialcan clearly only affect the outgoing portion; the effect is to change its coefficient(although, as we shall now see, only in phase):

1 → 1 + 2ikfl(k). (5.3.118)


Unitarity and phase shifts

An important though apparently trivial restriction that must be placed on theelastic scattering amplitude is that of unitarity or probability conservation. Statedsimply, the requirement is that the total outgoing flux must equal that of theincoming wave (since particles are neither created nor annihilated). The conditionis, moreover, strengthened by the fact that angular momentum is conserved andunitarity may thus be applied at the level of individual partial waves. Formally,we may apply Gauss’ theorem to the total current or flux at the surface of a sphere(or, indeed, any closed surface):

∫

S

dS ·j = 0. (5.3.119)

The above condition (at the partial-wave level) simply implies that the coefficientsof the outgoing (e ikr/r) and incoming (e− i (kr−lπ)/r) spherical waves are equal inmagnitude (though not necessarily in phase). Defining

Sl(k) ≡ 1 + 2ikfl(k), (5.3.120)we have then ∣∣Sl(k)

∣∣2 = 1. (5.3.121)

This is the unitarity condition for the l-th partial wave. We may therefore write

Sl(k) ≡ e2iδl(k), (5.3.122)

where the phase-shift δl must clearly be a real function of k (and only of k) whilethe factor 2 in the exponent is simply a matter of later convenience.

Equation (5.3.120) may be inverted to give

fl(k) =Sl(k)− 1

2ik, (5.3.123a)

which, in terms of δl, becomes

≡ e2iδl −1

2ik≡ e iδl sin δl

k≡ 1

k cot δl − ik. (5.3.123b)

The last equality is given for later use. The elastic scattering amplitude may thenfinally be expressed as

fk(θ) =∞∑

l=0

(2l + 1)e2iδl −1

2ikPl(cos θ)


=1

k

∞∑

l=0

(2l + 1) e iδl sin δl Pl(cos θ). (5.3.124)

Note that the derivation of fk(θ) so given is based solely on the requirements ofrotational invariance (leading to angular-momentum conservation) and unitarity(or probability conservation). It is therefore quite general and not necessarilylimited to non-relativistic quantum mechanics, for example.

We can now perform a standard (but more restricted) demonstration of theoptical theorem. As usual, the cross-section is obtained from

σtot =

∫dΩ

∣∣fk(θ)∣∣2

=1

k2

∫ +1

−1

dcos θ

∫ 2π

0

dφ∑

ll′

(2l + 1) (2l′ + 1)

× e iδl sin δl Pl(cos θ) e− iδ

l′ sin δl′ Pl

′(cos θ)

=4π

k2

∑

l

(2l + 1) sin2 δl. (5.3.125)

Returning to (5.3.124) and taking the imaginary part for θ=0 (forward scattering)leads to

Im fk(0) =1

k

∞∑

l=0

(2l + 1) Im e iδl sin δl Pl(1)

=1

k

∞∑

l=0

(2l + 1) sin2 δl =k

4πσtot, (5.3.126)

which is just the optical theorem already proven, see Eq. (5.3.57).

The Argand diagram: We may rewrite formula (5.3.123b) for fl in a rathersuggestive manner,

k fl(k) = 12

[i + al e

2iδl− iπ/2], (5.3.127)

where the positive real amplitude al≤1 allows for absorption (i.e., inelastic pro-cesses). As δl varies, the quantity kfl(k) thus describes a circle in the complexplane centred on the point (0, 1

2i ) and of radius 1

2(see Fig. 5.3), known as the

unitarity circle. Probability conservation requires kfl(k) to lie precisely on thiscircle; in the case of absorption (when probability is not conserved and flux is lost,al<1) it would lie inside the circle and (unless particles are somehow created fromthe vacuum) it may not lie outside.

The Argand diagram allows us to easily discern certain important aspects of


Rekfl

Imkfl

0

12

2δl

Figure 5.3: The Argand diagram depicting the possible variation of the quantity kfl(k)as the phase-shift δl varies. The permitted region lies inside the circle and elastic scat-tering lies precisely on the circle perimeter.

the behaviour of fl(k). Suppose first that δl is small, then fl(k) will be almostpurely real:

fl(k) =e iδl sin δl

k≃ (1 + iδl)δl

k≃ δl

k. (5.3.128)

If, on the other hand, δl∼π/2, then fl(k) will be almost purely imaginary and themodulus will take on its maximum value. The condition of maximum scatteringis a resonance condition, which we shall examine later. In any case, the maximumvalue for the partial cross-section is then

σmax

l = (2l + 1)4π

k2, (5.3.129)

which is attained whenever sin2δl=1 or δl=12, 3

2, 5

2etc.

The eikonal approximation: This approximation (introduced in Sec. 5.3.5) isvalid, in particular, in the high-energy limit, where the de Broglie wavelength ismuch less that the range of the potential. In this limit a very large number ofpartial waves are active; as E→∞, lmax→∞. Taking lmax ≈kR, we may makethe following substitutions in (5.3.124):

lmax∑

l=0

lmax→∞−−−−−→ k

∫ R

0

db , (5.3.130a)

Pl(cos θ)l→∞−−−→θ→0

J0(lθ) = J0(kbθ), (5.3.130b)


δll→∞−−−→ ∆(b)

∣∣∣b=l/k

. (5.3.130c)

Now, we have

e2iδl −1 = e2i∆(b) −1 = 0 for l > lmax (5.3.131)

and therefore

fk(θ) → k

∫ R

0

db2kb

2ik

[e2i∆(b)−1

]J0(kbθ)

= − ik

∫ R

0

db b J0(kbθ)[e2i∆(b)−1

]. (5.3.132)

The integral is to be performed using the explicit form of ∆(b) from Eq. (5.3.90b).

Phase-shift analysis: The phase-shifts δl(k) clearly, if not too numerous, rep-resent a very convenient experimental parametrisation of the scattering amp-litudes. Indeed, from measurements one obtains their values more-or-less directly.The passage back to the original scattering potential is, however, not straightfor-ward. Let us now examine the process of deriving the phase-shifts theoreticallystarting from the potential.

We shall assume a potential of finite range R; in other words, the interactionis essentially turned off for distances r>R. Outside this region the wave-functionis thus once again that of a free particle. However, since now by hypothesis theregion in the neighbourhood of the origin is not included, we must also admitcontributions proportional to nl (the Neumann functions, which are not regularat the origin). It then turns out to be more convenient to work in terms of theHankel functions:

h(1,2)l ≡ jl ± inl ≡ h±l . (5.3.133)

The asymptotic behaviour of these functions is

h±l (kr)r→∞−−−→ e± i (kr−lπ/2)

ikr, (5.3.134)

that is, they correspond to purely outbound and inbound waves respectively. Thesolution to the problem in the presence of the scattering potential, but for r>R,may be written as

〈x|ψ(+)〉 =1

(2π)32

∑

l

i l (2l + 1)Al(r)Pl(cos θ) (r > R), (5.3.135)

clearly here, all important details relating to the scattering potential are hidden


inside the radial wave-function Al(r):

Al(r) = c+l h+l (kr) + c−l h

−l (kr), (5.3.136)

where the coefficients of Al(r) are chosen such that, for the case V =0, the solutionjust corresponds to the original free case everywhere, i.e. simply jl(kr). We maynow compare the asymptotic behaviour with Eq. (5.3.117), which, with the simplesubstitution in accordance with (5.3.120 & 122),

1 + 2ikfl(k) = e2iδl, (5.3.137)

becomes

〈x|ψ(+)〉 =1

(2π)3/2

∞∑

l=0

(2l + 1)Pl(cos θ)

[e2iδl e ikr

2ikr− e− i (kr−lπ)

2ikr

]. (5.3.117′)

We thus see that the coefficients c±l are simply

c+l = 12e2iδl and c−l = 1

2. (5.3.138)

The radial wave-function Al(r) may then be rewritten as

Al(r) = e iδl[cos δl jl(kr)− sin δl nl(kr)

]. (5.3.139)

At this point we can calculate the logarithmic derivative of Al(r) with respect to r:

βl(k) =r

Al(r)

dAl(r)

dr

= kr

[j′l(kr) cos δl − n′

l(kr) sin δljl(kr) cos δl − nl(kr) sin δl

] ∣∣∣∣r=R

+

, (5.3.140)

where r=R+ indicates evaluation at the point r=R+, as approached from above(i.e. from outside the potential), and the prime indicates a derivative with respectto kr evaluated at the point kR. This equation can be easily inverted to obtain

tan δl(k) =kR j′l(kR)− βl(k) jl(kR)

kRn′l(kR)− βl(k)nl(kR)

. (5.3.141)

In other words, we simply need to determine βl(k).Inside the region of the potential range we have the one-dimensional reduced


Schrödinger equation:

d2ul(r)

dr2+

[k2 − 2m

~2 V (r)−

l(l + 1)

r2

]ul(r) = 0, (5.3.142)

where ul(r) = r Al(r), (5.3.143)

with boundary condition ul(r)∣∣r=0

= 0. (5.3.144)

This partial differential equation may be solved by direct integration (numericallyif necessary) from the lower limit r=0 up to the potential boundary at r=R. Thisthen provides the necessary value of βl(k) and hence δl(k).

Low-energy scattering and bound states

The partial wave expansion is particularly useful at low energies. As alreadyfrequently remarked, there is a direct connection between the incident particlemomentum p and the maximum associated orbital angular momentum. For a givenmomentum p, a finite range for the potential implies that the impact parameteris limited to b<R. The maximum classical orbital angular momentum is thenL∼Rp and so l~<∼ Rp=R/λ. The series thus actually truncates early. Indeed, ifλ∼R, then only the very first few waves will contribute to the scattering process.

Another way of seeing this is to consider the centrifugal term in the radial waveequation as an effective contribution to the potential:

Veff = V (r) +~2

2m

l(l + 1)

r2. (5.3.145)

Indeed, we can quantify the limiting behaviour. It is possible to construct anintegral equation for Al(r),

e iδl sin δlk

= −2m

~2

∫ ∞

0

r2 dr jl(kr) Veff(r)Al(r). (5.3.146)

Now, if the centrifugal barrier is dominant, then the wave-function will be veryclose to the free case, i.e. jl(kr)∼Al(r). Since jl(ρ)∼ρl for ρ→0, if kR≪1, thenthe right-hand side of the above relation will behave as k2l. The left-hand sidemust behave as δl/k for δl small. We therefore have

δl ∼ k2l+1 for δl and k → 0. (5.3.147)

This is known as threshold behaviour and shows the dominance of low angularmomenta (or partial waves) at low energies.


Exercise 5.3.2. Derive the integral equation (5.3.146) explicitly.

Scattering off a hard sphere: As a first concrete example of scattering, weshall examine the case of a perfectly rigid sphere (i.e. an infinite repulsive potential)of radius R. We thus have

V (r) =

∞ for r < R,

0 for r > R.(5.3.148)

Consider first low-energy scattering, i.e. such that kR≪1. In this case, as wehave just seen, all phase shifts except that of the s-wave (l=0) may be neglected.It is then sufficient to calculate δ0 by solving the radial part of the Schrödingerequation equation for l=0:

d2u0(r)

dr2+

[k2 − 2m

~2 V (r)

]u0(r) = 0, (5.3.149)

where, as usual, we have separated the wave-function as

ψ(r, ω) =ul(r)

rY lm(Ω), (5.3.150)

with boundary condition ul(r)∣∣r=R

= 0. (5.3.151)

Now, the general form in the large-r limit is

u0(r) = C sin(kr + δ0) (5.3.152)

and thus we immediately obtain δ0=−kR; i.e. the phase shift is negative and thewave-function is, so-to-speak, “pushed out”.

The differential cross-section is just

dσ(Ω)

dΩ≃ dσ0(Ω)

dΩ=

sin2 δ0

k2=

sin2 kR

k2≃ R2. (5.3.153)

Integrating over the full solid angle (4π) we obtain the answer:

σtot = 4πR2. (5.3.154)

We thus find that the total cross-section is independent of energy (for kR≪1)and equal to four times the geometrical cross-section of the hard sphere. That is,classical mechanics would give πR2. However, for low energy scattering, we shouldnot expect the same result in quantum mechanics.


Consider next the high-energy limit kR≫1. We must now include all partialwaves up to some lmax≈kR. The general form for u(r) in the large-r limit is now

ul(r) = C sin

(kr − lπ

2+ δl

)(5.3.155)

and so the boundary condition leads to δl=−kR+ lπ/2. The cross-section is then

σtot =4π

k2

lmax∑

l=0

(2l + 1) sin2 δl =4π

k2

lmax∑

l=0

(2l + 1) sin2

(−kR +

lπ

2

)

≈ 4π

k21

2l2max = 2πR2. (5.3.156)

The optical shadow: So, for high-energy scattering off a hard sphere, althoughwe might have expected to recover the classical result, we obtain 2πR2, twice thegeometrical cross-section; this is known as shadow scattering. If the wavelengthof the incident particle is very small compared to R, then the sphere casts ashadow; i.e. directly behind the sphere no scattered particles are found. However,experimentally, it is found that the shadow only extends up to some finite distance.Far from the sphere the shadow disappears and so it must be filled in by scatteringof waves from near the edge of the sphere. In order to fill in the shadow, thescattered flux must have the same magnitude as the flux removed from the incidentbeam; i.e. as the geometrical cross-section of the sphere. The total scattering cross-section must therefore have twice the magnitude of the geometrical cross-section.

Exercise 5.3.3. A particle is scattered by a totally black sphere (ie totally ab-sorbing) of radius R. The scattering amplitude may be expressed by the followingformula:

f(θ) =1

2ik

∞∑

l=0

(2l + 1) (ηl e2iδl −1)Pl(cos θ). (5.3.157)

Considering energies such that kR≫1 (where k≡2π/λ and λ is the de Brogliewavelength), describe the behaviour of ηl and δl as a function of l. Calculate thefollowing cross-sections:

(a) elastic,(b) absorption (or reaction),(c) total.

Hint: the finite radius R restricts the values of impact parameter and angular mo-mentum l that can contribute. In addition, the total cross-section may be obtainedvia the optical theorem.


The square well or barrier: This case is of particular importance in nuclearphysics for studies of the nucleon–nucleon potential. Since nuclear binding energiesare of the order of a few MeV, corresponding to momenta for nucleon (proton orneutron) projectiles of the order of tens of MeV, and the range of the nuclear forceis only of the order of 1 fm, in most cases then the scattering is entirely dominatedby s-waves. A similar situation also holds for low-energy pion scattering. A goodapproximation to the potential generated by the nuclear force is the square well,which we shall now examine to exemplify the foregoing discussion.

Let us therefore consider a potential of the form

V (r) =

V0 (constant) for r < R,

0 for r > R.(5.3.158)

Recall that V0<0 (V0>0) represents an attractive (repulsive) force. From (5.3.139),we know that the wave-function outside must behave as

A0(r) = e iδ0[cos δ0 j0(kr)− sin δ0 n0(kr)

]≃ e iδ0 sin(kr + δ0)

r. (5.3.159)

For constant V0, the solution inside is

A0(r) ∝ sin k′r, (5.3.160)with

k′ =

√2m(E − V0)

~. (5.3.161)

The sine dependence is determined by the requirement that u(r) vanish at theorigin (i.e. the n0 solution is excluded). Note that, in the case E<V0, the aboveequations for the solutions inside will have sinh substituted for sin, thus

A0(r) ∝ sinh κr, (5.3.162)with

κ =

√2m(V0 − E)

~. (5.3.163)

Now, the matching conditions at the boundary r=R indicate that the wavefronts are either pulled back into the well, if V0<0 and δ0>0, or pushed out of thewell, if V0>0 and δ0<0 (see Fig. 5.4). The curvature of the wave-function insidethe well will then be greater (lesser) for δ0>0 (δ0<0). Consider the attractive case;as V0 increases the curvature inside increases and thus too the phase-shift. Forany give k, there is some value of V0 for which the phase shift will be exactly π/2and the s-wave partial cross-section will attain its maximum value. If we continueincreasing the potential depth, there will come a point when δ0=π and the s-wave partial cross-section will therefore fall back to zero. If the incident energy


V (r)

r

u(r)

R

δ0/k

(a)

V (r)

r

u(r)

R δ0/k

(b)

Figure 5.4: Phase shifts inside and outside the potential region for (a) attractive and(b) repulsive potentials; the full (dashed) curves represent the non-zero potential (free)cases.

is still sufficiently low that only the s-wave effectively contributes, then there willbe no scattering of the incident wave. This apparently paradoxical phenomenonis known as the Ramsauer–Townsend effect—it was actually observed before theadvent of quantum mechanics in the scattering of electrons of around 0.7 eV offnoble-gas atoms, such as argon, krypton and xenon. A noble-gas atom has allelectron shells closed and is therefore a relatively small, sharp object. Moreover,the force it exerts on a charged particle is of finite and well-defined range.

