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Advanced Quantum Mechanics

Franz Schwabl

Advanced

Quantum Mechanics

Translated by Roginald Hilton and Angela Lahee

Third Edition

With 79 Figures, 4 Tables, and 103 Problems

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

Preface to the Third Edition

In the new edition, supplements, additional explanations and cross references have been added at numerous places, including new formulations of the prob- lems. Figures have been redrawn and the layout has been improved. In all these additions I have intended not to change the compact character of the book. 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 convey the 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 to thank all colleagues and students who have made suggestions to improve the book, as well as the publisher, Dr. Thorsten Schneider and Mrs. J. Lenz for the excellent cooperation.

Munich, May 2005 F. Schwabl

X Preface to the First Edition

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 teaching material already available. This book stems from lectures given regularly by the author at the Tech- 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, 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 made many valuable suggestions. I am indebted to Dr. A. Lahee for supplying the 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 who gave their help so generously, as well as to Dr. Hans-J¨urgen K¨olsch of Springer-Verlag, I wish to express my sincere gratitude.

Munich, March 1999 F. Schwabl

Table of Contents

Table of Contents XIII

Part II. Relativistic Wave Equations

XIV Table of Contents

• 1. Second Quantization Part I. Nonrelativistic Many-Particle Systems
     • and Permutation Symmetry 1.1 Identical Particles, Many-Particle States,
     • 1.1.1 States and Observables of Identical Particles
     • 1.1.2 Examples
  • 1.2 Completely Symmetric and Antisymmetric States
  • 1.3 Bosons - and Annihilation Operators 1.3.1 States, Fock Space, Creation
    • 1.3.2 The Particle-Number Operator
    • 1.3.3 General Single- and Many-Particle Operators
  • 1.4 Fermions - and Annihilation Operators 1.4.1 States, Fock Space, Creation
    • 1.4.2 Single- and Many-Particle Operators
  • 1.5 Field Operators
    • 1.5.1 Transformations Between Different Basis Systems
    • 1.5.2 Field Operators
    • 1.5.3 Field Equations
  • 1.6 Momentum Representation
    • 1.6.1 Momentum Eigenfunctions and the Hamiltonian
    • 1.6.2 Fourier Transformation of the Density
    • 1.6.3 The Inclusion of Spin
  • Problems
• 2. Spin-1/2 Fermions
  • 2.1 Noninteracting Fermions
    • 2.1.1 The Fermi Sphere, Excitations
    • 2.1.2 Single-Particle Correlation Function
    • 2.1.3 Pair Distribution Function
      • Density Correlation Functions, and Structure Factor ∗2.1.4 Pair Distribution Function,
        • of the Electron Gas 2.2 Ground State Energy and Elementary Theory
        • 2.2.1 Hamiltonian
          • in the Hartree–Fock Approximation 2.2.2 Ground State Energy
          • due to the Coulomb Interaction 2.2.3 Modification of Electron Energy Levels
    • 2.3 Hartree–Fock Equations for Atoms
    • Problems
• 3. Bosons
     • 3.1 Free Bosons - 3.1.1 Pair Distribution Function for Free Bosons
       • ∗3.1.2 Two-Particle States of Bosons
     • 3.2 Weakly Interacting, Dilute Bose Gas - 3.2.1 Quantum Fluids and Bose–Einstein Condensation - of the Weakly Interacting Bose Gas 3.2.2 Bogoliubov Theory
       • ∗3.2.3 Superfluidity
     • Problems
• 4. Correlation Functions, Scattering, and Response
     • 4.1 Scattering and Response
     • 4.2 Density Matrix, Correlation Functions
