advanced quantum mechanics

3 1.1 Identical Particles, Many-Particle States, and Permutation Symmetry.. 10 1.3.1 States, Fock Space, Creation and Annihilation Operators.. 16 1.4.1 States, Fock Space, Creation and A

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The true physics is that which will, one day,achieve the inclusion of man in his wholeness

in a coherent picture of the world

Pierre Teilhard de Chardin

To my daughter Birgitta

In the new edition, supplements, additional explanations and cross referenceshave been added at numerous places, including new formulations of the prob-lems Figures have been redrawn and the layout has been improved In allthese additions I have intended not to change the compact character of thebook The proofs were read by E Bauer, E Marquard–Schmitt and T Wol-lenweber It was a pleasure to work with Dr R Hilton, in order to conveythe spirit and the subtleties of the German text into the English translation.Also, I wish to thank Prof U T¨auber for occasional advice Special thanks

go to them and to Mrs J¨org-M¨uller for general supervision I would like tothank all colleagues and students who have made suggestions to improve thebook, as well as the publisher, Dr Thorsten Schneider and Mrs J Lenz forthe excellent cooperation

Preface to the First Edition

This textbook deals with advanced topics in the field of quantum mechanics,material which is usually encountered in a second university course on quan-tum mechanics The book, which comprises a total of 15 chapters, is dividedinto three parts: I Many-Body Systems, II Relativistic Wave Equations, andIII Relativistic Fields The text is written in such a way as to attach impor-tance to a rigorous presentation while, at the same time, requiring no priorknowledge, except in the field of basic quantum mechanics The inclusion

of all mathematical steps and full presentation of intermediate calculationsensures ease of understanding A number of problems are included at theend of each chapter Sections or parts thereof that can be omitted in a firstreading are marked with a star, and subsidiary calculations and remarks notessential for comprehension are given in small print It is not necessary tohave read Part I in order to understand Parts II and III References to otherworks in the literature are given whenever it is felt they serve a useful pur-pose These are by no means complete and are simply intended to encouragefurther reading A list of other textbooks is included at the end of each ofthe three parts

In contrast to Quantum Mechanics I, the present book treats relativisticphenomena, and classical and relativistic quantum fields

Part I introduces the formalism of second quantization and applies this

to the most important problems that can be described using simple methods.These include the weakly interacting electron gas and excitations in weaklyinteracting Bose gases The basic properties of the correlation and responsefunctions of many-particle systems are also treated here

The second part deals with the Klein–Gordon and Dirac equations portant aspects, such as motion in a Coulomb potential are discussed, andparticular attention is paid to symmetry properties

Im-The third part presents Noether’s theorem, the quantization of the Klein–Gordon, Dirac, and radiation fields, and the spin-statistics theorem The finalchapter treats interacting fields using the example of quantum electrodynam-ics: S-matrix theory, Wick’s theorem, Feynman rules, a few simple processessuch as Mott scattering and electron–electron scattering, and basic aspects

of radiative corrections are discussed

The book is aimed at advanced students of physics and related disciplines,and it is hoped that some sections will also serve to augment the teachingmaterial already available.

This book stems from lectures given regularly by the author at the nical University Munich Many colleagues and coworkers assisted in the pro-duction and correction of the manuscript: Ms I Wefers, Ms E J¨org-M¨uller,

Tech-Ms C Schwierz, A Vilfan, S Clar, K Schenk, M Hummel, E Wefers,

B Kaufmann, M Bulenda, J Wilhelm, K Kroy, P Maier, C Feuchter,

A Wonhas The problems were conceived with the help of E Frey and

W Gasser Dr Gasser also read through the entire manuscript and mademany valuable suggestions I am indebted to Dr A Lahee for supplyingthe initial English version of this difficult text, and my special thanks go to

Dr Roginald Hilton for his perceptive revision that has ensured the fidelity

of the final rendition

To all those mentioned here, and to the numerous other colleagues whogave their help so generously, as well as to Dr Hans-J¨urgen K¨olsch ofSpringer-Verlag, I wish to express my sincere gratitude