The effect is just the three-dimensional analogue of perfect transmission in thecase of a one-dimensional potential well or barrier of the correct width a: 2a=nλ,with n∈N and λ the wavelength of the incident particles. Such behaviour is alsowell known in optics, where it is exploited to obtain the selective transmissionof light of particular wavelengths by a thin layer of glass or dielectric; i.e. in theconstruction of so-called optical interference filters. Note, however, that in the


three-dimensional case such an effect cannot be produced by a repulsive potential;ka0 (where a0 is the so-called scattering length, defined in the next section) wouldneed to be at least of order unity and the corresponding potential would so strongas to induce important phase-shifts for higher-order waves, which would then maskthe s-wave zero.

Threshold behaviour and bound states: If we take the extreme case of k→0,then certainly only s-wave scattering can be important and, moreover, there is alink between the cross-section and the existence of bound states for the scatteringpotential. For l=0, outside the well, in the limit k∼0 we have

d2u(r)

dr2= 0, (5.3.164)

with the obvious and simple solution

u(r) ∝ r − a. (5.3.165)

Note that, since we are dealing with the solution outside, the boundary conditionat the origin cannot be applied and thus a is not yet determined. This linear beha-viour may be interpreted as the limiting case of a sine curve of infinite wavelength:

limk→0

sin(kr + δ0) = limk→0

sin

[k

(r +

δ0k

)]. (5.3.166)

We thus haveu′(r)

u(r)= k cot

[k

(r +

δ0k

)]k→0−−→ 1

r − a0, (5.3.167)

where the suffix zero on a0 now indicates explicit consideration of s-waves. In orderto determine the constant a0, we may take r with any value: it is most convenientto choose r=0 and thus

k cot δ0k→0−−→ − 1

a0. (5.3.168)

The quantity

a0 ≡ − limk→0

tan δ0k

(5.3.169)

is known as the scattering length. In the limit k→0 we then have for the scatteringcross-section

σtot

0 = 4π limk→0

[1

k cot δ0 − ik

]2= 4πa20. (5.3.170)

One might imagine that since a0 has dimensions of a length that its magnitude


should be roughly determined by the range R of the potential, especially since itseems to be equivalent to a size of the effective scattering centre. It turns out,however, that no such relation exists in general and that in the case of an attractivepotential a0 may even be orders of magnitude larger than R. Let us now find thephysical meaning of a0. From the definition (5.3.167), it is clear that a0 is noneother than the intercept of the limiting straight-line tangent to the wave-functionoutside in the case k→0 (see Fig. 5.5).

V (r)

r

u(r)

|a0|

(a)

V (r)

r

u(r)

|a0|

(b)

V (r)

r

u(r)

|a0|

(c)

Figure 5.5: A geometric description of the scattering length as the intercept of thelimiting straight-line behaviour outside the potential range for k→0, for: (a) a repuls-ive potential, (b) an attractive potential that does not support bound states and (c) apotential that supports bound states.

Indeed, from Fig. 5.5a, we see that in the specific case of a repulsive poten-


tial that the intercept must fall inside the potential region and therefore a0<R.However, the other two figures demonstrate the cases of potentials without andwith bound states respectively. To understand this behaviour, consider first thecase of a non-binding potential well: the wave-function can have no nodes inside oroutside the well and therefore must have positive slope at the outside boundary. Anon-binding potential is thus characterised by a negative scattering length. Now,imagine increasing the depth of the well: the curvature of the wave-function willincrease, the slope will decrease and thus (the negative) a0 will grow in size untilit reaches the limiting point of zero slope. This situation corresponds preciselyto a well depth that supports just a single bound state with zero binding energy.Henceforth, as the well depth increases, since u(r) must tend monotonically tozero as r→∞, u(r) and u′(r) will always have opposite signs at the boundary andthus the slope there will always be such that a0>0. The behaviour at the pointwhere a0 becomes infinite (or, rather, ill-defined) can be understood as a reson-ance condition (we shall examine this problem in detail shortly): the energy of thecorresponding bound state is zero, as too is the energy of the incident particle inthe k→0 limit. It is thus possible for the particle to actually become trapped bythe scattering potential.

The connection between the scattering length and the bound-state energy canbe made more quantitative. Examining Fig. 5.5 for the case V0<0, we see that ifa0 is large and positive then, since e−κr is also approximately a straight line justoutside the potential range (i.e. for r>∼ R), it is very similar to u(r)∝ r−a ande−κr, with κ≃0 is indeed the wave-function for a bound state of energy E=0−

for r>R. Now, the wave-function inside for the two cases E=0+ (zero-energyscattering) and E=0− (bound state of zero binding energy) are very similar sincein both cases the wave number k′ is determined by the equation

k′ =

√2m(E − V0)

~≃√2m|V0|~

. (5.3.171)

Matching the logarithmic derivatives for the two cases (bound state and thresholdscattering), we have

− κ e−κr

e−κr

∣∣∣∣r=R

=

(1

r − a0

)∣∣∣∣r=R

, (5.3.172)

which for a0≫R leads to

κ ≃ 1

a0(5.3.173)


Therefore, since the binding energy B is given by

B =~2κ2

2m≃ ~

2

2ma20, (5.3.174)

we thus have a direct connection between the scattering length a0 and the energylevel of the bound state. In other words, providing a0≫R, it is possible to directlydeduce the binding energy (for a weakly bound state) by performing scatteringmeasurements at very low energies.

We can actually take the approximation (5.3.147) a step further:

δl ≃ nl π − al k2l+1 + cl k

2l+3 for k → 0, (5.3.175)

where now we have also explicitly included the general ambiguity in δl up to anadditive, constant, integer multiple of π. Indeed, it can be shown that the integernl is just the number of bound states that the potential supports for a given waveof angular momentum l, (Levinson’s theorem, 1949). The low-energy expansion ofcotδ0 therefore becomes

k cot δ0k→0−−→ − 1

a0+r0k

2

2, (5.3.176)

where a0 has the same meaning as before and the new constant r0 introduced isknown as the effective range—one can show that it roughly corresponds to theradius at which the potential becomes negligible. The zero-energy, limiting cross-section then becomes

σtot0 = 4π lim

k→0

∣∣∣∣1

k cot δ0 − ik

∣∣∣∣2

≃ 4πa20(1− 1

2r0k

2a0)2

+ k2a20. (5.3.177)

Bound states as poles in Sl(k): An important mathematical property ofscattering amplitudes in quantum mechanics is analyticity. We shall thus nowbriefly examine the analytic properties of S0(k), the outgoing s-wave amplitude.From the definition, we have that the radial part of the wave-function at largedistances is proportional to

S0(k)e ikr

r− e− ikr

r. (5.3.178)

For a bound state the corresponding expression is

e−κr

r. (5.3.179)


Bound-states solutions exist only for discrete values of κ. The form is, however,just that of the outgoing wave in the scattering case with k→ iκ. We may thussay that the essential difference between the scattering and bound-state solutionsis the respective presence or absence of an incoming wave. In the case of scatteringthe quantity of physical interest is just the ratio of the coefficients of the outgoingto incoming waves and this is precisely S0(k). From this viewpoint, one wouldthen say that the bound state corresponds to an approximate pole in S0(k), seenas a function of the complex variable k. This pole lies on the imaginary axis inthe complex k plane. On the other hand, k real and positive corresponds to thephysical scattering region.

We thus have a function S0(k), which on the real axis has the form e2iδ0 (withδ0 necessarily real). Moreover, for vanishing wave number we have

limk→0

k cot δ0 = − 1

a0, (5.3.180)

which is finite. That is, for k→0, the limiting behaviour of δ0 must then beδ0→n0π (n0∈Z) and that of S0(k), S0(k)=e2iδ0 →1. It is interesting to attemptto construct a function S0(k) satisfying the above requirements: namely, it must

1. have a pole for k= iκ (the bound state),2. have unit modulus for k real and positive,3. be real and unity in the limit k→1.

The simplest expression satisfying all these conditions is the following:

S0(k) =−k − iκ

k − iκ. (5.3.181)

Note that such a choice implicitly excludes the existence of other important nearbysingularities besides the bound state.

Given this form, we may now obtain the scattering amplitude:

f0 =S0 − 1

2ik=

1

−κ− ik. (5.3.182)

We therefore have

1

k cot δ0 − ik=

1

−κ− ik, (5.3.183)

which leads tolimk→0

k cot δ0 = − 1

a0= −κ. (5.3.184)

This is precisely the relation between scattering and bound states already obtained.


We thus see that it is sufficient to impose analyticity on the solutions of theSchrödinger equation without necessarily explicitly solving it. Such a procedureis extremely useful in those cases where little is known about the form of thepotential. Note that the description of strong-interaction physics of the 1960s waslargely based on such a principle.

Resonant scattering and the Breit–Wigner formula

In various physical situations, from the relatively low energies of atomic physics,through nuclear and up to high-energy particle physics, a common phenomenonis that of resonance formation, in both scattering and decay. The visible result ofthis is a pronounced peak in a scattering cross-section as a function of energy, incorrespondence with the resonant energy. In general, the effect is due to the forma-tion of a metastable intermediate (possibly bound) state. For the type of situationwe are studying, this simply implies that the scattering potential has a boundstate of energy corresponding to the kinetic energy of the incident particle. Forthe simplified, square-well potentials considered so far, this is clearly not possible.Recall, however, that the radial equation in three dimensions, for l >0, has aneffective positive (i.e. repulsive) contribution coming from the so-called centrifugalbarrier,

Vcb =~2

2m

l(l + 1)

r2. (5.3.185)

The resultant potential, while retaining an attractive central core, gains a repulsiveouter shell and therefore effectively a barrier (see Fig. 5.6).

V (r)

r

~2

2m

l(l+1)

r2

boundstateenergies

Figure 5.6: A schematic representation of the effective centrifugal barrier surroundinga three-dimensional potential well for l>0. The horizontal dashed lines schematicallydemarcate the energy region in which there exist bound states.


The significance of Fig. 5.6 is that there exists a range of external kineticenergies that are below the barrier height, but which, via tunnelling, would allowthe particle to penetrate to the inner core. Once inside, if the energy correspondsto a bound-state level, the particle will be (temporarily) trapped. By the sametunnelling mechanism, it will at some later time escape, leading to “decay” of theintermediate state. One can easily imagine that the metastable state so formedwill have a well-defined (and calculable), finite, mean lifetime. Now, according tothe Heisenberg principle of uncertainty, a state that only exists for a finite timeτ , say, cannot have a perfectly well-defined energy. The induced energy smearingwill be Γ≡∆E=~/τ . Provided Γ is not too large (i.e. τ not too small), we thusexpect the phenomenon to occur for a small range of energies around the centralresonant value. The (BW) resonant peak might therefore be expected to have awidth of order Γ in energy.

We shall now quantify such behaviour. Consider the problem wave-by-wave:since the barrier height varies with l, it is clear that for any given incident energyonly a single partial wave is likely to meet the resonance condition. Assuming theresult to be that the corresponding partial-wave amplitude attains its maximumfor this energy, then, since the amplitude is proportional to sinδl, the phase shiftδl is expected to pass through the value π/2 (or some odd multiple thereof). Thepeak partial cross-section should therefore be

σmax

l = (2l + 1)4π

k2. (5.3.186)

This can, of course, be verified by explicit calculation in a simple model. Forexample, the curves shown in Fig. 5.7 are calculated for a three-dimensional squarewell in the case l=1. The phase-shift δ0 is seen to vanish (as it should) for k→0,but to grow with k and to make a relatively rapid transition (when k∼1/R) fromzero to π, passing quickly through the resonant value π/2.

Let us now examine in more detail the variation of the phase-shift as a functionof k (or incident energy E). As we have already noted, the resonance conditionof an amplitude maximum must correspond to δl(k)→ (n+ 1

2)π (n∈Z), for some

partial wave l and for some resonant value of k, given by E=Er say. As shown inthe example (Fig. 5.7), the motion of δl in the resonance region is relatively rapid(and rising), we may thus expand cotδl in powers of (E−Er) and approximate asfollows:

cot δl = −c (E −Er) +O((E −Er)

2), (5.3.187)

with c a constant. In terms of the scattering amplitude, this leads to

fl(k) =1

k[cot δl − i ]≃ 1

k[−c(E − Er)− i ], (5.3.188)


σ1

k

12π

k2

kres

δ1(k)

π

12π

kkres

Figure 5.7: Schematic representation of the appearance of a BW resonance in terms ofthe partial cross-section (upper plot) and phase shift (lower plot). The curves correspondto the case l=1 in a three-dimensional square well.

which we may then rewrite as

= −12Γ

k[(E −Er) +12iΓ]

, (5.3.189)

where we have formally introduced the resonance width Γ, defined as

dcot δldE

∣∣∣∣E=Er

= −c ≡ − 2

Γ. (5.3.190)


Note that cotδl varying rapidly corresponds to Γ small, i.e. a narrow resonance.If we now assume, as is most often the case, that in the neighbourhood of such

a resonance the contributions of the other partial waves may be neglected, thenthe form of the cross-section in this region is

σl ≃ 4π

k2(2l + 1) (1

2Γ)2

(E − Er)2 + (1

2Γ)2

. (5.3.191)

This is just the BW formula derived earlier in Eq. (5.2.76). The peak value isjust 4π(2l+1)/k2, the maximum possible value allowed by unitarity. Providing Γis sufficiently narrow it is thus indeed the FWHM of the ensuing spectrum. Thecondition is clearly that Γ≪Er; Γ should also be sufficiently small that no otherpartial waves contribute appreciably in the region bounded by Er± 1

2Γ and that its

own variation with k may be neglected. Moreover, comparison with the discussionpreceding (5.2.76) shows that the choice of sign of the constant c above is notarbitrary: it is fixed by the fact that it must describe the decay of the metastableintermediate state.

5.3.8 The scattering of identical particles

A situation in which the scattering phase can have a non-trivial influence (viainterference) is in the scattering of identical particles. Consider first the case ofspinless particles and a spherically symmetric potential. Since the particles areidentical, after the collision it will be impossible to distinguish the projectile fromthe recoiling target. In the centre–of–mass frame the scattered wave must thereforebe of the form

e ik·x+e− ik·x+[fk(θ) + fk(π − θ)

] e ikrr, (5.3.192)

where x=x1−x2 is the relative position of the two particles. This leads to thefollowing cross-section:

dσ

dΩ=∣∣fk(θ) + fk(π − θ)

∣∣2

=∣∣fk(θ)

∣∣2 +∣∣fk(π − θ)

∣∣2 + 2Re[fk(θ)f

∗k (π − θ)

]. (5.3.193)

Note the constructive interference for θ=π/2: the cross-section is four times itsnaïve value. At other angles, however, no simple statement can be made.

Now, consider instead the case of two identical spin-1/2 particles. The Pauli ex-clusion principle obliges us to take either the spin-singlet combination in a spatiallysymmetric wave function or the spin-triplet combination in a spatially antisymmet-ric wave function. If the beam is not polarised, then the random spin alignments


will so combine to provide a ratio of 3 : 1—triplet : singlet, leading to

dσ

dΩ= 1

4

∣∣fk(θ) + fk(π − θ)∣∣2 + 3

4

∣∣fk(θ)− fk(π − θ)∣∣2

=∣∣fk(θ)

∣∣2 +∣∣fk(π − θ)

∣∣2 − Re[fk(θ)f

∗k (π − θ)

]. (5.3.194)

In this case then, in contrast, the interference is destructive for θ=π/2. This effectis an observable consequence of the different spin statistics obeyed by bosons andfermions and, as such, it is purely quantum mechanical in nature.


5.4 Bibliography

Caldirola, P., Cirelli, R. and Prosperi, G.M. (1982), Introduzione alla Fisica Teor-ica (UTET).