     • 4.3 Dynamical Susceptibility
     • 4.4 Dispersion Relations
     • 4.5 Spectral Representation
     • 4.6 Fluctuation–Dissipation Theorem
     • 4.7 Examples of Applications
  • ∗4.8 Symmetry Properties - 4.8.1 General Symmetry Relations - for Hermitian Operators 4.8.2 Symmetry Properties of the Response Function
    • 4.9 Sum Rules - 4.9.1 General Structure of Sum Rules - 4.9.2 Application to the Excitations in He II
    • Problems
• Bibliography for Part I
  • and their Derivation 5. Relativistic Wave Equations
  • 5.1 Introduction
  • 5.2 The Klein–Gordon Equation
    • 5.2.1 Derivation by Means of the Correspondence Principle
    • 5.2.2 The Continuity Equation
    • 5.2.3 Free Solutions of the Klein–Gordon Equation
  • 5.3 Dirac Equation
    • 5.3.1 Derivation of the Dirac Equation
    • 5.3.2 The Continuity Equation
    • 5.3.3 Properties of the Dirac Matrices
    • 5.3.4 The Dirac Equation in Covariant Form
      • to the Electromagnetic Field 5.3.5 Nonrelativistic Limit and Coupling
  • Problems
  • and Covariance of the Dirac Equation 6. Lorentz Transformations
  • 6.1 Lorentz Transformations
  • 6.2 Lorentz Covariance of the Dirac Equation
    • 6.2.1 Lorentz Covariance and Transformation of Spinors
    • 6.2.2 Determination of the Representation S(Λ)
    • 6.2.3 Further Properties of S
    • 6.2.4 Transformation of Bilinear Forms
    • 6.2.5 Properties of the γ Matrices
  • 6.3 Solutions of the Dirac Equation for Free Particles
    • 6.3.1 Spinors with Finite Momentum
    • 6.3.2 Orthogonality Relations and Density
    • 6.3.3 Projection Operators
  • Problems
• 7. Orbital Angular Momentum and Spin
  • 7.1 Passive and Active Transformations
  • 7.2 Rotations and Angular Momentum
  • Problems
• 8. The Coulomb Potential
  • 8.1 Klein–Gordon Equation with Electromagnetic Field
    • 8.1.1 Coupling to the Electromagnetic Field
    • 8.1.2 Klein–Gordon Equation in a Coulomb Field
  • 8.2 Dirac Equation for the Coulomb Potential
  • Problems
    • and Relativistic Corrections 9. The Foldy–Wouthuysen Transformation
    • 9.1 The Foldy–Wouthuysen Transformation - 9.1.1 Description of the Problem - 9.1.2 Transformation for Free Particles - 9.1.3 Interaction with the Electromagnetic Field
    • 9.2 Relativistic Corrections and the Lamb Shift - 9.2.1 Relativistic Corrections - 9.2.2 Estimate of the Lamb Shift
    • Problems
    • of the Solutions to the Dirac Equation 10. Physical Interpretation
    • 10.1 Wave Packets and “Zitterbewegung” - 10.1.1 Superposition of Positive Energy States - 10.1.2 The General Wave Packet - in the Heisenberg Representation ∗10.1.3 General Solution of the Free Dirac Equation
      • ∗10.1.4 Potential Steps and the Klein Paradox
    • 10.2 The Hole Theory
    • Problems
    • of the Dirac Equation 11. Symmetries and Further Properties - Transformations of Vectors ∗11.1 Active and Passive Transformations,
    • 11.2 Invariance and Conservation Laws - 11.2.1 The General Transformation - 11.2.2 Rotations - 11.2.3 Translations - 11.2.4 Spatial Reflection (Parity Transformation)
    • 11.3 Charge Conjugation
    • 11.4 Time Reversal (Motion Reversal) - 11.4.1 Reversal of Motion in Classical Physics - 11.4.2 Time Reversal in Quantum Mechanics - 11.4.3 Time-Reversal Invariance of the Dirac Equation
      • ∗11.4.4 Racah Time Reflection
  • ∗11.5 Helicity
  • ∗11.6 Zero-Mass Fermions (Neutrinos)
    • Problems
• Bibliography for Part II
• 12. Quantization of Relativistic Fields Part III. Relativistic Fields
  • 12.1 Coupled Oscillators, the Linear Chain, Lattice Vibrations - 12.1.1 Linear Chain of Coupled Oscillators - 12.1.2 Continuum Limit, Vibrating String - Relationship to the Klein–Gordon Field 12.1.3 Generalization to Three Dimensions,