Table of Contents

Part I Nonrelativistic Many-Particle Systems

1. Second Quantization 3

1.1 Identical Particles, Many-Particle States, and Permutation Symmetry 3

1.1.1 States and Observables of Identical Particles 3

1.1.2 Examples 6

1.2 Completely Symmetric and Antisymmetric States 8

1.3 Bosons 10

1.3.1 States, Fock Space, Creation and Annihilation Operators 10

1.3.2 The Particle-Number Operator 13

1.3.3 General Single- and Many-Particle Operators 14

1.4 Fermions 16

1.4.1 States, Fock Space, Creation and Annihilation Operators 16

1.4.2 Single- and Many-Particle Operators 19

1.5 Field Operators 20

1.5.1 Transformations Between Different Basis Systems 20

1.5.2 Field Operators 21

1.5.3 Field Equations 23

1.6 Momentum Representation 25

1.6.1 Momentum Eigenfunctions and the Hamiltonian 25

1.6.2 Fourier Transformation of the Density 27

1.6.3 The Inclusion of Spin 27

Problems 29

2 Spin-1/2 Fermions 33

2.1 Noninteracting Fermions 33

2.1.1 The Fermi Sphere, Excitations 33

2.1.2 Single-Particle Correlation Function 35

2.1.3 Pair Distribution Function 36

∗2.1.4 Pair Distribution Function, Density Correlation Functions, and Structure Factor 39