Gordon, J.P., Zeiger, H.J. and Townes, C.H. (1954), Phys. Rev. 95, 282.

Itano, W.M., Heinzen, D.J., Bollinger, J.J. and Wineland, D.J. (1990), Phys. Rev.A41, 2295.

Levinson, N. (1949), Phys. Rev. 75, 1445.

Lippmann, B.A. and Schwinger, J. (1950), Phys. Rev. 79, 469.

Maiman, T.H. (1960), Nature 187, 493.

Misra, B. and Sudarshan, E.C.G. (1977), J. Math. Phys. 18, 756.

Chapter 6

Relativistic Quantum Mechanics

In this final chapter we shall follow the extension of quantum mechanics to theinclusion of special relativity. So far, the theory we have developed is manifestlynot relativistically invariant. The Schrödinger equation depends on space andtime differently: it is quadratic in the former but linear in the latter. This is notsurprising since the quantisation procedure followed was, after all, based on non-relativistic classical mechanics. The natural step then is to apply our well-triedmethods to a relativistic classical equation of motion. Let us stress that such adevelopment exemplifies a principle that has become something of a guiding lightin fundamental physics: the notion of unification. In many physical situationswe find that different regimes require different treatments; it is then natural toattempt to produce an all-encompassing theory, which can be applied equally indifferent circumstances. When this is achieved often new and unexpected aspectsemerge.

6.1 The Klein–Gordon equation

6.1.1 Derivation of the Klein–Gordon equation

The first attempt to unify quantum mechanics and special relativity is due to Klein(1927) and Gordon (1926). The natural starting point is the standard energy–momentum relation for a free particle in special relativity:

E2 = p2c2 +m2c4, or E2/c2 − p

2 = m2c2, (6.1.1)

where m is the rest mass of the particle we wish to describe and c is the velocityof light. In a more compact four-vector notation, this becomes

p2 = pµpµ = m2c2, (6.1.2)

179

180 CHAPTER 6. RELATIVISTIC QUANTUM MECHANICS

where nowpµ ≡ (E/c,p) and pµ ≡ (E/c,−p). (6.1.3)

In the standard approach to quantisation, kinematical variables such as E and pare replaced with the corresponding operators:

E → i~∂

∂tand p → − i~∇. (6.1.4)

Such a procedure can be represented conveniently and compactly in four-vectornotation as follows

pµ ≡ (E/c,p) → i~ ∂µ ≡ i~∂

∂xµ≡ i~

(1

c

∂

∂t,−∇

). (6.1.5)

Note the negative sign in front of the spatial components.Substituting this into (6.1.2) leads to the following relativistic wave equation:

(∂µ∂µ +

m2c2

~2

)φ = 0, (6.1.6)

which is known as the Klein–Gordon (KG) equation (Klein, 1927 and Gordon,1926). For an even more compact notation, it is customary to define the four-dimensional Laplacian (or d’Alembertian) ≡∂µ∂µ, and thus write

(+

m2c2

~2

)φ = 0. (6.1.7)

We immediately see that the plane-wave solutions have precisely the same formas those of the Schrödinger equation: namely,

φ(x) = φ0 e− i

~p·x, (6.1.8)

where (and henceforth) it is understood that quantities such as p and x are fourvectors. Note however, that unfortunately the connection between E and p givenby (6.1.1) is ambiguous in the sign for E. At this stage we might be tempted tosimply ignore the negative-energy solutions as spurious. We shall see, however,that they do indeed have a profound physical significance.

6.1.2 Interpretation of the Klein–Gordon equation

We now wish to recover the usual interpretation of φ in terms of a probabilityamplitude. Recall the procedure for determining the form of the probability dens-

6.1. THE KLEIN–GORDON EQUATION 181

ity and corresponding current in the case of the Schrödinger equation for a freeparticle,

i~∂ψ

∂t= − ~

2

2m∇

2ψ. (6.1.9)

We multiply the given wave equation by ψ∗ and subtract its complex conjugate(taking care with the factor “ i ” on the left-hand side) multiplied by ψ to obtain

i~∂(ψ∗ψ)

∂t= − ~

2

2m

[ψ∗

∇2ψ − ψ∇2ψ∗]

= − ~2

2m∇·[ψ∗

∇ψ − ψ∇ψ∗] . (6.1.10)

Comparison of the above with the standard continuity equation,

∂ρ

∂t+∇·j = 0, (6.1.11)

where ρ is a density and j the corresponding current, leads to the following naturalidentification (the subscript S indicates Schrödinger):

ρS ≡ ψ∗ψ = |ψ|2 (6.1.12a)

jS ≡ ~

2m i

[ψ∗

∇ψ − ψ∇ψ∗] . (6.1.12b)

This also holds true in the presence of a potential represented by an Hermitianoperator, i.e. corresponding to a real, classical (probability conserving) potential.

Applying the same procedure to the KG equation leads to

ψ∗ψ − ψψ∗ = 0. (6.1.13)

Separating the space and time parts of , we find the same current as in theSchrödinger case, but the following candidate for the probability density:

ρKG =i~

2mc2

[ψ∗∂ψ

∂t− ψ

∂ψ∗

∂t

]. (6.1.14)

Exercise 6.1.1. Using the procedure outlined above derive the explicit expressionsfor the KG density and current.

Such an expression is evidently not positive definite and therefore does not lenditself to interpretation as a probability density. Note, however, that it does givethe correct form in the non-relativistic limit, providing we only consider positive-energy solutions. Let us rewrite the wave-function by separating out the dominant


part of the time dependence in this limit (E≃mc2),

ψ(x) = e−i~mc

2t Ψ(x, t), (6.1.15)

where we have implicitly taken a positive-energy solution and shall now assumethe time dependence of the first factor on the right-hand side to be much strongerthan that of the second; that is,

mc2

~Ψ ≫ ∂Ψ

∂t. (6.1.16)

Eq. (6.1.14) then reduces to

ρKG =i~

2mc2

[− i

~mc2Ψ∗Ψ+Ψ∗∂Ψ

∂t− i

~mc2ΨΨ∗ −Ψ

∂Ψ∗

∂t

]

non-rel.−−−−−→limit

ρS. (6.1.17)

where, in the limit, we have neglected the two terms Ψ∗∂Ψ/∂t and Ψ∂Ψ∗/∂t.It is then reasonable to assume that the problems of negative-energy solutionsand negative probabilities are related. In any case, historically, owing to thesedifficulties the Klein–Gordon equation was essentially abandoned until the correctinterpretation was uncovered through the Dirac∗ equation in 1928.

6.2 The Dirac equation

The observation that positive definiteness is lost owing to the quadratic energydependence in the starting point of E2=p

2c2+m2c4, with consequent quadratictime derivative, led Dirac (1928) to seek a linear, or first-order differential, equa-tion. To maintain the desired relativistic covariance, the equation should, indeed,be linear in both E and p or, equivalently, first order in both temporal and spatialderivatives.

6.2.1 A modern derivation

In a nutshell, Dirac’s idea was an attempt to avoid negative energies by effectivelytaking the square-root of the KG equation and thus write

γµpµ ϕ(x) = mcϕ(x), (6.2.1)

∗ The 1933 Nobel Prize for physics was awarded equally to Erwin Schrödinger and Paul AdrienMaurice Dirac “for the discovery of new productive forms of atomic theory.”

6.2. THE DIRAC EQUATION 183

where γµ is some new and as yet unknown dimensionless vector object (to bedetermined), necessary to render the left-hand side a scalar quantity, just as theright-hand side. Then, in order that the operator version γ·p =mc should agreewith the operator form of Einstein’s relation p2 =m2c2 (or, equivalently, the KGequation), we require

(γ·p)2 = p2 (6.2.2)

and therefore the γµ must be constant (otherwise [p,γ] 6=0 and the above equalitywould be impossible) and satisfy a Clifford algebra: γµ,γν= gµν . Indeed, it iseasy to see that with such an algebra we have

(γ·p)2 = γµpµγνpν = 1

2γµ, γνpµpν = gµνpµpν = p2. (6.2.3)

Finally then, Eq. (6.2.1) is just the Dirac equation in its simplest, operator, form.

6.2.2 The historical derivation

The approach actually followed by Dirac was to write

i~∂ψ

∂t= − i~cα·∇ψ + βmc2ψ ≡ Hψ, (6.2.4)

where the normalisation is chosen such that the objects α and β be dimension-less. These new quantities should also be constant if we are not to introducespurious forces into the dynamics. Moreover, the related energy–momentum equa-tion should also satisfy the Einstein relation or, equivalently, (6.2.4) should notcontradict the Klein–Gordon equation.

We may now apply E =H=− i~cα·∇+βmc2 once again to the above equationto obtain

( i~)2∂2ψ

∂t2=(− i~cα·∇+ βmc2

)2ψ ≡ H2 ψ. (6.2.5)

Expanding and writing out explicitly the middle bracket above gives

− ~2 ∂

2ψ

∂t2= −~

2c2[

α21

∂2

∂x12 + α2

2

∂2

∂x22 + α2

3

∂2

∂x32

]

+

[(α1α2 + α2α1)

∂2

∂x1 ∂x2+ · · ·

]ψ

− i~mc3(α1β + βα1)

∂

∂x1+ · · ·

ψ + β2m2c4ψ. (6.2.6)


Comparison with the KG equation (6.1.7) then leads to the following conditions:

αi, αj = 2δij , (6.2.7a)

αi, β = 0 (6.2.7b)

and β2 = 1. (6.2.7c)

It is evident that this algebra (a Clifford algebra, often called the Dirac algebra byphysicists) cannot be satisfied by ordinary numbers; a convenient representationmay be constructed using matrices.

From (6.2.7a & c), we see that all the matrices αi and β have eigenvalues ±1.They must also all be traceless, as we shall now demonstrate.

Proof. The proof exploits property (6.2.7c):

Tr(αi) = Tr(αiβ2) = Tr(βαiβ) = −Tr(αiββ) = −Tr(αi), (6.2.8)

where the second equality is obtained via the cyclic property of the trace operation(TrAB=TrBA).

Let us then, arbitrarily, take β as the following diagonal matrix (this willpartially determine our chosen representation):

β =

(1 00 −1

), (6.2.9)

where the ±1 represent unit sub-matrices respectively of rank n and m, say. Con-sider now the condition αi,β=0, from (6.2.7b); the kl element gives (with noimplicit summation)

(αi)kl(βkk + βll) = 0. (6.2.10)

Thus, if both k and l correspond to the same eigenvalue ±1, then (αi)kl=0,otherwise it may be non-zero. The α matrices must therefore take the form

α =

(0 α

′

α′′ 0

). (6.2.11)

The algebra then requires that

α′iα

′′j = 1n×n δij and α′′

i α′j = 1m×m δij . (6.2.12)

It is not difficult to convince oneself that this is not possible (for three differentmatrices αi) if either n or m is unity. However, for n=m=2, we already know of


a set of matrices satisfying such relations: the Pauli spin matrices:

σx =

(0 11 0

), σy =

(0 − ii 0

), σz =

(1 00 −1

). (6.2.13)

We thus find that the lowest-dimension representation of the Clifford algebradefined in (6.2.7) is of dimension four. One explicit form is

β =

(1 00 −1

)and α =

(0 σσ 0

), (6.2.14)

where the sub-matrices are now all 2×2.The novel implication is then that the wave-function must be represented by

a four-component spinor (not to be confused with a Lorentz four-vector). Theindices on such a spinor, as too those on the matrices α and β are often referredto as Dirac indices and the space in which they act, Dirac space. Note that itcannot be thought of as any sort of Lorentz vector since, for example, as we shallsee later, a rotation through 2π reverses its sign. Note further that there arethen naturally four possible distinct solutions to the Dirac equation for any givensystem. The spinor structure already encountered in the Pauli construction forincluding the spin of the electron in the Schrödinger formulation and the presenceof the Pauli matrices provide the clue that at least two components may, indeed,have to do with the spin of the electron.

It is convenient to rewrite the wave equation in a more manifestly Lorentzcovariant form, by defining the following set of matrices:

γ0 ≡ β and γ ≡ βα, (6.2.15)

which in the present representation are

γ0 =

(1 00 −1

)and γ =

(0 σ

−σ 0

). (6.2.16)

Expressed in terms of these matrices, the Clifford algebra is defined by

γµ, γν = 2gµν 1, (6.2.17)

where gµν is the usual Lorentz metric tensor. The Dirac equation then becomes∗

i~γµ∂µψ −mcψ = 0. (6.2.18)

∗ The version iγ·∂ψ=mψ (in natural units, i.e. c=1 and ~=1) is inscribed on a plaque honouringDirac at Westminster Abbey in London.


Or, introducing another convenient shorthand due to Feynman γ·a≡/a (where a isany four-vector), (

i~/∂ −mc)ψ = 0; (6.2.19a)

in operator language, this is just(/p−mc

)ψ = 0. (6.2.19b)

6.2.3 Probability current in the Dirac equation

Let us now examine the expressions for the probability density and current. Ap-plying the standard procedure to (6.2.4), we should multiply from the left by ψ†

(the Hermitian conjugate since it is a four-component spinor) and subtract theHermitian-conjugate equation multiplied on the right by ψ. Let us, however, at-tempt to make immediate use of the compact notation in (6.2.19a) by noticingthat this is the same equation as (6.2.4) multiplied on the left by γ0=β. Sinceγ0γ0=1, our procedure thus involves multiplication on the left by ψ†γ0. Indeed,it will soon become apparent that this quantity is rather common and thereforedeserves its own symbol: we shall define then

ψ ≡ ψ†γ0. (6.2.20)

Our standard procedure thus leads directly to

0 = ∂µ ψγµψ. (6.2.21)

Noting that, in four-vector notation, the continuity equation is

∂µ jµ = 0, (6.2.22)

with jµ := (cρ,j), we are thus led to identify the Dirac probability-density four-current with

c ψγµψ = jµD, (6.2.23)

where the factor c guarantees the correct dimensions. That is, the probabilitydensity and three-current corresponding to the Dirac equation are respectively

ρD = ψ†γ0γ0ψ and jD = c ψ†γ0γψ

= ψ†ψ = c ψ†αψ. (6.2.24)

Finally then, we have obtained an expression for the probability density that ismanifestly positive definite. Note also that the specific form of the spatial part ofthe current suggests that cα be interpreted as the velocity operator for the particle


so described.

6.2.4 Interpretation of the Dirac equation

Free-particle solutions

Let us now examine in detail the free-particle solutions to the Dirac equation.Inspecting the set of (four) equations component-by-component, we find that thenegative-energy solutions are still present. For a free zero-momentum particle, wehave (the indices 1,... ,4 label the solution, not the component)

ψ1,2 = e−i~mc

2t

(φ0

)and ψ3,4 = e+

i~mc

2t

(0χ

). (6.2.25)

In this representation then the upper (lower) two components correspond to pos-itive (negative) energy solutions. It will help to make sense of this situation if wecouple the spinor to some external force and examine its behaviour. The simplestnon-trivial example is a classical electromagnetic field—the interaction terms willbe constructed by applying the so-called principle of minimal coupling.

Coupling to an EM field

From classical electromagnetic theory, we know that the motion of a particle sub-jected to an external classical electromagnetic field (E and B) may be described byapplying the following transformations to the energy and momentum (the minimal-coupling principle):

E → E − eΦ and p → p− e

cA, (6.2.26)

with the usual relations for the electromagnetic fields

E = −1

c

∂A

∂t−∇Φ and B = ∇∧A, (6.2.27)

defining the scalar and vector potentials Φ and A, respectively. We may thendefine an equivalent four -potential,

Aµ ≡ (Φ,A), (6.2.28)

with which the minimal-coupling principle becomes

pµ → pµ − e

cAµ. (6.2.29)


For an electron in the presence of an electromagnetic field, we thus expect theDirac equation to take on the following (operator) form:

(/p−

e

c/A−mc

)ψ = 0. (6.2.30)

Let us now separate (6.2.30) into its spatial and temporal parts and rewrite itas

i~∂ψ

∂t= [H0 +Hint]ψ, (6.2.31)

with Hint=−eα·A+eΦ. Comparison with the corresponding classical expression−e

cv·A+eΦ again suggests the velocity interpretation for cα. Indeed, if we apply

Ehrenfest’s relations, for example Eq. (2.2.13), to the relativistic case, we obtainthe following operator equations:

dx

dt=

i

~[H,x] = cα ≡ v (6.2.32a)

anddπ

dt=

i

~[H,π]− e

c

∂

∂tA = e

[E +

1

cv∧B

], (6.2.32b)

with the usual kinetic momentum π≡p− ecA.