2.2 Ground State Energy and Elementary Theory

of the Electron Gas 41

2.2.1 Hamiltonian 41

2.2.2 Ground State Energy in the Hartree–Fock Approximation 42

2.2.3 Modification of Electron Energy Levels due to the Coulomb Interaction 46

2.3 Hartree–Fock Equations for Atoms 49

Problems 52

3. Bosons 55

3.1 Free Bosons 55

3.1.1 Pair Distribution Function for Free Bosons 55

∗3.1.2 Two-Particle States of Bosons 57

3.2 Weakly Interacting, Dilute Bose Gas 60

3.2.1 Quantum Fluids and Bose–Einstein Condensation 60

3.2.2 Bogoliubov Theory of the Weakly Interacting Bose Gas 62

∗3.2.3 Superfluidity 69

Problems 72

4. Correlation Functions, Scattering, and Response 75

4.1 Scattering and Response 75

4.2 Density Matrix, Correlation Functions 82

4.3 Dynamical Susceptibility 85

4.4 Dispersion Relations 89

4.5 Spectral Representation 90

4.6 Fluctuation–Dissipation Theorem 91

4.7 Examples of Applications 93

∗4.8 Symmetry Properties 100

4.8.1 General Symmetry Relations 100

4.8.2 Symmetry Properties of the Response Function for Hermitian Operators 102

4.9 Sum Rules 107

4.9.1 General Structure of Sum Rules 107

4.9.2 Application to the Excitations in He II 108

Problems 109

Bibliography for Part I 111

Table of Contents XIII

Part II Relativistic Wave Equations

and their Derivation 115

5.1 Introduction 115

5.2 The Klein–Gordon Equation 116

5.2.1 Derivation by Means of the Correspondence Principle 116 5.2.2 The Continuity Equation 119

5.2.3 Free Solutions of the Klein–Gordon Equation 120

5.3 Dirac Equation 120

5.3.1 Derivation of the Dirac Equation 120

5.3.2 The Continuity Equation 122

5.3.3 Properties of the Dirac Matrices 123

5.3.4 The Dirac Equation in Covariant Form 123

5.3.5 Nonrelativistic Limit and Coupling to the Electromagnetic Field 125

Problems 130

6 Lorentz Transformations and Covariance of the Dirac Equation 131

6.1 Lorentz Transformations 131

6.2 Lorentz Covariance of the Dirac Equation 135

6.2.1 Lorentz Covariance and Transformation of Spinors 135

6.2.2 Determination of the Representation S(Λ) 136

6.2.3 Further Properties of S 142

6.2.4 Transformation of Bilinear Forms 144

6.2.5 Properties of the γ Matrices 145

6.3 Solutions of the Dirac Equation for Free Particles 146

6.3.1 Spinors with Finite Momentum 146

6.3.2 Orthogonality Relations and Density 149

6.3.3 Projection Operators 151

Problems 152

7. Orbital Angular Momentum and Spin 155

7.1 Passive and Active Transformations 155

7.2 Rotations and Angular Momentum 156

Problems 159

8. The Coulomb Potential 161

8.1 Klein–Gordon Equation with Electromagnetic Field 161

8.1.1 Coupling to the Electromagnetic Field 161

8.1.2 Klein–Gordon Equation in a Coulomb Field 162

8.2 Dirac Equation for the Coulomb Potential 168

Problems 179

9 The Foldy–Wouthuysen Transformation

and Relativistic Corrections 181

9.1 The Foldy–Wouthuysen Transformation 181

9.1.1 Description of the Problem 181

9.1.2 Transformation for Free Particles 182

9.1.3 Interaction with the Electromagnetic Field 183

9.2 Relativistic Corrections and the Lamb Shift 187

9.2.1 Relativistic Corrections 187

9.2.2 Estimate of the Lamb Shift 189

Problems 193

10 Physical Interpretation of the Solutions to the Dirac Equation 195

10.1 Wave Packets and “Zitterbewegung” 195

10.1.1 Superposition of Positive Energy States 196

10.1.2 The General Wave Packet 197

∗10.1.3 General Solution of the Free Dirac Equation in the Heisenberg Representation 200