Negative-energy solutions and antimatter

First of all, let us examine the electrostatic term (∝Φ) in Eq. (6.2.30): it has theform

− eγ0Φ ψ. (6.2.33)

Recalling the structure of the matrix γ0, see Eq. (6.2.16), we immediately seethat this term (corresponding to the Coulomb interaction) has opposite signs forthe upper and lower pairs of components, i.e. the lower components behave verymuch like the electron: i.e. a particle having the same mass but positive charge.These two types of solutions are naturally interpreted as representing particles andantiparticles. It was just this observation that led Dirac to propose the existenceof antimatter before it had been detected experimentally. Shortly after Anderson(1933) discovered the antielectron, or, as it is now known, the positron.

Magnetic moments and spin states

Since we already know how to describe the EM interactions of an electron in thenon-relativistic case, in order to uncover the physical significance of the spinor, itis natural to examine the non-relativistic limit of the above equation. This is mostconveniently achieved by decomposing the four-component spinor into upper and


lower two-component parts, thus

ψ ≡(φχ

). (6.2.34)

The Dirac equation coupled to an electromagnetic field then becomes

i~∂

∂t

(φχ

)= cσ·π

(χφ

)+ eΦ

(φχ

)+mc2

(φ

−χ

). (6.2.35)

In the non-relativistic limit the rest-mass energy dominates the temporal depend-ence of the wave-functions and we thus once again write

(φχ

)≡ e−

i~mc

2t

(φχ

), (6.2.36)

where now the two-component spinors φ and χ are relatively slowly varying func-tions of time. The Dirac equation then becomes

i~∂

∂t

(φχ

)= cσ·π

(χφ

)+ eΦ

(φχ

)− 2mc2

(0χ

). (6.2.37)

Neglecting all terms that are small compared to mc2 (i.e. all kinetic and interactionenergies), the lower equation here may be approximated as

χ ≃ σ·π2mc

φ (6.2.38)

We thus see that the lower components χ in this approximation are to be consideredsmall as compared to the upper components φ. Roughly speaking, we have thatχ/φ∼v/c. Note, moreover, that the upper and lower components are thereforenot, in fact, independent.

Inserting the expression for χ into the upper equation leads to

i~∂

∂tφ =

[(σ·π)22m

+ eΦ

]φ. (6.2.39)

Now, a simple identity for the Pauli matrices is

σ·aσ·b = a·b+ iσ·(a∧b). (6.2.40)Thus,

σ·πσ·π = π2 + iσ·(π∧π)

= π2 − e~

cσ·B. (6.2.41)


Finally then, we obtain the following non-relativistic two-component equationfor φ:

i~∂φ

∂t=

[(p− e

cA)2

2m− e~

2mcσ·B + eΦ

]φ, (6.2.42)

which we immediately recognise as Pauli’s extension of the Schrödinger equation toinclude the spin degree-of-freedom of the electron and, in particular, its magneticmoment. The two components of φ clearly correspond to the two spin projections(up and down, say) of the spin-1/2 electron. Indeed, simplifying the equation furtherand keeping only the leading terms of the interaction with the magnetic field (whichis normally weak owing to the factor 1/c), we have

i~∂φ

∂t=

[p2

2m− e

2mc(L + 2S)·B + eΦ

]φ, (6.2.43)

where, as usual, L=r∧p is just the orbital angular momentum of the electronand S= 1

2~σ is the electron spin operator with eigenvalues ±1

2~. The remarkable

observation is that the normalised gyromagnetic ratio g for the electron is correctlypredicted to be precisely 2.∗†

Exercise 6.2.1. Derive Eq. (6.2.43) explicitly starting from (6.2.42).

6.2.5 The γ-matrix algebra

It is instructive at this point to consider the γ-matrix algebra a little more in detail.In calculations it will often turn out necessary to handle products and, in partic-ular, traces of products of such matrices. Not surprisingly perhaps, since physicalquantities must be independent of the representation adopted, there are also manyuseful identities for γ-matrix products that do not depend on the representation.Let us begin with the simplest trace of all: namely, Tr[1]=4. This number, beingthe dimensionality of the matrices, is clearly representation dependent; however,since it also appears in the normalisation of the states (see later), it will alwayscancel in any physical calculation.

The first non-trivial trace is Tr[γµγν ]. To evaluate it, we may exploit the cyclic∗ “When I realised that the equation contained the spin of the electron, and also the magneticmoment – everything needed for the properties of the electron – it was a surprise. A completesurprise.” P.A.M. Dirac

† More accurate experimental determinations reveal that the gyromagnetic ratio is not, in fact,exactly two but slight larger. This can be accounted for by considering the effects of vacuumfluctuations—this requires, however, a complete quantum field theory.


property of the trace: namely, Tr[AB· · ·C]=Tr[B · · ·CA]. We thus have

Tr[γµγν

]= Tr

[γνγµ

]= 1

2Tr[γµγν + γνγµ

]

= 12Tr[2gµν · 1

]= gµν Tr

[1

]= 4gµν. (6.2.44)

We might have anticipated such a form for the result by noting that the trace mustbe an invariant tensor and the only such two-index tensor available is precisely gµν .Thus, Tr[γµγν ]= cgµν ; the constant c could then have been found by contractionwith gµν . Using the same technique, it is possible to show that

Tr[γµγνγργσ

]= 4

[gµν gρσ − gµρ gνσ + gµσ gνρ

](6.2.45)

Exercise 6.2.2. Derive the previous formula and extend it (by iteration) to productsof arbitrary but even numbers of γ-matrices.

We now introduce a further important matrix, usually denoted γ5 (though itis not a true γ-matrix):

γ5 ≡ γ5 ≡ i γ0 γ1 γ2 γ3 =

(0 1

1 0

)

= 14!iεµνρσ γ

µ γν γρ γσ, (6.2.46)

where we have also introduced εµνρσ the alternating (or Levi-Civita) tensor of rankfour, defined by

εµνρσ =

+1 if (µνρσ) is an even permutation of (0123),−1 if (µνρσ) is an odd permutation of (0123),0 otherwise.

(6.2.47)

Note that ε0123=−ε0123=−1 with this convention.∗ Note too that the definitionof γ5 is dependent on the overall dimensionality of space–time and its extension todifferent dimensionalities can be problematic. A significant property of γ5 is thatit anticommutes with each of the other four matrices:

γ5, γ

µ

= 0 for µ = 0, 1, 2, 3. (6.2.48)

Moreover, γ5γ5=1 is a trivial, representation-independent, identity. We shall seethat this matrix is associated with pseudoscalar and pseudovector (or axial-vector)quantities; it is therefore important in the description of spin.∗ Beware: in some textbooks the opposite sign convention is adopted.


We now can prove an important γ-matrix identity:

Tr[γµ1 γµ2 · · · γµn

]= Tr

[γ5 γ5 γ

µ1 γµ2 · · · γµn]

= (−1)nTr[γ5 γ

µ1 γµ2 · · · γµn γ5]

(anticommutation)

= (−1)nTr[γ5 γ5 γ

µ1 γµ2 · · · γµn]

(cyclic property)

= (−1)nTr[γµ1 γµ2 · · · γµn

]. (6.2.49)

From this it immediately follows that

Tr[γµ1 γµ2 · · ·γµn ] = 0 for n odd ; (6.2.50)

i.e. when tracing, only products of even numbers of γ-matrices (not counting anyappearances of γ5) survive. Note also that

Tr[γ5γ

µγνγργσ]= 4iεµνρσ. (6.2.51)

The following is a useful identity (due to Chisholm, 1963):

γµ γν γρ = gµν γρ − gµρ γν + gρν γµ + iεµνρσ γ5 γσ. (6.2.52)

Products of large numbers of γ-matrices may be reduced to shorter strings by re-peated application of this relation. It is also easy to derive the following simplifyingidentities:

γµ /a γµ = −2 /a, (6.2.53a)

γµ /a /b γµ = 4 a·b, (6.2.53b)

γµ /a /b /c γµ = −2 /c /b /a. (6.2.53c)

The numerical factors above depend on both the space–time dimensionality andthat of the representation. However, up to a constant factor, all of the relationsderived in this section depend only on the defining algebra and not on the particularrepresentation.

A final object, useful to define, is the antisymmetric product of two γ-matrices,

σµν ≡ i

2

[γµ, γν

]. (6.2.54)

In the chosen representation we have

σ0i = iαi = i

(0 σiσi 0

)and σij = iεijk

(σk 00 σk

). (6.2.55)


Using this new matrix, we can write another useful relation,

γµ γν = gµν − iσµν . (6.2.56)

A final property of the γ-matrices themselves, that again is representationindependent, is their behaviour under Hermitian conjugation:

γ0† = γ0 (6.2.57a)

and γi† = −γi for i = 1, 2, 3 (6.2.57b)

or, more compactly, γµ† = γµ = γ0γµγ0. (6.2.57c)

This property ensures that the Hamiltonian is Hermitian, as it should be.

6.2.6 Lorentz transformations

Covariance of the Dirac equation

We should now verify the Lorentz covariance of the Dirac equation.∗ That is, theequation must retain its form under a Lorentz transformation. Such a transform-ation is effected as follows:

x′ν ≡ Λνµ x

µ, (6.2.58)

where the transformation matrix Λ depends on the relative displacements, velo-cities and spatial orientations of the two coordinate systems. Invariance of theinfinitesimal proper-time interval dτ 2≡gµνdxµdxν =dxµdxµ requires that

Λσµ Λσν = gµν . (6.2.59)

Such a relation define both proper and improper Lorentz transformations: theformer have det(Λ)=+1 and are standard rotations and Lorentz boosts while thelatter have det(Λ)=−1 and are space and time reflections (or inversions).

To prove invariance we must now show that it is possible to construct a Lorentztransformation from one reference frame to another while preserving the form ofthe Dirac equation. In the transformed system we should thus have

[i~ γ ′µ ∂

∂x′µ−mc

]ψ′(x′) = 0. (6.2.60)

∗ While this is, indeed, a somewhat academic exercise (since the equation was constructed pre-cisely with the aim of maintaining Lorentz covariance), it is nonetheless important and usefulto understand the construction of the operators generating Lorentz boosts and rotations in thecase of a Dirac spinor.)


The transformed γ-matrices, γ ′, must satisfy the same defining anticommutationrelations; i.e., as before, γ ′†

0 =γ ′0 and γ

′†=−γ′. The transformed Hamiltonian

is thus still Hermitian. Now, it can be shown (though the proof is non-trivial;see, e.g. Schweber, 1962) that all such sets of matrices are related by unitarytransformations U :

γ ′µ = U † γµ U with U † = U−1; (6.2.61)

the distinction between γµ and γ ′µ is therefore superfluous and so we may write(i~/∂

′ −mc)ψ′(x′) = 0. (6.2.62)

Let us now introduce the required transformation:

ψ′(x′) = ψ′(Λx) = S(Λ)ψ(x) = S(Λ)ψ(Λ−1x′), (6.2.63)

where S(Λ) is a 4×4 matrix acting in Dirac space. Since the inverse must alsoexist, we may write

ψ(x) = S−1(Λ)ψ′(x′) = S−1(Λ)ψ′(Λx), (6.2.64)

which must clearly be equivalent to

ψ(x) = S(Λ−1)ψ′(Λx). (6.2.65)

And so we have the natural identification

S(Λ−1) = S−1(Λ). (6.2.66)

We must now find S. To do this, we re-express the original Dirac equation(6.2.19a) in terms of the transformed wave-function,

[i~ γµ

∂

∂xµ−mc

]S−1(Λ)ψ′(x′) = 0, (6.2.67)

and multiply from the left with S(Λ); this leads to[i~S(Λ) γµS−1(Λ)

∂

∂xµ−mc

]ψ′(x′) = 0. (6.2.68)

Now,∂

∂xµ=

∂x′ν

∂xµ∂

∂x′ν= Λν

µ

∂

∂x′ν(6.2.69)


and we therefore have[i~S(Λ) γµS−1(Λ) Λν

µ

∂

∂x′ν−mc

]ψ′(x′) = 0. (6.2.70)

We thus require that

S(Λ) γµS−1(Λ) Λνµ = γν , (6.2.71a)

or, equivalently,Λν

µ γµ = S−1(Λ) γνS(Λ). (6.2.71b)

This then is the fundamental defining relation for S.Let us first construct S for an infinitesimal transformation,

Λνµ = δνµ +∆ων

µ (6.2.72a)with

∆ωνµ = −∆ωµν , (6.2.72b)

as required by (6.2.59). For example, an infinitesimal Lorentz boost of velocity∆β=∆v/c along the x-axis is generated by

∆ω01 = ∆β (6.2.73a)while

∆ω12 = −∆φ (6.2.73b)

generates a rotation about the z-axis through an infinitesimal angle ∆φ.

Exercise 6.2.3. Verify explicitly the statements just made with respect to the formof boosts and rotations.

We may expand S in powers of the infinitesimal parameter ∆ωµν and write

S ≃ 1− i4Σµν ∆ω

µν , (6.2.74)

where Σµν is necessarily a 4×4 matrix in Dirac space with Σνµ=−Σµν . Insertingthe infinitesimal forms for S and Λν

µ into the defining equation for S, we have

∆ωµνγ

ν = − i4∆ωρσ

(γµΣρσ − Σρσγ

µ). (6.2.75)

Since the generators ∆ωµν are arbitrary and antisymmetric, we finally obtain

2i[δνρ γσ − δνσ γρ

]=[γν ,Σρσ

]. (6.2.76)

It is not difficult to see that such a relation is satisfied by Σρσ =σρσ. The standard


form for a general infinitesimal Lorentz transformation may thus be taken as

S ≃ 1− i4σµν ∆ω

µν = 1+ 18

[γµ, γν

]∆ωµν . (6.2.77)

Boosts and rotations

It is then straightforward to construct finite transformations by forming suitableproducts of the infinitesimal forms above. Let us now write

∆ωµν = ∆ω Iµν , (6.2.78)

where, for a boost, ∆ω=∆β is the infinitesimal velocity or, for a rotation, ∆ω=∆φis the infinitesimal angle and Iµν will be determined by the direction of the boostor rotation axis. Indeed, there are clearly six such independent (antisymmetric)matrices: generating the three possible boosts and three rotations. For example,if we choose to perform a boost along the z-axis, then

Iνµ =

0 0 0 −10 0 0 00 0 0 0

−1 0 0 0

, (6.2.79)

whereby we haveI03 = I30 = −I03 = +I30 = −1. (6.2.80)

The finite transformation is then constructed as follows (∆ω=ω/N):

Λνµ = lim

N→∞

(g +

ω

NI)ν

σ1

(g +

ω

NI)σ1

σ2

· · ·(g +

ω

NI)σN−1

µ

=(eωI)ν

µ,

which, noting that I3 = I (although I2 6= 1), may be rewritten as

=[cosh(ωI) + sinh(ωI)

]νµ

=[1− I2 + I2 coshω + I sinhω

]νµ

=

coshω 0 0 − sinhω

0 1 0 0

0 0 1 0

− sinhω 0 0 coshω

, (6.2.81)


with tanhω=β, coshω=γ and sinhω=βγ (γ=1/√1−β2). The generalisation to

any direction should be obvious.The corresponding spinor transformations may then be constructed in a similar

manner:ψ′(x′) = S ψ(x) (6.2.82)

with

S = limN→∞

(1− i

4

ω

NσµνI

µν

)N

= exp

(− i

4ω σµνI

µν

). (6.2.83)

For a boost along the z-axis we have

S = SB(z) = exp

(i

2ω σ03

)(6.2.84)

and analogously, for a rotation about the z-axis (with I12=−I21=−1),

S = SR(z) = exp

(i

2ω σ12

). (6.2.85)

Note the similarity with the corresponding form for the two-component (non-relativistic) Pauli-spinor case: namely,

φ′(x′) = ei2ω σ3 φ(x). (6.2.86)

In the case of rotations applied to a spinor, the factor 1/2 in the exponenthas the consequence that, as we have already seen earlier, a rotation through 2πintroduces a minus sign. Therefore, to return the spinor to its original value, arotation through 4π must be performed. This property is related to the Fermi–Dirac statistics (or Pauli exclusion principle) obeyed by fermions. Recall thatthere is no incongruity here since a spinor is not a physical quantity. Indeed, theimplicit demand is that any observable depend on spinor bilinears, whence the signproblem disappears.

Note, moreover, that S is Hermitian for rotations but not for boosts:

S†R = e−

i4σ†ij ω

ij

= e−i4σij ω

ij

= S−1R (6.2.87a)

whileS†B = e−

i4σ†0i ω

0i

= e+i4σ0i ω

0i

6= S−1B (6.2.87b)


The following does hold, however, for general Lorentz transformations:

S−1 = γ0 S† γ0. (6.2.88)

This may easily be demonstrated via a power-series expansion of the exponentialforms just given, exploiting the commutation properties

γ0, γi

= 0 and

[γ0, σij

]= 0. (6.2.89)

Covariance of the continuity equation

We can now demonstrate the Lorentz covariance of the continuity equation. Todo so, it is sufficient to show that the current defined for the Dirac equation(6.2.23) has the correct transformation properties. We first note that the quantityψ introduced earlier has a natural transformation:

ψ′(x′) = ψ′†(x′) γ0

= ψ†(x)S† γ0

= ψ†(x) γ0 S−1 = ψ(x)S−1. (6.2.90)

Starting then from the transformed current, we find

j′µ(x′) = c ψ′(x′) γµ ψ′(x′)

= c ψ(x)S−1 γµ S ψ(x)

= c ψ(x) Λµν γ

ν ψ(x) = Λµν j

ν(x), (6.2.91)

as required. From this it also immediately follows that the left-hand side of

∂µ jµ(x) = 0 (6.2.92)

is a Lorentz invariant (as indeed it should be).It is important to recall that the individual components of jµ(x) depend on

the reference frame; in particular, the density ρ(x)=ψ†ψ=ψγ0ψ transforms as atemporal component.


6.2.7 Discrete transformations (CPT)

Spatial inversion (parity)

As already noted, the parity transformation is an improper Lorentz transformation.It is defined by x

′=−x and t′= t; that is,∗

Λµν =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

. (6.2.93)

The problem now posed is to find the corresponding transformation P for the Diracspinor ψ(x) and γ-matrices. The obvious requirements are

P−1 γ0 P = +γ0 (6.2.94a)

andP−1 γi P = −γi. (6.2.94b)

From the γ-matrix algebra, we find that the following has the required properties:

P ≡ e iφ γ0, (6.2.95)

where φ is an arbitrary phase of no physical significance; it is usually conventionallyset to zero.

The spinor parity transformation is thus

ψ′(x′) = ψ′(−x, t) = e iφ γ0 ψ(x, t). (6.2.96)

Since γ0=diag(1,1,−1,−1), we see that the upper and lower components (corres-ponding to particle and antiparticle solutions) must have opposite intrinsic parit-ies.†

Time-reversal

Recall that in non-relativistic quantum mechanics the time-reversal operator is an-tilinear ; we shall find that so too is the version appropriate for the Dirac equation.We seek the transformation T such that

ψ′(x′) ≡ T ψ(x), (6.2.97)∗ In view of its appearance, it is perhaps worth warning the reader not to confuse this matrixwith the tensor gµν .

† Note that this statement holds only for fermions and not for bosons.


with x′=x and t′=−t. Applying such a transformation to the Dirac equation

(6.2.4), we obtain (suppressing the superfluous x dependence)

−T i T−1~∂ψ′(t′)

∂t′= −T i T−1

~cα′·∇ψ′(t′) + β ′mc2ψ′(t′)

≡ H ′ ψ′(t′). (6.2.98)

Since we cannot arrange for H ′=THT−1=−H (especially if we consider the coup-

ling to an external classical electromagnetic field), we are thus forced to takeT i T−1=− i ; i.e. the transformation includes complex conjugation. And we there-fore write

ψ′(x′) = T ψ(x) = Tψ∗(x). (6.2.99)

To maintain H ′=H , we require

α′ = Tα

∗T

−1 = −α (6.2.100a)and

β ′ = T β∗T

−1 = β, (6.2.100b)

thus, in the standard representation T commutes with α2 (imaginary) and β (real)while it anticommutes with α1 and α3 (both real). The natural choice is therefore

T = e iφ α1 α3 = − e iφ γ1 γ3, (6.2.101)

with again an arbitrary phase φ. It is conventional to take

T = i γ1 γ3. (6.2.102)

Charge conjugation

Finally, since we have interpreted the lower components as corresponding to anantielectron (or positron), we are forced to examine the role of the transformationthat renders the Dirac equation for antimatter. Indeed, we could clearly havestarted from an equation aimed at describing the positron; in which case in placeof

(i /∂ − e

c/A−mc

)ψ = 0, (6.2.103a)

we should have written (i /∂ +

e

c/A−mc

)ψ = 0, (6.2.103b)

where the sign in front of the term ec/A has simply been negated.

The charge conjugation operation should then transform between the two cases.To change the relative sign between the terms i /∂ and e

c/A (with A real—a classical


external field), it is sufficient to take the complex conjugate of the entire equationand thus (6.2.103b) becomes

[(− i∂µ +

e

cAµ)γ∗µ −mc

]ψ∗ = 0. (6.2.104)

We thus now require a matrix K, say, such that

K γ∗µK−1 = −γµ. (6.2.105)

In the standard representation only γ2 is imaginary and we may thus clearly use

K = e iφ γ2, (6.2.106)

where, as always, there is an arbitrary phase φ. The spinor transformation is then

Cψ ≡ Cψ T = Kψ∗ = K γ0 ψ T, (6.2.107)

where we have exploited the fact that ψ∗=γ0ψT. The transformation sought isthus just

C = K γ0 = i γ2 γ0, (6.2.108)

where we have applied the conventional phase choice (C is real).

6.2.8 Spinor bilinears

Using the γ-matrices and their products, 16 linearly independent 4×4 matricesmay be constructed:∗

ΓS ≡ 1, ΓP ≡ γ5, ΓV ≡ γµ, ΓA ≡ γ5γµ, ΓT ≡ σµν , (6.2.109)

where the suffixes S, P , V , A and T indicate the tensorial nature of the objectsunder Lorentz transformations: S= scalar, P =pseudoscalar, V =vector, A=axialvector (or pseudovector) and T = tensor. Note that the numbers of linearly inde-pendent objects represented by each category are NS =1, NP =1, NV =4, NA=4and NT =6 (the total being 16).

Let us now demonstrate that the set of matrices listed above is indeed completeand linearly independent:

1. For each ΓX , Γ2X =±1.

2. (a) For each ΓX (excluding ΓS), there exists ΓY with ΓX ,ΓY =0.∗ Note that only ΓS , the identity matrix, has non-vanishing trace. Moreover, the hermiticity,anti-hermiticity requirements (6.2.57c) reduce the 32 complex independent matrices to 16.


(b) Therefore, Tr[ΓX ]=0 (excluding ΓS). To see this, consider the followingtrace: ±Tr[ΓX ]=Tr[Γ2

Y ΓX ]=−Tr[ΓY ΓXΓY ]=0.

3. For each pair ΓX ,ΓY (X 6=Y ), there exists ΓZ 6=ΓS such that ΓXΓY =ΓZ ;this can be seen by inspection.

4. Now, suppose there exist coefficients cX such that∑

X cXΓX =0. Multiplyingby ΓY and tracing, we find cY =0.

We may therefore take the above set as a complete basis for the 4×4 matrices.Moreover, the following set then exhausts the possibilities for spinor bilinears:

ψΓSψ, ψΓPψ, ψΓV ψ, ψΓAψ, ψΓTψ. (6.2.110)

The third is just the vector current already constructed while the others havepossible analogous roles in other theories (e.g. the weak interaction or effectivetheories of the strong interaction). Noting that

[γ5, σµν ] = 0, (6.2.111)

we see that γ5 commutes with the matrix S for any proper Lorentz transformationbut anticommutes with the parity transformation matrix P. The general Lorentztransformation properties of the spinor bilinears are thus

ψ′(x′)ψ′(x′) = ψ(x)ψ(x) (scalar), (6.2.112a)

ψ′(x′)γ5ψ′(x′) = det(Λ)ψ(x)γ5ψ(x) (pseudoscalar), (6.2.112b)

ψ′(x′)γµψ′(x′) = Λµν ψ(x)γ

νψ(x) (vector), (6.2.112c)

ψ′(x′)γ5γµψ′(x′) = det(Λ) Λµ

ν ψ(x)γ5γµψ(x) (axial vector), (6.2.112d)

ψ′(x′)σµνψ′(x′) = Λµρ Λ

νσ ψ(x)σ

ρσψ(x) (rank-2 tensor). (6.2.112e)

We should note here that the only known method at present for constructinginteractions is to use current–current combinations of the above five types. It is aninteresting and important consequence of their properties under the three discretetransformations of C, P or T that, while it is possible to construct interaction termsthat violate the corresponding symmetries individually (or the product of any pair),the triple product CPT is always a symmetry. For example, an interaction thatcouples a vector and an axial-vector will violate parity invariance, however theCPT transformation properties of ψγνψ and ψγ5γ

νψ are identical. Equally, animaginary phase (e.g. in a complex coupling constant) will violate time-reversalinvariance or, equivalently, the product CP . An example of the relevance of CPTinvariance is that particle and antiparticle inevitably have the same mass; i.e.


we cannot decouple the upper and lower components while maintaining CPTinvariance.

6.2.9 General free-particle solutions

We have already written down the solutions to the Dirac equation for a free particleat rest in (6.2.25); they may be written collectively as

ψa(x) = wa(0) e−i~ǫa mc

2t, (6.2.113)

where, recall, a=1,... ,4 labels the chosen solution (and not a component), thezero argument of the wa(0) spinor indicates zero three-momentum and

ǫa =

+1 a = 1, 2,

−1 a = 3, 4.(6.2.114)

It will turn out convenient (for reasons of covariance) to choose the spinor nor-malisation such that the natural definition of the density w†w= wγ0w actuallycorresponds to an energy density. For a particle at rest, we thus set w†w=2mc2

(the two is a conventional choice).∗ The four associated spinors are then simply

w1(0) =√

2mc2

1000

, w2(0) =

√2mc2

0100

,

w3(0) =√

2mc2

0010

, w4(0) =

√2mc2

0001

.

(6.2.115)

Note that these are all eigenstates of the 4×4 matrix σ3≡σ12; to understand thisdefinition recall (6.2.55).

The solutions for a particle of arbitrary velocity and spin may now be easilyconstructed by using the Lorentz transformations derived in the previous section.Boosting to a system with relative velocity −v will generate the spinor for aparticle of velocity +v. Likewise, the prior application (i.e. before any boost:boosts and rotations clearly do not commute) of a rotation will allow an arbitraryspin direction.

∗ The only apparently more natural convention adopted in many older texts is w†w=1. However,this creates unfortunate differences between the natural definitions of Lorentz-invariant phase-space measures for fermions and bosons. It leads, moreover, to spurious singularities in the caseof massless fermions (such as neutrinos in the standard model).


The space–time dependence is most easily revealed by writing (6.2.113) in amore explicitly covariant form, using

mc2 t = p0 x0 = pµ x

µ, (6.2.116)

where p and x indicate the particle in its rest frame while pµ≡Λµν p

ν and xµ≡Λµ

ν xν refer to the boosted system. We choose (here and henceforth) p0≡E/c=√

p2+m2c2. Note that the positive- and negative-energy solutions do not mix un-

der either proper Lorentz transformations or spatial inversion. This follows fromthe fact that since p2=m2c2>0 the four-momenta pµ lie inside the forward andbackward light cones respectively in p-space. Under the aforementioned trans-formations the forward and backward light cones (and hence the positive- andnegative-energy solutions) remain distinct. We should stress that the non-mixingrefers to the sign in the exponential; on the other hand, the components do mix,as we shall now see.

Boosted solutions—arbitrary momentum direction

To boost the spinors, along the positive i-axis say, we apply the generalisation ofthe transformation defined by (6.2.84), where now though tanhω=−β since weare boosting to a system with velocity −v. We thus have

SB(−v) = exp(i2ω σ0i

)= cosh ω

2+ iσ0i sinh ω

2, (6.2.117)

where we have exploited the identity ( iσ0i)2=1 in the power-series expansions.Since the matrix iσ0i is given by

− iσ0i = αi =

(0 σiσi 0

), (6.2.118)

we see that the upper and lower components of the spinor are mixed by a boost.Using standard trigonometric identities, we may re-express the above matrix

in terms of the energy and/or momentum:

tanh ω2=

tanhω

1 +√

1− tanh2 ω=

−β1 +

√1− β2

= −

√E −mc2

E +mc2, (6.2.119a)

thus

cosh ω2=

√coshω + 1

2=

√γ + 1

2=

√E +mc2

2mc2(6.2.119b)


and

sinh ω2= −

√E −mc2

2mc2. (6.2.119c)

In order to generalise to an arbitrary direction, we take the boost tensor Iµν as

Iµν =

0 − cos θ1 − cos θ2 − cos θ3− cos θ1 0 0 0

− cos θ2 0 0 0

− cos θ3 0 0 0

, (6.2.120)

where the cosθi are the direction cosines of the velocity vector v. The transform-ation matrix thus becomes

σµνIµν(v) = 2

∑

i

σ0i cos θi = −2iα·v =−2iα·pc√E2 −m2c4

, (6.2.121)

which gives

SB(−v) = exp

(−ω2

α·pc√E2 −m2c4

)

=

√E +mc2

2mc2

1

σ·pcE +mc2

σ·pcE +mc2

1

. (6.2.122)

The general form of a free-particle state is then

ψa(x) = wa(p) e−i~ǫap·x, (6.2.123)

where the spinor w(p) is

w(p) =

√E +mc2 1√E −mc2 σ·p

⊗ χ (positive energy),

√E −mc2 σ·p√E +mc2 1

⊗ χ (negative energy),

(6.2.124)


with two-component spinor χ=(10

)or(01

).

One can then show that the spinor w(p) satisfies the following relations:

(/p− ǫamc)wa(p) = 0, (6.2.125a)

wa(p) (/p− ǫamc) = 0, (6.2.125b)

wa(p)wb(p) = 2mc2 ǫa δab, (6.2.125c)4∑

a=1

ǫa waα(p) w

aβ(p) = 2mc2 δαβ, (6.2.125d)

where the adjoint spinor wa(p)≡wa†(p)γ0 has been introduced and α, β in thelast line are Dirac indices. The relations (6.2.125a & b) are just the Dirac equationand its Hermitian conjugate for the spinors wa(p) and wa(p). The normalisationof the boosted spinor is given by

wa†(p)wb(p) = 2E δab, (6.2.126)

i.e. such a bilinear transforms as the temporal component of a four-vector, as itshould.

Note also that (6.2.125c) is an orthogonality condition between spinors of pos-itive and negative energy corresponding to the same spatial momentum p. Again,one sees that the positive- and negative-energy solutions correspond to physicallydistinct states. Finally, (6.2.125d) is just a completeness relation for the four Diracspinors wa(p), for a=1,... ,4. To prove this in general, note that it holds triviallyfor the spinors at rest and thus a suitable Lorentz transformation leads to thedesired result.

Exercise 6.2.4. Derive explicitly the relations (6.2.125a–d).