∗10.1.4 Potential Steps and the Klein Paradox 202

10.2 The Hole Theory 204

Problems 207

11 Symmetries and Further Properties of the Dirac Equation 209

∗11.1 Active and Passive Transformations, Transformations of Vectors 209

11.2 Invariance and Conservation Laws 212

11.2.1 The General Transformation 212

11.2.2 Rotations 212

11.2.3 Translations 213

11.2.4 Spatial Reflection (Parity Transformation) 213

11.3 Charge Conjugation 214

11.4 Time Reversal (Motion Reversal) 217

11.4.1 Reversal of Motion in Classical Physics 218

11.4.2 Time Reversal in Quantum Mechanics 221

11.4.3 Time-Reversal Invariance of the Dirac Equation 229

∗11.4.4 Racah Time Reflection 235

∗11.5 Helicity 236

∗11.6 Zero-Mass Fermions (Neutrinos) 239

Problems 244

Bibliography for Part II 245

Table of Contents XV

Part III Relativistic Fields

12 Quantization of Relativistic Fields 249

12.1 Coupled Oscillators, the Linear Chain, Lattice Vibrations 249

12.1.1 Linear Chain of Coupled Oscillators 249

12.1.2 Continuum Limit, Vibrating String 255

12.1.3 Generalization to Three Dimensions, Relationship to the Klein–Gordon Field 258

12.2 Classical Field Theory 261

12.2.1 Lagrangian and Euler–Lagrange Equations of Motion 261 12.3 Canonical Quantization 266

12.4 Symmetries and Conservation Laws, Noether’s Theorem 266

12.4.1 The Energy–Momentum Tensor, Continuity Equations, and Conservation Laws 266

12.4.2 Derivation from Noether’s Theorem of the Conservation Laws for Four-Momentum, Angular Momentum, and Charge 268

Problems 275

13 Free Fields 277

13.1 The Real Klein–Gordon Field 277

13.1.1 The Lagrangian Density, Commutation Relations, and the Hamiltonian 277

13.1.2 Propagators 281

13.2 The Complex Klein–Gordon Field 285

13.3 Quantization of the Dirac Field 287

13.3.1 Field Equations 287

13.3.2 Conserved Quantities 289

13.3.3 Quantization 290

13.3.4 Charge 293

∗13.3.5 The Infinite-Volume Limit 295

13.4 The Spin Statistics Theorem 296

13.4.1 Propagators and the Spin Statistics Theorem 296

13.4.2 Further Properties of Anticommutators and Propagators of the Dirac Field 301

Problems 303

14 Quantization of the Radiation Field 307

14.1 Classical Electrodynamics 307

14.1.1 Maxwell Equations 307

14.1.2 Gauge Transformations 309

14.2 The Coulomb Gauge 309

14.3 The Lagrangian Density for the Electromagnetic Field 311

14.4 The Free Electromagnatic Field and its Quantization 312

14.5 Calculation of the Photon Propagator 316

Problems 320

15 Interacting Fields, Quantum Electrodynamics 321

15.1 Lagrangians, Interacting Fields 321

15.1.1 Nonlinear Lagrangians 321

15.1.2 Fermions in an External Field 322

15.1.3 Interaction of Electrons with the Radiation Field: Quantum Electrodynamics (QED) 322

15.2 The Interaction Representation, Perturbation Theory 323

15.2.1 The Interaction Representation (Dirac Representation) 324 15.2.2 Perturbation Theory 327

15.3 The S Matrix 328

15.3.1 General Formulation 328

15.3.2 Simple Transitions 332

∗15.4 Wick’s Theorem 335

15.5 Simple Scattering Processes, Feynman Diagrams 339

15.5.1 The First-Order Term 339

15.5.2 Mott Scattering 341

15.5.3 Second-Order Processes 346

15.5.4 Feynman Rules of Quantum Electrodynamics 356

∗15.6 Radiative Corrections 358

15.6.1 The Self-Energy of the Electron 359

15.6.2 Self-Energy of the Photon, Vacuum Polarization 365

15.6.3 Vertex Corrections 366

15.6.4 The Ward Identity and Charge Renormalization 368

15.6.5 Anomalous Magnetic Moment of the Electron 371

Problems 373

Bibliography for Part III 375

Appendix 377

A Alternative Derivation of the Dirac Equation 377

B Dirac Matrices 379

B.1 Standard Representation 379

B.2 Chiral Representation 379

B.3 Majorana Representations 380

C Projection Operators for the Spin 380

C.1 Definition 380

C.2 Rest Frame 380

C.3 General Significance of the Projection Operator P (n) 381 D The Path-Integral Representation of Quantum Mechanics 385

E Covariant Quantization of the Electromagnetic Field, the Gupta–Bleuler Method 387

E.1 Quantization and the Feynman Propagator 387

Table of Contents XVII

E.2 The Physical Significance of Longitudinal

and Scalar Photons 389E.3 The Feynman Photon Propagator 392E.4 Conserved Quantities 393

F Coupling of Charged Scalar Mesons

to the Electromagnetic Field 394

Index 397

Nonrelativistic Many-Particle Systems

quantiza-is proved within relativquantiza-istic quantum field theory (the spin-statquantiza-istics rem) An important consequence in many-particle physics is the existence ofFermi–Dirac statistics and Bose–Einstein statistics We shall begin in Sect.1.1 with some preliminary remarks which follow on from Chap 13 of Quan-tum Mechanics1 For the later sections, only the first part, Sect 1.1.1, isessential.

theo-1.1 Identical Particles, Many-Particle States,

and Permutation Symmetry

1.1.1 States and Observables of Identical Particles

We consider N identical particles (e.g., electrons, π mesons) The Hamiltonian

is symmetric in the variables 1, 2, , N Here 1 ≡ x1, σ1denotes the positionand spin degrees of freedom of particle 1 and correspondingly for the otherparticles Similarly, we write a wave function in the form

as follows by renaming the integration variables.

(iii) The adjoint permutation operator P † is defined as usual by

same in the states ψ i and in the permutated states P ψ i

2 It is well known that every permutation can be represented as a product of cyclesthat have no element in common, e.g., (124)(35) Every cycle can be written as

a product of transpositions,

P124 ≡ (124) = (14)(12) even

Each cycle is carried out from left to right (1→ 2, 2 → 4, 4 → 1), whereas the

products of cycles are applied from right to left

1.1 Identical Particles, Many-Particle States, and Permutation Symmetry 5

(v) The converse of (iv) is also true The requirement that an exchange ofidentical particles should not have any observable consequences implies

that all observables O must be symmetric, i.e., permutation invariant Proof ψ| O |ψ = P ψ| O |P ψ = ψ| P † OP |ψ holds for arbitrary ψ Thus, P † OP = O and, hence, P O = OP

Since identical particles are all influenced identically by any physical

pro-cess, all physical operators must be symmetric Hence, the states ψ and P ψ

are experimentally indistinguishable The question arises as to whether all

these N ! states are realized in nature.

In fact, the totally symmetric and totally antisymmetric states ψ s and ψ a

do play a special role These states are defined by

where T is the time-ordering operator.3

(ii) For arbitrary permutations P , the states introduced in (1.1.9) satisfy

P ψ a = (−1) P ψ a , with ( −1) P =



1 for even permutations

−1 for odd permutations Thus, the states ψ s and ψ a form the basis of two one-dimensional repre-

sentations of the permutation group S N For ψ s , every P is assigned the number 1, and for ψ a every even (odd) element is assigned the number1(−1) Since, in the case of three or more particles, the P ij do not all com-

mute with one another, there are, in addition to ψ s and ψ a, also states

for which not all P ij are diagonal Due to noncommutativity, a

com-plete set of common eigenfunctions of all P ij cannot exist These statesare basis functions of higher-dimensional representations of the permu-tation group These states are not realized in nature; they are referred to

3 QM I, Chap 16

as parasymmetric states.4 The fictitious particles that are described bythese states are known as paraparticles and are said to obey parastatis-tics.