Rotated solutions—arbitrary spin direction

Thus far we have shown how to construct spinors for arbitrary spatial momenta,now let us examine the problem of spin polarisation. The states defined in (6.2.115)correspond to particles having spin vectors aligned parallel or antiparallel to thez-axis. Again, we may first apply a rotation to the rest-frame spinors to obtain anyspin direction and then apply a boost (as already remarked, boosts and rotationsdo not commute). A rotation is applied via the operator

SR = ei2φσ·n, (6.2.127)

where here σ is the 4×4 matrix with the corresponding Pauli σ matrices as thediagonal blocks. The spin vector s is defined to be a unit vector. The states


w(p,s) so generated satisfy

σ·sw(p, s) = w(p, s). (6.2.128)

It is convenient to introduce a spin four-vector sµ and thus define a spinor u(p,s)corresponding to a positive-energy solution of the Dirac equation of momentum pµ

and spin sµ. That is,(/p−mc) u(p, s) = 0. (6.2.129)

The extension of the spin vector to four dimensions is made by taking sµ=(0,s)in the rest frame and applying a boost: sν =Λν

µsµ. Now, since pµ=(mc,0) then

clearly p·s=0. We thus have the following useful conditions on sµ:

p·s = 0

s·s = −1(6.2.130)

And in the rest frame we have

σ·s u(p, s) = u(p, s). (6.2.131)

Similarly, we can introduce the spinor v(p,s) for negative-energy solutions withpolarisation −s in the rest frame, such that

(/p +mc) v(p, s) = 0 (6.2.132a)and

σ·s v(p, s) = −v(p, s). (6.2.132b)

In terms of the original spinors wa(p) the new spinors u(p,s) and v(p,s) are

w1(p) = u(p,+ηz),

w2(p) = u(p,−ηz),w3(p) = v(p,−ηz),w4(p) = v(p,+ηz),

(6.2.133)

where ηz is the four-vector (0,0,0,1) in the particle rest frame. An arbitrary spinoris thus determined by its four-momentum pµ, its covariant spin vector sµ and thesign of its energy.

It is also convenient to construct projection operators for well-defined mo-mentum and spin states. It is left as an exercise to demonstrate∗ (or verify) that

∗ A useful identity is /a/a=a2, for any simple (i.e. self-commuting) four-vector aµ.


the following operators

Λa ≡ (ǫa/p +mc)/2mc,

or Λ± ≡ (±/p+mc)/2mc (6.2.134a)

and Σ(s) ≡ 12(1 + γ5 /s), (6.2.134b)

where Λ± projects onto positive- and negative-energy states, are precisely what weseek. That is, they satisfy

Λ2±(p) = Λ±(p), (6.2.135a)

Λ+(p) Λ−(p) = 0 (6.2.135b)

and Λ+(p) + Λ−(p) = 1, (6.2.135c)

while Σ(s) u(p, s) = u(p, s), (6.2.135d)

Σ(s) v(p, s) = v(p, s) (6.2.135e)

and Σ(−s) u(p, s) = 0 = Σ(−s) v(p, s), (6.2.135f)

Exercise 6.2.5. Verify that Λ±(p) and Σ(±s) commute.

Using Λ±(p) and Σ(±ηz), we may then construct the four complete spin–momentum projectors:

P1(p) = Λ+Σ(+ηz),

P2(p) = Λ+Σ(−ηz),P3(p) = Λ−Σ(−ηz),P4(p) = Λ−Σ(+ηz).

(6.2.136)

The negative-energy states that we have been forced to introduce thus corres-pond to eigenfunctions of momentum with eigenvalue −p, while the “spin-up” and“spin-down” states have rest-frame eigenvalues for the operator σz of −1 and +1respectively.

6.2.10 Wave-packets

Now, there are still surprises in store when we study the construction of wave-packets for Dirac particles. Let us start by attempting to construct a packet usingonly positive-energy states:

ψ(+)(x, t) =

∫d3p√

(2π~)32E

∑

±s

b(p, s)u(p, s) e−i~p·x . (6.2.137)


The normalisation of the coefficients b(p,s) is determined by our conventionalnormalisation of the spinors u(p,s) and by requiring:

1 ≡∫

d3xψ(+)†(x, t)ψ(+)(x, t)

=

∫d3p

2E

∑

±s,±s′

b∗(p, s′) b(p, s) u†(p, s′) u(p, s)

=

∫d3p∑

±s

∣∣b(p, s)∣∣2 . (6.2.138)

We should now evaluate the mean current carried by such a wave-packet. Todo this, we shall need a useful relation for a generalised current, known as theGordon decomposition. Let ψ1,2(x) be two solutions of the free Dirac equation,(/p−mc)ψ(x)=0, then (here p is an operator)

ψ2γµψ1 =

1

2mc

[ψ2 (p

µψ1)− (pµψ2)ψ1

]− i

2mcpν(ψ2σ

µνψ1

), (6.2.139)

which in terms of the spinors u may be written, equivalently (here p is now theeigenvalue),

u(p′)γµu(p) = u(p′)

[(p′ + p)µ

2mc− iσµν(p′ − p)ν

2mc

]u(p). (6.2.140)

To derive this relation, we take the Dirac equations for u(p) and u(p′), multiply onthe left with u(p)γµ and u(p′)γµ respectively; then, adding the complex conjugateof the second to the first leads to

0 = u(p′) γµ [/p−mc] u(p) + u(p′) [/p′ −mc] γµ u(p) (6.2.141)

and finally application of (6.2.56) to the products γµ/p and /p′γµ provides the desired

result. Note that decomposition (6.2.140) reveals the Dirac current to be a sum ofcontributions, the first of which corresponds to that of the non-relativistic theorywhile the second is a new term, a spin current.

Exercise 6.2.6. Derive explicitly the Gordon decomposition (6.2.140).

Let us now evaluate the mean current:

J (+)µ =

∫d3x c ψ(+)(x, t) γµ ψ(+)(x, t)


=

∫d3p

2E

∑

±s

∣∣b(p, s)∣∣2 c u†(p, s) γµ u(p, s)

and, applying (6.2.140) with p′ = p, this becomes

=

∫d3p

2E

∑

±s

∣∣b(p, s)∣∣2 p

µ

mu†(p, s) u(p, s)

=

∫d3p∑

±s

∣∣b(p, s)∣∣2 c

2pµ

E. (6.2.142)

With the chosen normalisation this leads to the natural identification

〈cα〉+ = J(+) =

⟨c2p

E

⟩

+

= 〈vgroup〉+. (6.2.143)

In other words, in the case of purely positive-energy solutions the mean current(or mean velocity 〈cα〉+) is just the classical group velocity.

At this point an important difference between the Schrödinger and Dirac the-ories emerges. For a free particle, in the Schrödinger theory the velocity operatorp/m is a constant of the motion (as one might expect) while in the Dirac the-ory this is not the case—the velocity operator cα does not commute with theHamiltonian although dp/dt=0. Indeed, the eigenvalues of cαi are ±c while themagnitude of the group velocity above must be strictly less than the speed of light.We are thus forced to admit both positive- and negative-energy solutions into theconstruction.

Let us therefore construct a more general current, including both positive- andnegative-energy contributions:

ψ(x, t) =

∫d3p√

(2π~)32E

∑

±s

[b(p, s)u(p, s) e−

i~p·x+d∗(p, s)v(p, s) e+

i~p·x].

(6.2.144)Once again, normalising to unit probability (i.e. just one particle), we obtain

1 ≡∫

d3xψ†(x, t)ψ(x, t) =

∫d3p∑

±s

[∣∣b(p, s)∣∣2 +

∣∣d(p, s)∣∣2]

(6.2.145)

and, denoting pµ=(E/c,−p) with E≡+√

p2c2+m2c4, the current is now


Jk =

∫d3p

∑

±s

[∣∣b(p, s)∣∣2 +

∣∣d(p, s)∣∣2] c2pkE

+ i∑

±s,±s′

b∗(p, s′) d∗(p, s) e+i~2Et u(p, s′) σk0 v(p, s)

− i∑

±s,±s′

b(p, s′) d(p, s) e−i~2Et v(p, s′) σk0 u(p, s)

. (6.2.146)

The first term (in the square brackets) is, as one might expect, proportional tothe constant group velocity. There are, however, two more terms, arising frominterference between the positive- and negative-energy components, that oscillateviolently (owing to the exponential phase factors), with frequencies

2E

~≥ 2mc2

~= 2× 1021 s−1. (6.2.147)

This zitterbewegung (or jittering), being proportional to the amplitude of thenegative-energy contribution, is suppressed at low energies but becomes importantfor relativistic particle velocities or when the particle is confined or localised in avery small region of space.

Consider an initial (Gaussian) solution

ψ(0, r, s) =e−r

2/2r

20

(πr20)3/4

w1(0)

=

∫d3p√

(2π~)32E

∑

±s

[b(p, s)u(p, s) e+

i~p·x +d∗(p, s)v(p, s) e−

i~p·x].

(6.2.148)

Applying a Fourier transform and exploiting the spinor orthogonality relations toproject out the different components leads to

b(p, s) =1√2E

r20

π~2 e− p

2r20

2~2 u†(p, s)w1(0) (6.2.149a)

and

d∗(p, s) =1√2E

r20

π~2 e− p

2r20

2~2 v†(p, s)w1(0). (6.2.149b)

In other words, the ratio of the coefficients is just the ratio of the small upper


components of v over the large upper components of u:

∣∣∣∣d

b

∣∣∣∣ =|p|c

E +mc2=

√E −mc2

E +mc2. (6.2.150)

From this we see explicitly that the coefficient d will become important for |p|∼mc. In the case of such a Gaussian wave-packet this will occur for r0∼~/mc,the Compton wavelength. We are thus forced to deal with the negative energysolutions; they cannot simply be banished by fiat .

6.2.11 The Klein paradox

As further proof of this last statement, let us examine the case of a Dirac particleconstrained to the left semi-axis by a step-like potential, which we assume to bedue to some electromagnetic interaction, such as eΦ=V0 (see Fig. 6.1). For the

V (z)

V0

E

zregion 1 region 2

Figure 6.1: A step-like potential constraining an initially right-moving Dirac particleto the left semi-axis.

incident particle we take a positive-energy right-moving solution:

ψinc = a e+ik1z

1

0+~ck1E+mc

2

0

. (6.2.151)

The reflected wave will then be of the form

ψr = b e− ik1z

1

0−~ck1E+mc

2

0

+ b′ e− ik1z

0

1

0+~ck1E+mc

2

. (6.2.152)


In region 2 the wave has energy p0 c = E−V0 and we thus have

(~c)2k22 = (E − V0)2 −m2c4. (6.2.153)

For E > V0, the transmitted wave thus takes on the form

ψt = d e+ik2z

1

0+~ck2

E−V0+mc2

0

+ d′ e+ik2z

0

1

0−~ck2

E−V0+mc2

. (6.2.154)

The coefficients are fixed, as usual, by the requirements of continuity at the bound-ary. In particular, one finds b′=0=d′ (i.e. there is no spin-flip—the scattering po-tential, as hypothesised, is purely scalar). We must now contemplate the variouspossibilities for the height of the barrier relative to the incident energy.

V0<E: In this case one has transmission and reflection as usual and no neweffects arise.

V0<E+mc2: Here k2 is purely imaginary and the transmitted wave is evanescentwith a “skin-depth” d>~/mc, again as usual.

V0>E+mc2: For this case k2 is again purely real, indicating transmission! (Notethat this cannot occur for the non-relativistic Schrödinger equation.) The situationis, in fact, even more startling when we calculate the reflection and transmissioncoefficients. Indeed, we find

jrjinc

=(1 + r2)

(1 + r)2and

jtjinc

=4r

(1 + r)2, (6.2.155)

where

r =k2k1

E +mc2

E − V0 +mc2= −

√|E − V0| −mc2√|E − V0|+mc2

√E +mc2√E −mc2

. (6.2.156)

Note that jinc= jr+jt, as expected (this is just a consequence of the continuityconditions). However, since r is now negative, we have the surprising result thatjt<0 and jr>jinc. At first sight, it would thus appear that the transmitted currentre-emerges from the barrier and the reflected current is greater than that incident.


The result can be understood by noticing that a potential V0>E+mc2≥2mc2

is capable of producing e+e− pairs and that a right-moving positron generatesthe same current as a left-moving electron. In fact, the effect can be traced tothe requirement that the “skin-depth” should now be d<~/mc, i.e. less than theCompton wavelength.

Exercise 6.2.7. Verify the above relations for reflection and transmission in thecase V0>E+mc2≥2mc2.

6.2.12 The negative-energy solutions

The Dirac sea interpretation

A first attempt at interpreting the negative-energy solutions of the Dirac equationwas proposed by Dirac himself in 1930. Exploiting the fermionic nature of thestates under consideration, he suggested that the spectrum should be viewed as inFig. 6.2. Borrowing from Fermi and condensed-matter physics, Dirac hypothesised

E

0

mc2

−mc2

...

...

Eγ>2mc2

Figure 6.2: The Dirac-sea picture of electrons and hole states (called positrons).

a sea of negative-energy states filled up to the bottom of the gap at E=−mc2.There would be no direct effect of this infinitely charged sea on, say, a test

charge in the vacuum (which is now no longer empty); rather as a boat on thesurface of a deep lake, we are only sensitive to fluctuations. A photon with E>2mc2 might thus excite an electron occupying one of the negative-energy statesand raise it to an unoccupied positive-energy state. The overall effect would bethe creation of a positive-energy electron and a negative-energy hole. The latterwould have all the opposite quantum numbers and kinematical variables of theassociated electron state and therefore corresponds to a positive-energy positronwith opposite spin and momentum to the vacant state. Such a “hole” theoryled Dirac to predict the existence of the positron. Almost immediately, in 1933,


Anderson duly discovered a particle of the same mass and general properties asthe electron but with the opposite charge.

The new concept of anti-matter still took some time to be fully accepted. Thediscovery of the positron notwithstanding, Pauli still had this to say of Dirac’santimatter:

“Dirac has tried to identify holes with antielectron . . . We do not believe thatthis explanation can be seriously considered.”

Wolfgang Pauli

Such opposition was not entirely unwarranted: an infinitely charged sea is morethan a little uncomfortable and somewhat asymmetric with respect to matter andantimatter, but worse still, such an interpretation could not be applied to the caseof bosons, (of which there are now many known examples) since they do not obeythe Pauli exclusion principle and should therefore inevitably cascade down throughthe infinite set of negative-energy states.

The Stückelberg–Feynman interpretation

The infinite sea is avoided in the interpretation due to Stückelberg (1941) andFeynman (1949). Consider the effect of the combined CPT transformation:

ψCPT (−x,−t) ≡ CPT ψ(x)

= CP iγ1γ3ψ∗(x)

= C γ0 iγ1γ3ψ∗(x)

= iγ2[γ0 iγ1γ3ψ∗(x)]∗

= − iγ5ψ(x), (6.2.157)

where we have suppressed (or rather fixed) the overall arbitrary phase. We thussee that a positron wave-function corresponds to the wave-function describing anelectron moving backwards in space and time and multiplied by γ5, which simplyinterchanges upper and lower components. We can also see this by rewriting

e+i[Et−p·x] = e− i [E(−t)−p·(−x)] . (6.2.158)

Exercise 6.2.8. Examine the effect of the same transformation on the Dirac equa-tion in the presence of a real electromagnetic potential and verify the equivalencestated above in terms too of interactions.

Considering also earlier observations on the relativistic electron current; thepicture to which we are led is shown in Fig. 6.3: even a free electron does not


t

x

Figure 6.3: The Stückelberg–Feynman picture of electron motion in space–time.

move smoothly through space; that is, the Dirac equation also predicts zitterbewe-gung . According to the Stückelberg–Feynman picture, this jittering motion maybe interpreted as due to the spontaneous creation of electron–positron pairs inthe vacuum. The positron of such a pair may annihilate with the electron we arestudying, leaving the electron of the pair to take its place but in a slightly differentspace–time point. Such spontaneous pair production in the vacuum is continuousand is possible by virtue of the uncertainty principle (the pair may only survivefor a time inversely proportional to its total energy).

The inevitable conclusion to be drawn is that the theory can no longer be con-sidered as that of a single particle, even when Dirac’s infinite sea of electrons isavoided: the vacuum no longer appears the simple concept of emptiness it oncewas. At this point we should move on to a quantum field theory for electro-magnetic interactions, where all such effects may be properly described. This isbeyond the scope of the present exposition and will be the subject of the followingvolume. It is, however, still possible to discuss the implications of the relativ-istic formulation presented, in the context of, e.g. the hydrogen-atom levels (see,e.g. Bjorken et al., 1964) or elastic electron–nucleus scattering (see Sec. 6.3.1) andeven simple electron–positron annihilation processes (see, e.g. Bjorken et al., op.cit.). For a slightly more modern treatment of relativistic scattering in quantummechanics (that is, without recourse to a full quantum field theory), see alsoAitchison and Hey (1982, first edition only).

6.3. RELATIVISTIC SCATTERING THEORY 217

6.3 Relativistic scattering theory

An important application of the Dirac theory of the electron is the calculation ofscattering cross-sections in the relativistic limit. If we wish to study, e.g. nuclearstructure via scattering processes, then the momentum transfers involved must beat least of the order of a few MeV, or a few 100MeV to map out the internal chargedistribution; one of the cleanest probes is just the electron (being an elementary,point-like, particle with no internal structure). For energies of this order, theelectron is ultra-relativistic and we should expect the classical Rutherford formulato be inadequate. We shall thus now pursue the calculation of the cross-sectionfor the elastic scattering of an electron off a classical potential Aµ in the form of aconstant Coulomb potential, such as might be generated by a static nucleus.

In order to simplify the appearance of the expressions in the following, we shallhenceforth adopt the standard particle-physics convention of so-called natural unitsby setting ~ and c to unity. This leads to the equivalence of the dimensionalitiesassociated with the following quantities:

[E] = [p] = [m] = [t−1] = [l−1].

The powers of ~ and c to be inserted in any final physical expression may always berecovered on dimensional grounds. A possible procedure might be to multiply alldimensional quantities by powers of ~ and c in order to arrive at dimensions thatare simple powers of energy. Thus, E→E, p→pc, m→mc2, t→ t/~, x→x/~cetc. The overall dimensions of the object calculated will then be some power ofE, which can easily be converted to any other dimension using ~ and c. In thisregard, it is useful to bear in mind the value of the product

~c ≃ 197 MeV· fm ≃ 2 keV· Å (6.3.1)

and that, e.g. the electron rest mass is equivalent to 511keV while the time scaleassociated with 1 fm is roughly 10−23 s and with 1Å, 10−18 s.

6.3.1 The Mott cross-section

The standard procedure (i.e., by which the Fermi golden rule was derived in thenon-relativistic formalism earlier), leads to a transition amplitude between initialand final states ψi,f(x) in the presence of an interaction potential e /A(x):

Sfi = − ie

∫d4xψf(x) /A(x)ψi(x), (6.3.2)


where the integral is over all space–time and, to render manifest the Lorentz cov-ariance of the amplitude, we have specifically retained the integration in t (whichnormally provides the energy-conservation δ-function).

The physical situation of interest typically consists of incoming and outgoingplane-wave spinors ψi,f while the interaction potential Aµ is that generated by thepresence of a static, charged, nucleus. We thus take

ψi(x) =1√V

1√2Ei

u(pi, si) e− ipi·x (6.3.3a)

andψf(x) =

1√V

1√2Ef

u(pf, sf) e+ipf·x, (6.3.3b)

which corresponds to the standard normalisation of 2E electrons in a box of volumeV . The four-potential generated by a static charge Ze located at the origin is just

Aµ = (Φ, 0) with Φ(x) =−Ze4π|x| . (6.3.4)

The scattering amplitude then becomes

Sfi =iZe2

4π

1

V

1√2Ef2Ei

u(pf, sf) γ0 u(pi, si)

∫d4x

e i (pf−pi)·x

|x| . (6.3.5)

The integral over the time coordinate provides the usual energy-conservation δ-function, 2πδ(Ef−Ei) while the remaining spatial integral is just a Fourier trans-form of the Coulomb potential:

∫d3xe iq·x

|x| =4π

|q|2, (6.3.6)

with q=pi−pf, the three-momentum transfer. And thus the scattering amplitudeis finally

Sfi = iZe21

V

1√2Ef2Ei

Mfi

1

|q|22πδ(Ef − Ei), (6.3.7)

where we have defined the spinor product or reduced amplitude

Mfi ≡ u(pf, sf) γ0 u(pi, si) = u†(pf, sf) u(pi, si). (6.3.8)

The transition probability is just the squared amplitude weighted by the num-ber of final states in the relevant momentum interval. With the chosen normalisa-


tion, the latter is

dn = Vd3pf

(2π)3. (6.3.9)

Therefore, the transition probability is

Pfi = |Sfi|2 dn

= Z2(4πα)21

2Ei

1

|q|4∣∣Mfi

∣∣2 d3pf

(2π)32Ef

[2πδ(Ef − Ei)

]2. (6.3.10)

As usual, we encounter the square of an energy δ-function as the result of the timeintegration. Using (see Sec. 5.2.4)

2πδ(Ef − Ei) = limT→∞

∫ +T/2

−T/2

dt e i (Ef−Ei)t

= limT→∞

2 sin[12T (Ef −Ei)

]

Ef − Ei

∼ limT→∞

T, (6.3.11)

this singular object may be recast as[2πδ(Ef − Ei)

]2= lim

T→∞T 2πδ(Ef − Ei). (6.3.12)

Dividing by the total time T to obtain a rate, we thus have

Φinc dσ := Rfi = Pfi/T

=4Z2α2

V 2Ei

∣∣Mfi

∣∣2

|q|4d3pf

2Ef

δ(Ef − Ei), (6.3.13)

where Φinc=∣∣ψi(x)γψi(x)

∣∣=∣∣vinc

∣∣/V is the incident flux and dσ is the differentialcross-section we seek to calculate. Using the relativistic relation pdp=EdE toperform the integration over the δ-function, we finally obtain

dσ

dΩ=

Z2α2

|q|4∣∣Mfi

∣∣2 . (6.3.14)

Exercise 6.3.1. Verify that the previous expressions corresponds to the Rutherfordcross-section in the non-relativistic limit.

The spinor product Mfi can, in fact, be evaluated with no reference to thechosen γ-matrix representation and no explicit calculation of matrix products.


First of all, let us note that, for any γ-matrix product Γ,∣∣u(pf, sf) Γ u(pi, si)

∣∣2 =[u(pf, sf) Γ u(pi, si)

] [u(pi, si) Γ u(pf, sf)

], (6.3.15)

where Γ≡γ0Γ†γ0. Now, it is not difficult, using (6.2.57c), to show that

γµγν · · ·γρ = γρ · · · γνγµ (6.3.16a)and

γ5γµγν · · ·γρ = −γρ · · · γνγµγ5. (6.3.16b)

Moreover, an external product such as u(p,s)u(p,s) may be reduced to a productof γ-matrices using the projection operators in Eqs. (6.2.136):

uα(p, s) uβ(p, s) =∑

a

[(/p+m)(1 + γ5/s)]αδ

4mǫaw

aδ (0, nz)w

aβ(0, nz), (6.3.17)

where, we remind the reader that a and b label the rest-frame spinors while α, βand δ are Dirac-spinor indices. By virtue of the orthogonality relation (6.2.125d),we then have

u(p, s) u(p, s) = (/p+m)12(1 + γ5/s) (6.3.18)

and therefore∣∣Mfi

∣∣2 = Tr[(/pi

+m)12(1 + γ5/si

)γ0(/pf+m)1

2(1 + γ5/sf

)γ0]. (6.3.19)

The form thus obtained depends on the momenta and polarisation states of theincoming and outgoing electron states, i.e. it expresses the probability of scatteringfrom and to definite initial and final momenta and spin states. However, in mostexperimental situations nothing is known about the spins of the electrons and thuswe can simplify by averaging (hence the factor 1/2 below) over the two possibleinitial spin states and summing over the final two:

∣∣Mfi

∣∣2 → 12

∑

si,sf

∣∣Mfi

∣∣2

= 12Tr[(/pi

+m)γ0(/pf+m)γ0

]. (6.3.20)

Applying the trace theorems, we finally have

dσ

dΩ=

Z2α2

|q|42[2EiEf − pi·pf +m2

]. (6.3.21)


The usual kinematical relations give

pi·pf = E2 − p2 cos θ = m2 + 2β2E2 sin2 θ

2(6.3.22a)

and|q|2 = 4|p|2 sin2 θ

2. (6.3.22b)

And thus the final expression for what is then known as the Mott cross-section is

dσ

dΩ=

Z2α2

4|p|2β2 sin4 θ2

(1− β2 sin2 θ

2

). (6.3.23)

Observe that this is just the original Rutherford formula multiplied by a factor(1−β2 sin2 θ

2

), which arises owing to helicity conservation. In the non-relativistic

limit (whence β→0) it reduces to unity while for an ultra-relativistic electron(with β≃1) it suppresses scattering through 180. This additional suppressionreflects the combined effect of standard angular-momentum conservation and thehelicity-conserving nature of the vector interaction. By examining the relevantbilinear spinor products in the ultra-relativistic limit E≫m, it is not difficult toshow that the quantity h := p·s (the helicity) is conserved by the form of interactioninvoked here (∝ /A). However, for an electron scattered through 180 this wouldrequire inversion of the spin direction, which could not then be balanced by anyother angular-momentum variation (since, in this problem by hypothesis, there areno other angular momenta with non-zero projection along the z-axis).

As a corollary to this observation, we note that the helicity-flip amplitude isproportional to the particle mass—for a massless particle β=1. In other words,in the ultra-relativistic limit (1−β)∝m2/E2. This has important repercussionsfor the neutrino and also for those decay channels in which angular-momentumconservation requires helicity flip (such as π→µν, eν). This leads, for example,to the surprising result that decay into a heavier particle may be favoured over alighter particle! Indeed, cf. the decay rates for π−→µ−νµ and π−→ e−νe; helicityconservation here would require the final (back-to-back) lepton and antineutrinoto have opposite helicity in the massless limit whereas they must be the same toconserve angular momentum. As a result the muon decay mode is dominant.


6.4 Bibliography

Aitchison, I.J.R. and Hey, A.J.G. (1982), Gauge Theories in Particle Physics: APractical Introduction (Adam Hilger), 1st. edition.

Anderson, C.D. (1933), Phys. Rev. 43, 491.

Bjorken, J.D. and Drell, S.D. (1964), Relativistic Quantum Mechanics (McGraw–Hill).

Chisholm, J.S.R. (1963), Nuovo Cim. 30, 426.




Gordon, W. (1926), Z. Phys. 40, 117.

Klein, O. (1927), Z. Phys. 41, 407.

Schweber, S.S., ed. (1962), Introduction to Relativistic Quantum Field Theory(Harper & Row).

Stückelberg, E.C.G. (1941), Helv. Phys. Acta 14, 588.

Appendix A

Answers to Selected Exercises

A.1 Exercises to Chapter 2

Ex. 2.1.3

Starting from Eq. (2.1.24), derive Eq. (2.1.25).We have

U(t, t0) = 1+

∞∑

n=1

(− i

~

)n∫ t

t0

dt1

∫ t1

t0

dt2 · · ·∫ tn−1

t0

dtnH(t1)H(t2) · · ·H(tn).

(2.1.24)The n-th term involves and integral of the form

∫ t

t0

dt1 · · ·∫ t(i−1)

t0

dti · · ·∫ t(j−1)

t0

dtj · · ·∫ tn−1

t0

dtn

×H(t1) · · ·H(ti) · · ·H(tj) · · ·H(tn). (A.1.1)

If we simply interchange the labels on the i-th and j-th integrals, the only problemwill be the different ordering of the H(ti) and H(tj) and, as by hypothesis they donot anticommute, the result would be different. We thus may write

=

∫ t

t0

dt1 · · ·∫ t(j−1)

t0

dtj · · ·∫ t(i−1)

t0

dti · · ·∫ tn−1

t0

dtn

× T(H(t1) · · ·H(ti) · · ·H(tj) · · ·H(tn)

), (A.1.2)

where T is the time-ordering operator introduced in the main text and which herewould have the action of interchanging the positions ofH(ti) andH(tj). Thus, withthe proviso that we apply this time-ordering to the operator product, any order ofintegration provides the same result. There are precisely n! different orderings and

223

224 APPENDIX A. ANSWERS TO SELECTED EXERCISES

any point in the n-dimensional hypercube, with each dimension bounded by [0,t],belongs to one an only one of these orderings. Finally then, the desired integralmay be replaced by a time-ordered integral over the full hypercube divided by n!.

Ex. 2.3.2

Derive the form of the propagator in the case of a charged massive particle (e.g.the electron) in a constant and uniform electric field.

The direct calculation of the free-particle propagator makes use of the so-calledBaker–Campbell–Hausdorff formula:

eA eB = eC , (A.1.3)

where

C = A+B + 12[A,B] + 1

12

([A, [A,B]] + [[A,B], B]

)+ 1

24[A, [[A,B], B]] + . . .

(A.1.4)

Ex. 2.3.4

Rederive result 2.3.9 for the free-particle propagator using the path-integral form-alism.

We may start directly from the explicit form of the Feynman path integralexpression Eq. (2.3.39) for the free-particle propagator:


( m

2π i~ δt

)n/2 ∫dxn−1

∫dxn−2 · · ·

∫dx1

n∏

m=1

exp[i~Sm

],

(2.3.39)where δt≡ (tf− ti)/n and the infinitesimal action is given by

Sr =

∫ tr

tr−1

dtmx2

2≈ δt

m(xr − xr−1)2

2δt2. (A.1.5)

We may now change variables to ξr≡xr−xr−1. There are, however, only n−1integrals and the n variables ξr are not independent:

0 = xf − xi −n∑

1

ξr. (A.1.6)

A.1. EXERCISES TO CHAPTER 2 225

This restriction may be implemented via the following trick:

∫dξn δ

(xf − xi −

n∑

1

ξr

)≡ 1 (A.1.7)

and

δ

(xf − xi −

n∑

1

ξr

)=

1

2π~

∫dp′ exp

i~p′

[xf − xi −

n∑

1

ξr

], (A.1.8)

with which we may rewrite the path integral as


1

2π~

∫dp′

( m

2π i~ δt

)n/2 ∫dξn

∫dξn−1 · · ·

∫dξ1

× exp

[i

~p′(xf − xi)

] n∏

r=1

exp

[i

~

mξ2r2δt

− i

~p′ξr

]. (A.1.9)

The n identical integrals over the ξr can now be performed by completing thesquare in the exponents. Recalling that nδt≡ tf− ti, one then finds precisely theremaining integral over p′ given in Eq. (2.3.7) and thus with answer (2.3.9).

Ex. 2.3.5

Using the path-integral formalism, rederive the form of the propagator found inEx. 2.3.2 in the case of a charged massive particle in a constant and uniformelectric field.

The starting point is naturally the explicit form of the path integral (2.3.39):


( m

2π i~ δt

)n/2 ∫dxn−1

∫dxn−2 · · ·

∫dx1

n∏

m=1

exp[i~Sm

],

(2.3.39)where we work in one dimension and now the infinitesimal action is given by

Sr =

∫ tr

tr−1

dt

[mx2

2− V (x)

]

≃ δt

m

2

[xr − xr−1

δt

]2− qExr

. (A.1.10)

for a particle of charge q in a constant electric field E. Note that we have chosento replace the average position 1

2(xr+xr−1) in the argument of the potential with

simply xr, as the difference vanishes in the limit.We now make the change of variables to ξ≡xf−x′ and write (using the trick


of the previous exercise)


1

2π~

∫dp′

( m

2π i~ δt

)n/2 ∫dξn−1

∫dξn−2 · · ·

∫dξ1

× exp

[i

~p′(xf − xi)

] n∏

r=1

exp

[i

~δt

(12mξ2r − p′ξrδt− qExrδt

2)], (A.1.11)

where, of course, xr=xi+∑r

s=1ξs. Writing the product of exponentials as theexponential of a sum in the product on the right-hand side above, we have

n∏

r=1

exp[· · · ] = exp

[i

~δt

n∑

r=1

(12mξ2r − p′ξrδt− qExrδt

2)]. (A.1.12)

The implicit double sum in∑n

r=1xr can be simplified by reordering and relabelling:

n∑

r=1

r∑

s=1

ξs =

n∑

s=1

n∑

r=s

ξs =

n∑

s=1

(n− s)ξs =

n∑

r=1

(n− r)ξr. (A.1.13)

And thus the exponent may be rewritten as

i

~δt

n∑

r=1

(12mξ2r − p′ξrδt− qE[xi + (n− r)ξr]δt

2). (A.1.14)

The term in xi can be evaluated immediately:

− i

~

n∑

r=1

qExiδt = − i

~nqExiδt = − i

~qExi∆t (A.1.15)

where, since the temporal dependence is limited to the time interval itself, we haveintroduced ∆t≡nδt= tf− ti. This leads to an overall factor

exp

[− i

~qExi∆t

], (A.1.16)

As usual, in the remaining piece we complete the square for the ξr dependence:

12mξ2r − p′ξrδt− qE(n− r)ξrδt

2

=m

2

(ξr −

p′δt+ qE(n− r)δt2

m

)2

−(p′δt + qE(n− r)δt2

)2

2m. (A.1.17)


The first term on the right-hand side, after a shift in the ξr variables, leads to thesame integral as in the previous exercise. The second term leads to another overallp′-dependent factor; the sums to be evaluated are

n∑

r=1

(n− r) =

n−1∑

r=0

r = 12n(n− 1) (A.1.18)

and n∑

r=1

(n− r)2 =

n−1∑

r=0

r2 = 16n(n− 1)(2n− 1). (A.1.19)

In the limit the final sums contribute as 12n2 and 1

3n3. The resulting exponent is

then

− i

2m~

(np′2δt+ n2p′qEδt2 + 1

3n3q2E2δt3

)

= − i

2m~

(p′2∆t + p′qE(tf − ti)

2 + 13q2E2∆t3

). (A.1.20)

The remaining integral is now

〈xf,tf|xi,ti〉 = exp

[− i

~qExi∆t

]1

2π~

∫dp′ exp

[i

~p′(xf − xi)

]

× exp

[− i

2m~

(p′2∆t+ p′qE∆t2 + 1

3q2E2∆t3

)]

= exp

[− i

~qExi∆t

]exp

[− i

~

q2E2∆t3

6m

]√m

2π i~∆t

× exp

[im(xf − xi − qE∆t2/2m)2

2~∆t

]

=

√m

2π i~∆texp

[im(xf − xi)

2

2~∆t

]

× exp

[−iqE(xi + xf)∆t

2~

]exp

[− iq2E2∆t3

24m~

]. (A.1.21)

Ex. 2.5.2

A beam of neutrons is polarised parallel to the direction B of a uniform magneticfield B. The beam is separated into two equal parts, of which one passes through theuniform field B while the other passes through a field of the same magnitude butof slowly varying direction. After a certain distance the two beams are recombined.