1.1.2 Examples

(i) Two particles

Let ψ(1, 2) be an arbitrary wave function The permutation P12leads to P12ψ(1, 2)

(ii) Three particles

We consider the example of a wave function that is a function only of the spatialcoordinates

ψ(1, 2, 3) = ψ(x1, x2, x3).

Application of the permutation P123 yields

P123ψ(x1, x2, x3) = ψ(x2, x3, x1),

i.e., particle 1 is replaced by particle 2, particle 2 by particle 3, and

parti-cle 3 by partiparti-cle 1, e.g., ψ(1, 2, 3) = e −x2(x2−x2)2, P12ψ(1, 2, 3) = e −x2(x2−x2)2,

to integrals over products of spatial wave functions

Let us assume that we have the state

1.1 Identical Particles, Many-Particle States, and Permutation Symmetry 7

with ψ(x1, x2, x3) =x1|1x2|2x3|3|ψ In |x i  jthe particle is labeled by the

num-ber j and the spatial coordinate is x i The effect of P123, for example, is defined asfollows:

If the state|ψ has the wave function ψ(x1, x2, x3), then P |ψ has the wave function

P ψ(x1, x2, x3) The particles are exchanged under the permutation Finally, we

discuss the basis vectors for three particles: If we start from the state |α |β |γ and

apply the elements of the group S3, we get the six states

If we assume that α, β, and γ are all different, then the same is true of the six

states given in (1.1.15) One can group and combine these in the following way toyield invariant subspaces5:

In the bases 3 and 4, the first of the two functions in each case is even under

P12 and the second is odd under P12 (immediately below we shall call these twofunctions|ψ1 and |ψ2) Other operations give rise to a linear combination of the

This fact implies that the basis vectors|ψ1 and |ψ2 span a two-dimensional

repre-sentation of the permutation group S3 The explicit calculation will be carried out

in Problem 1.2

1.2 Completely Symmetric and Antisymmetric States

We begin with the single-particle states|i: |1, |2, The single-particle states of the particles 1, 2, , α, , N are denoted by |i1,|i2, , |i α,

, |i N These enable us to write the basis states of the N -particle system

|i1, , i α , , i N  = |i11 |i α  α |i N  N , (1.2.1)where particle 1 is in state |i11 and particle α in state |i α  α, etc (Thesubscript outside the ket is the number labeling the particle, and the indexwithin the ket identifies the state of this particle.)

Provided that the {|i} form a complete orthonormal set, the product states defined above likewise represent a complete orthonormal system in the

1.2 Completely Symmetric and Antisymmetric States 9

space of N -particle states The symmetrized and antisymmetrized basis states

are then defined by

In other words, we apply all N ! elements of the permutation group S N of N

objects and, for fermions, we multiply by (−1) when P is an odd permutation.

The states defined in (1.2.2) are of two types: completely symmetric andcompletely antisymmetric

Remarks regarding the properties of S ± ≡ √1

N !



P(±1) P P : (i) Let S N be the permutation group (or symmetric group) of N quantities.

Assertion: For every element P ∈ S N , one has P S N = S N

Proof The set P S N contains exactly the same number of elements as S Nand these,

due to the group property, are all contained in S N Furthermore, the elements of

P S N are all different since, if one had P P1 = P P2, then, after multiplication by

P −1 , it would follow that P1= P2

If P is even, then even elements remain even and odd ones remain odd If

P is odd, then multiplication by P changes even into odd elements and vice

1!n2! are different, each of these

terms occurs with a multiplicity of n1!n2!

i1, , i N | S+† S+|i1, , i N  = 1

N ! (n1!n2! )

n !n ! = n1!n2!