The two magnetic fields may be represented as:

B1 = B0

[sin θ |x〉+ cos θ |z〉

](2.5.19a)

andB2 = B0

[sin θ cosφ(t) |x〉+ sin θ sin φ(t) |y〉+ cos θ |z〉

], (2.5.19b)

where φ(t) varies from 0 to 2π just once during the entire journey. Note that B1

is just B2 for φ(t)=0. Evaluate:

1. the ground-state wave-function in the basis of eigenstates of Sz for a constantfield B2 (in other words, for t fixed);

2. the phase difference between the two beams when they are recombined andthus the final intensity.

1: Let us write the most general form for the eigenstates as

|α,β〉 = cosα |+z〉+ e iβ sinα |−z〉. (A.1.23)

The interaction Hamiltonian is

Hint = −µ·B = −γS·B. (2.1.34)

For beam two we thus have

Hint = −γB0

[sin θ cos φ(t)Sx + sin θ sin φ(t)Sy + cos θ Sz

]. (A.1.24)

We may rewrite Sx,y in terms of S± :=Sx± iSy:

Sx = 12S+ + 1

2S− (A.1.25a)

andSy = − i

2S+ + i

2S−. (A.1.25b)

The interaction Hamiltonian therefore becomes

Hint = −γB0

[12sin θ e− iφ(t) S+ + 1

2sin θ e+iφ(t) S− + cos θSz

]. (A.1.26)

The eigenvalue equation Hint|α〉=Eα|α〉, for φ(t) constant, is then

Eα

[cosα |+z〉+ e iβ sinα |−z〉

]

= −γB0

[12sin θ e− iφ S+ + 1

2sin θ e+iφ S− + cos θSz

]

×[cosα |+z〉+ e iβ sinα |−z〉

]

= −12γB0~

[sin θ

(e iβ sinα e− iφ |+z〉+ cosα e+iφ |−z〉

)

+ cos θ(cosα |+z〉 − e iβ sinα |−z〉

)](A.1.27)


Therefore,

Eα cosα = −12γB0~

[sin θ e iβ sinα e− iφ+cos θ cosα

](A.1.28a)

andEα e

iβ sinα = −12γB0~

[sin θ cosα e+iφ− cos θ e iβ sinα

](A.1.28b)

Taking the ratio of these two equations gives

e iβ sinα[sin θ e iβ sinα e− iφ +cos θ cosα

]

= cosα[sin θ cosα e+iφ − cos θ e iβ sinα

], (A.1.29)

which leads to

cos θ sin 2α = sin θ(cos2 α e+i(φ−β) − sin2 α e− i (φ−β)

).

= sin θ(cos 2α cos(φ− β) + i sin(φ− β)

). (A.1.30)

Since all angles are implicitly assumed real we must have β=φ together with eitherα= θ/2 or (θ−π)/2 and so the initial energy eigenstates may be written as

|+B2〉 = cos θ2|+z〉+ e iφ sin θ

2|−z〉. (A.1.31a)

and|−B2〉 = sin θ

2|+z〉 − e iφ cos θ

2|−z〉, (A.1.31b)

where the ±B2 notation in the kets on the left-hand side indicates spin paralleland antiparallel to the direction of B2. The energies of these two states are then

E± = ∓12~γB0. (A.1.32)

2: The time-dependent Schrödinger equation is

i~∂

∂t|a;t〉 = Hint|a;t〉. (A.1.33)

And we expand |a;t〉 over the time-dependent basis set |±B2;t〉, where we thusinclude the time dependence implicit in φ(t):

|a;t〉 = c+(t)|+B2;t〉+ c−(t)|−B2;t〉. (A.1.34)

If we now assume adiabatic variation, so that the instantaneous frequencyassociated with φ(t) is small compared to E±/~, we then have

i~[c±(t) + c±(t) i φ(t) sin

2 θ2 ± c∓(t) i φ(t) sin

θ2cos θ

2

]= E±c±(t), (A.1.35)


where we cannot simply neglect the time derivatives of |±B2;t〉 as this wouldthen be the same as the static solution and we would lose the geometric phase.The spirit of the adiabatic approximation is really that the transitions should benegligible. Therefore, if we assume that the initial configuration is in the groundstate (c+(0)=1 and c−(0)=0), then the moduli of the coefficients should remainapproximately constant: |c+(t)|≃1 and |c−(t)|≃0. It is not difficult to show thatfor |φ/φ|≪ |E±/~| this does indeed hold. We may therefore neglect the c− termin the equation for c+ and the solution is then

c+(t) ≃ e−i~E+t e− i sin

2 θ2φ(t) . (A.1.36)

The first term is the usual dynamical phase and as such is the same as in the casefor constant φ=0 while the second is the geometric phase, which is not presentfor the first beam.

For φ(t) varying from zero to 2π, the phase difference is 2π sin2 θ2

(it is actuallyjust the solid angle swept out by the vector B). As θ lies between zero and π, thisthen lies between zero and π/2. That is, e.g., for θ=π the two beams recombinecompletely destructively. The intensity of the recombined beams is

I(θ) ≃ 4 cos2(π sin2 θ

2

). (A.1.37)

A.2 Exercises to Chapter 3

Ex. 3.4.1

Derive the commutation relations (3.4.8) from Eq. (3.4.6).Let us try to construct Rij(n,ǫ) using n and the tensor εijk. Examining the

cases when n is directed along the x-, y- or z-axis, it is easy to see that we have

Rij(n, ǫ) = δij − ǫ nk εijk. (A.2.1)

Eq. (3.4.6) may thus be written as

Vi − i~ǫ[Vi,J ·n] = (δij − ǫ nk εijk)Vj

= Vi − ǫ nk εijk Vj. (A.2.2)

Rearranging this gives

[Vi, Jknk] = − i~ nk εijk Vj . (A.2.3)


and, since this is valid for any direction n, we obtain

[Vi, Jk] = − i~ εijk Vj, (A.2.4)

which, finally rearranged, leads to Eq. (3.4.8).

Appendix B

Supplementary Topics

B.1 The Hellmann–Feynman theorem

The Hellmann–Feynman theorem (Güttinger, 1932; Pauli, 1933, p. 162; Hellmann,1937, p. 285 and Feynman, 1939) bears some resemblance to the Ehrenfest theoremdiscussed in Sec. 2.2.3. Put simply, it states that for an energy eigenstate thederivative of the energy with respect to some generic (continuous) parameter isjust the expectation value of the derivative of the Hamiltonian with respect tothe same parameter; i.e. the variation of the wave-function with respect to to theparameter has no effect. This implies, in particular, that once the Schrödingerequation has been solved for a given system, the dynamics may be studied viaclassical mechanics.

Theorem B.1.1. If |ψλ〉 is a properly normalised energy eigenstate with energyeigenvalue Eλ then

dEλ

dλ=

⟨ψλ

∣∣∣∣dH

dλ

∣∣∣∣ψλ

⟩. (B.1.1)

Proof. The proof is straightforward:

dEλ

dλ=

d

dλ〈ψλ |H |ψλ〉

=

⟨dψλ

dλ

∣∣∣∣H∣∣∣∣ψλ

⟩+

⟨ψλ

∣∣∣∣dH

dλ

∣∣∣∣ψλ

⟩+

⟨ψλ

∣∣∣∣H∣∣∣∣dψλ

dλ

⟩

= Eλ

⟨dψλ

dλ

∣∣∣∣ψλ

⟩+

⟨ψλ

∣∣∣∣dH

dλ

∣∣∣∣ψλ

⟩+ Eλ

⟨ψλ

∣∣∣∣dψλ

dλ

⟩

233

234 APPENDIX B. SUPPLEMENTARY TOPICS

the first and last terms can be combined to give

=

⟨ψλ

∣∣∣∣dH

dλ

∣∣∣∣ψλ

⟩+ Eλ

d

dλ〈ψλ |ψλ〉

the last terms vanishes by the normalisation condition, 〈ψλ |ψλ〉 ≡ 1, and so

=

⟨ψλ

∣∣∣∣dH

dλ

∣∣∣∣ψλ

⟩. (B.1.2)

A simple example application of the Hellmann–Feynman theorem is the de-termination of the expectation value of 1/r2 for hydrogenoid atoms. As usual,the radial Schrödinger equation for such a case is given in terms of the followingreduced Hamiltonian:

Hl = − ~2

2µr2

[d

dr

(r2

d

dr

)− l(l + 1)

]− Ze2

r, (B.1.3)

where µ is the reduced electron mass and l the orbital quantum number. Theenergy levels are given by

En = −Z2µe4

2~2n2 with n = nr + l + 1, (B.1.4)

where n and nr are the principal and radial quantum numbers respectively. Bytreating l as a continuous free parameter, we have

∂Hl

∂l=

~2

2µr2(2l + 1) (B.1.5)

and thus obtain⟨ψnl

∣∣∣∣1

r2

∣∣∣∣ψnl

⟩=

2µ

~2

1

(2l + 1)

⟨ψnl

∣∣∣∣∣∂Hl

∂l

∣∣∣∣∣ψnl

⟩

=2µ

~2

1

(2l + 1)

∂En

∂l

=2µ

~2

1

(2l + 1)

∂En

∂n

∂n

∂l

=2µ

~2

1

(2l + 1)

Z2µe4

~2n3 =

2Z2µ2e4

~4n3(2l + 1)

. (B.1.6)

B.2. THE PATH INTEGRAL 235

B.2 The path integral

We give here a more complete derivation of the Feynman path integral (2.3.41).The starting point is the transition matrix element

M(xf , tf ; xi, ti) = 〈xf |e− iH(tf−ti) |xi〉 = 〈xf ,tf |xi,ti〉, (B.2.1)

which describes the probability amplitude for a state initially having position ei-genvalue xi at some instant ti to find itself with xf at a later instant tf . It encodesall important information of the quantum theory. The idea then is to expressthis object as a path integral. The first step is to divide the finite interval [ti,tf ]into n+1 subintervals of infinitesimal duration δt=(tf − ti)/(n+1) by inserting ncomplete sets of states (ti<t1< · · ·<tn<tf):

M(xf , tf ; xi, ti) =

∫dq1 dq2 . . .dqn 〈xf ,tf |xn,tn〉 · · · 〈x2,t2|x1,t1〉〈x1,t1|xi,ti〉.

(B.2.2)Consider a generic infinitesimal interval:

〈x′,t+δt|x,t〉 = 〈x′|e− iHδt |x〉

= δ(x′ − x)− iδt〈x′|H|x〉+O((δt)2

). (B.2.3)

The Hamiltonian H=H(p,x) is a function of the operators for position and con-jugate momentum x and p. The simplest, but most useful and most common, caseis a massive particle subject to a potential depending only on position x:

H =p2

2m+ V (x). (B.2.4)

Using the fact that 〈x|p〉=e ipx and that∫dp e ip(x

′−x)= δ(x′−x), we have

〈x′|H(p,x)|x〉 =

∫dp′ dp

[〈x′|p′〉〈p′|p2|p〉〈p|x〉2m

]+ 〈x′|V (x)|x〉

=

∫dp e ip(x

′−x)

[p2

2m+ V (x)

]

=

∫dp e ip(x

′−x) H(p, x), (B.2.5)

where H(p,x) is now simply the classical Hamiltonian function.∗

∗ In some texts one finds the more symmetric 1

2(x+x′) in place of x as the argument of H , but


To first order in δt, we can thus write

〈x′,t+δt|x,t〉 ≃∫

dp e i[p(x′−x)−H(p,x)δt] (B.2.6)

and substituting this expression into that above for M(xf ,tf ;xi,ti), we have

M(xf , tf ; xi, ti) = limn→∞

∫ n∏

k=1

dxk

n+1∏

k=1

dpk

× exp

i

n+1∑

j=1

[pj(xj − xj−1)−H(pj, xj) δt

], (B.2.7)

with the identification x0≡xi and xn+1≡xf . In the limit the sum over temporalslices becomes an integral and we may simplify the notation by rewriting the (nowinfinite) products as∗

M(xf , tf ; xi, ti) =

∫DxDp exp

i

∫ tf

ti

dt[p x−H(p, x)

]. (B.2.8)

This passage then defines the Feynman path integral.With the previous choice of Hamiltonian, the complex integrals in p are Gauss-

ian and may therefore be performed via the usual Wick rotation:

∫ ∞

−∞dp e iδt (p x−p

2/2m) =

[2π iδt

m

]−1/2

e iδtmx2/2 . (B.2.9)

For the transition matrix element, we finally obtain

M(xf , tf ; xi, ti) = limn→∞

∫ n∏

k=1

dxk[2π iδt/m

]1/2

× exp

i

n+1∑

j=1

[m

2

(xj − xj−1

δt

)2

− V (xj)

]δt

=

∫ n∏

k=1

dxk[2π iδt/m

]1/2 expi

∫ tf

ti

dt Lcl(x, x)

, (B.2.10)

this really is unnecessary in view of the limit to be taken.∗ In general, any numerical factors accompanying the measures dxk and dpk may simply beabsorbed into an irrelevant overall normalisation constant.

B.2. THE PATH INTEGRAL 237

where Lcl =12mx2−V (x) is simply the classical Lagrangian. We see then that the

exponent is precisely the classical action

Scl :=

∫ tf

ti

dt Lcl(x, x), (B.2.11)

which thus determines the temporal evolution in quantum mechanics. The im-portant point is that while the expression for the transition amplitude presentedhere has been derived starting from the Schrödinger equation, we may now equallytake the path integral as the starting point and, as shown in the main text, derivefrom it the Schrödinger equation.

It must be stressed, however, that the final simple form is a direct consequenceof the particular choice of Hamiltonian, which, while rather simple too, is, infact, generally sufficient for our purposes. If, on the other hand, the Hamiltonianinvolves products of p and x, then the final form does not contain the classicalaction, but what might be called an effective action, which can be calculated fromEq. (B.2.7).


B.3 Bibliography


Güttinger, P. (1932), Z. Phys. 73, 169.

Hellmann, H. (1937), Einführung in die Quantenchemie (Franz Deuticke).

Pauli, W. (1933), Principles of Wave Mechanics (Springer–Verlag).

Appendix C

Glossary of Acronyms

BW: Breit–Wigner

EDM: electric dipole moment

FWHM: full-width at half maximum

KG: Klein–Gordon

NMR: nuclear magnetic resonance

JWKB: Jeffreys–Wentzel–Kramers–Brioullin

239

240 APPENDIX C. GLOSSARY OF ACRONYMS

Quantum Physics - Università degli Studi...

Documents

Transcript of Quantum Physics - Università degli Studi...