Thus, the normalized Bose basis functions are

and further application of √1

N ! S ±, with the identity (1.2.6a), results in

1.3.1 States, Fock Space, Creation and Annihilation Operators

The state (1.2.5) is fully characterized by specifying the occupation numbers

|n1, n2,  = S+|i1, i2, , i N  √ 1

n1!n2! . (1.3.1)Here, n1 is the number of times that the state 1 occurs, n2 the number of

times that state 2 occurs, Alternatively: n1is the number of particles in

state 1, n2is the number of particles in state 2, The sum of all occupation numbers n i must be equal to the total number of particles:

∞



We now combine the states for N = 0, 1, 2, and obtain a complete

orthonormal system of states for arbitrary particle number, which satisfy theorthogonality relation6

n1, n2, |n1 , n

2 ,  = δ n1,n 1
δ n2,n 2
(1.3.3a)and the completeness relation



n1,n2,

|n1, n2,  n1, n2, | =11 (1.3.3b)

This extended space is the direct sum of the space with no particles (vacuum

state|0), the space with one particle, the space with two particles, etc.; it is known as Fock space.

The operators we have considered so far act only within a subspace of

fixed particle number On applying p, x etc to an N -particle state, we obtain

again an N -particle state We now define creation and annihilation operators, which lead from the space of N -particle states to the spaces of N ± 1-particle

 , n i  , | a i | , n i ,  = √ n i δ n i
+1,n i

Expressed in words, the operator a i reduces the occupation number by 1

Assertion:

a i | , n i ,  = √ n i | , n i − 1,  for n i ≥ 1 (1.3.6)and

a i | , n i = 0,  = 0

6 In the states |n1, n2, , the n1, n2 etc are arbitrary natural numbers whosesum is not constrained The (vanishing) scalar product between states of differingparticle number is defined by (1.3.3a)

[a i , a j ] = 0, [a †

i , a †

j ] = 0, [a i , a †

j ] = δ ij (1.3.7a,b,c)

Proof It is clear that (1.3.7a) holds for i = j, since a i commutes with itself For

i = j, it follows from (1.3.6) that

hence also proving (1.3.7c)

Starting from the ground state ≡ vacuum state

1.3.2 The Particle-Number Operator

The particle-number operator (occupation-number operator for the state|i)

opera-1.3.3 General Single- and Many-Particle Operators

Let us consider an operator for the N -particle system which is a sum of

single-particle operators

T = t1+ t2+ + t N ≡

α

e.g., for the kinetic energy t α= p2

α /2m, and for the potential V (x α) For one

particle, the single-particle operator is t Its matrix elements in the basis |i

Our aim is to represent this operator in terms of creation and annihilation

op-erators We begin by taking a pair of states i, j from (1.3.19) and calculating their effect on an arbitrary state (1.3.1) We assume initially that j = i

It is possible, as was done in the third line, to bring the S+to the front, since

it commutes with every symmetric operator If the state j is n j-fold occupied,

it gives rise to n j terms in which |j is replaced by |i Hence, the effect of

S+is to yield n j states| , n i + 1, , n j − 1, , where the change in the

normalization should be noted Equation (1.3.20) thus leads on to

j (y)f(2)(x, y)ϕ k (x)ϕ m (y) (1.3.25)

In (1.3.23), the condition α = β is required as, otherwise, we would have

only a single-particle operator The factor 12 in (1.3.23) is to ensure that eachinteraction is included only once since, for identical particles, symmetry im-

We now investigate the action of one term of the sum constituting F :

A somewhat shorter derivation, and one which also covers the case offermions, proceeds as follows: The commutator and anticommutator for

bosons and fermions, respectively, are combined in the form [a k , a j]∓ = δ kj

1.4.1 States, Fock Space, Creation and Annihilation Operators

For fermions, one needs to consider the states S − |i1, i2, , i N  defined in

(1.2.2), which can also be represented in the form of a determinant:

1.4 Fermions 17

The determinants of one-particle states are called Slater determinants If any

of the single-particle states in (1.4.1) are the same, the result is zero This

is a statement of the Pauli principle: two identical fermions must not occupy

the same state On the other hand, when all the i α are different, then thisantisymmetrized state is normalized to 1 In addition, we have

S − |i2, i1,  = −S − |i1, i2,  (1.4.2)This dependence on the order is a general property of determinants

Here, too, we shall characterize the states by specifying their occupation

numbers, which can now take the values 0 and 1 The state with n1particles

in state 1 and n2 particles in state 2, etc., is

|n1, n2, 

The state in which there are no particles is the vacuum state, represented by

|0 = |0, 0, 

This state must not be confused with the null vector!

We combine these states (vacuum state, single-particle states, two-particle

states, ) to give a state space In other words, we form the direct sum

of the state spaces for the various fixed particle numbers For fermions, this

space is once again known as Fock space In this state space a scalar product

is defined as follows:

n1, n2, |n1 , n

2 ,  = δ n1,n 1
δ n2,n 2
; (1.4.3a)i.e., for states with equal particle number (from a single subspace), it is iden-tical to the previous scalar product, and for states from different subspaces

it always vanishes Furthermore, we have the completeness relation

Here, we wish to introduce creation operators a †

i once again These must

be defined such that the result of applying them twice is zero Furthermore,the order in which they are applied must play a role We thus define the

which also implies the impossibility of double occupation

The anticommutator encountered in (1.4.5a) and the commutator of two

operators A and B are defined by

{A, B} ≡ [A, B]+ ≡ AB + BA

Given these preliminaries, we can now address the precise formulation If onewants to characterize the states by means of occupation numbers, one has tochoose a particular ordering of the states This is arbitrary but, once chosen,must be adhered to The states are then represented as

number of anticommutations necessary to bring the a †

i to the position i.

The adjoint relation reads:

 , n i , | a i= (1− n i)(−1)Pj<i n j  , n i + 1, | (1.4.9)This yields the matrix element

δ n i +1,n i
= 0 always gives zero The factor n i also ensures that the right-hand

side cannot become equal to the state| , n i  − 1,  = | , −1, .

To summarize, the effects of the creation and annihilation operators are

a †

i | , n i ,  = (1 − n i)(−1)Pj<i n j | , n i + 1, 

a | , n ,  = n(−1)Pj<i n j | , n − 1,  (1.4.12)

of the property (1.4.13b) one can regard a †

i a i as the occupation-number erator for the state |i By taking the sum of (1.4.13a,b), one obtains the

sum-[a i , a j]+= 0, [a †

i , a †

j]+= 0, [a i , a †

1.4.2 Single- and Many-Particle Operators

For fermions, too, the operators can be expressed in terms of creation andannihilation operators The form is exactly the same as for bosons, (1.3.21)and (1.3.24) Now, however, one has to pay special attention to the order ofthe creation and annihilation operators

The important relation

X

α

from which, according to (1.3.26), one also obtains two-particle (and many-particle)

operators, can be proved as follows: Given the state S − |i1, i2, , i N , we assume,

without loss of generality, the arrangement to be i1 < i2 < < i N Application

of the left-hand side of (1.4.15) gives

The symbol| j →i implies that the state|j is replaced by |i In order to bring the

i into the right position, one has to carry outP

k<j n k+P

k<i n k permutations of

rows for i ≤ j andPk<j n k+P

k<i n k − 1 permutations for i > j.

This yields the same phase factor as does the right-hand side of (1.4.15):

a † i a j | , n i , , n j ,  = n j(−1)Pk<j n k a † i | , n i , , n j − 1, 

= n i(1− n i)(−1)Pk<i n k+ P

k<j n k −δ i>j | , n i + 1, , n j − 1, 

In summary, for bosons and fermions, the single- and two-particle operators

can be written, respectively, as

From this point on, the development of the theory can be presented neously for bosons and fermions

simulta-1.5 Field Operators

1.5.1 Transformations Between Different Basis Systems

Consider two basis systems{|i} and {|λ} What is the relationship between the operators a i and a λ?

The state|λ can be expanded in the basis {|i}:

where ϕ i(x) is the single-particle wave function in the coordinate

representa-tion The creation and annihilation operators corresponding to the eigenstates

of position are called field operators

The operator ψ † (x) (ψ(x)) creates (annihilates) a particle in the position

eigenstate|x, i.e., at the position x The field operators obey the following

where the upper sign applies to fermions and the lower one to bosons

We shall now express a few important operators in terms of the fieldoperators



d3xd3x  ψ † (x)ψ †(x )V (x, x  )ψ(x  )ψ(x) (1.5.6d)

Particle density (particle-number density)

The particle-density operator is given by

This representation is valid in any basis and can also be expressed in terms

of the field operators

many-ψ(x) However, the analogy is no more than a formal one since the former is

an operator and the latter a complex function This formal correspondence

has given rise to the term second quantization, since the operators, in the

cre-ation and annihilcre-ation operator formalism, can be obtained by replacing the

1.5 Field Operators 23

wave function ψ(x) in the single-particle densities by the operator ψ(x) This

immediately enables one to write down, e.g., the current-density operator(see Problem 1.6)

j(x) =

2im [ψ

†(x)∇ψ(x) − (∇ψ † (x))ψ(x)] (1.5.11)

The kinetic energy (1.5.12) has a formal similarity to the expectation value

of the kinetic energy of a single particle, where, however, the wavefunction isreplaced by the field operator

Remark The representations of the operators in terms of field operators that we

found above could also have been obtained directly For example, for the number density

particle-Z

d3ξd3ξ  ψ † ξ) ξ| δ(3)(x− bξ) ˛˛ξ ¸

ψ( ξ  ) = ψ † (x)ψ(x), (1.5.12)

where bξ is the position operator of a single particle and where we have made use of

the fact that the matrix element within the integral is equal to δ(3)(x−ξ)δ(3)(ξ−ξ ).

In general, for a k-particle operator V k:

∂t ψ(x, t) = −[H, ψ(x, t)] = −e iHt/
[H, ψ(x, 0)] e −iHt/
. (1.5.16)

Using the relation

[AB, C] − = A[B, C] ± ∓ [A, C] ± B Fermi

one obtains for the commutators with the kinetic energy:

In this last equation, (1.5.17) and (1.5.5c) are used to proceed from the second

line Also, after the third line, in addition to ψ(x  )ψ(x ) =∓ψ(x  )ψ(x ), the

symmetry V (x, x  ) = V (x  , x) is exploited Together, these expressions give

the equation of motion (1.5.15) of the field operator, which is also known as

the field equation.

The equation of motion for the adjoint field operator reads:

i
˙ψ † (x, t) = −



−
22m ∇2+ U (x)

where it is assumed that V (x, x )∗ = V (x, x ).

If (1.5.15) is multiplied from the left by ψ † (x, t) and (1.5.18) from the right

by ψ(x, t), one obtains the equation of motion for the density operator

where j(x) is the particle current density defined in (1.5.11) Equation (1.5.19)

is the continuity equation for the particle-number density

1.6 Momentum Representation 25

1.6 Momentum Representation

1.6.1 Momentum Eigenfunctions and the Hamiltonian

The momentum representation is particularly useful in translationally variant systems We base our considerations on a rectangular normalization

in-volume of dimensions L x , L y and L z The momentum eigenfunctions, which

are used in place of ϕ i(x), are normalized to 1 and are given by

ϕk (x) = eik·x /

√

with the volume V = L x L y L z By assuming periodic boundary conditions

the allowed values of the wave vector k are restricted to

For two-particle potentials V (x − x ) that depend only on the relative

coor-dinates of the two particles, it is useful to introduce their Fourier transform

For the matrix element of the two-particle potential, one then finds

The interaction term allows a pictorial interpretation It causes the

annihila-tion of two particles with wave vectors k and p and creates in their place two particles with wave vectors k− q and p + q This is represented in Fig 1.1a.

The full lines denote the particles and the dotted lines the interaction

po-tential Vq The amplitude for this transition is proportional to Vq This

Fig 1.1 a) Diagrammatic representation of the interaction term in the

Hamilto-nian (1.6.6) b) The diagrammatic representation of the double scattering of two

particles

1.6 Momentum Representation 27

grammatic form is a useful way of representing the perturbation-theoreticaldescription of such processes The double scattering of two particles can berepresented as shown in Fig 1.1b, where one must sum over all intermediatestates

1.6.2 Fourier Transformation of the Density

The other operators considered in the previous section can also be expressed

in the momentum representation As an important example, we shall look

at the density operator The Fourier transform of the density operator8 isdefined by

pap It will always be clear from the context which one of the two

meanings is meant The operator for the total number of particles (1.3.13) inthe momentum representation reads

1.6.3 The Inclusion of Spin

Up until now, we have not explicitly considered the spin One can think of

it as being included in the previous formulas as part of the spatial degree

8 The hat on the operator, as used here for ˆnqand previously for the number operator, will only be retained where it is needed to avoid confusion