  • 12.2 Classical Field Theory - 12.2.1 Lagrangian and Euler–Lagrange Equations of Motion
  • 12.3 Canonical Quantization
  • 12.4 Symmetries and Conservation Laws, Noether’s Theorem - and Conservation Laws 12.4.1 The Energy–Momentum Tensor, Continuity Equations, - Angular Momentum, and Charge of the Conservation Laws for Four-Momentum,
  • Problems
• 13. Free Fields
  • 13.1 The Real Klein–Gordon Field - and the Hamiltonian 13.1.1 The Lagrangian Density, Commutation Relations, - 13.1.2 Propagators
  • 13.2 The Complex Klein–Gordon Field
  • 13.3 Quantization of the Dirac Field - 13.3.1 Field Equations - 13.3.2 Conserved Quantities - 13.3.3 Quantization - 13.3.4 Charge
    • ∗13.3.5 The Infinite-Volume Limit
  • 13.4 The Spin Statistics Theorem - 13.4.1 Propagators and the Spin Statistics Theorem - and Propagators of the Dirac Field 13.4.2 Further Properties of Anticommutators
  • Problems
• 14. Quantization of the Radiation Field
  • 14.1 Classical Electrodynamics - 14.1.1 Maxwell Equations - 14.1.2 Gauge Transformations
  • 14.2 The Coulomb Gauge
  • 14.3 The Lagrangian Density for the Electromagnetic Field
  • 14.4 The Free Electromagnatic Field and its Quantization
    • 14.5 Calculation of the Photon Propagator XVI Table of Contents
    • Problems
• 15. Interacting Fields, Quantum Electrodynamics
      • 15.1 Lagrangians, Interacting Fields
        • 15.1.1 Nonlinear Lagrangians
        • 15.1.2 Fermions in an External Field
          • Quantum Electrodynamics (QED) 15.1.3 Interaction of Electrons with the Radiation Field:
      • 15.2 The Interaction Representation, Perturbation Theory
        • 15.2.1 The Interaction Representation (Dirac Representation)
        • 15.2.2 Perturbation Theory
      • 15.3 The S Matrix
        • 15.3.1 General Formulation
        • 15.3.2 Simple Transitions
  • ∗15.4 Wick’s Theorem
    • 15.5 Simple Scattering Processes, Feynman Diagrams
      • 15.5.1 The First-Order Term
      • 15.5.2 Mott Scattering
      • 15.5.3 Second-Order Processes
      • 15.5.4 Feynman Rules of Quantum Electrodynamics
  • ∗15.6 Radiative Corrections - 15.6.1 The Self-Energy of the Electron - 15.6.2 Self-Energy of the Photon, Vacuum Polarization - 15.6.3 Vertex Corrections - 15.6.4 The Ward Identity and Charge Renormalization - 15.6.5 Anomalous Magnetic Moment of the Electron
    • Problems
• Bibliography for Part III
• Appendix - A Alternative Derivation of the Dirac Equation - B Dirac Matrices - B.1 Standard Representation - B.2 Chiral Representation - B.3 Majorana Representations - C Projection Operators for the Spin - C.1 Definition - C.2 Rest Frame - C.3 General Significance of the Projection Operator P (n) - D The Path-Integral Representation of Quantum Mechanics - the Gupta–Bleuler Method E Covariant Quantization of the Electromagnetic Field, - E.1 Quantization and the Feynman Propagator

Table of Contents XVII

E.2 The Physical Significance of Longitudinal and Scalar Photons............................... 389 E.3 The Feynman Photon Propagator.................. 392 E.4 Conserved Quantities............................. 393 F Coupling of Charged Scalar Mesons to the Electromagnetic Field............................. 394

Index......................................................... 397

1. Second Quantization

In this first part, we shall consider nonrelativistic systems consisting of a large number of identical particles. In order to treat these, we will introduce a particularly efficient formalism, namely, the method of second quantiza- tion. Nature has given us two types of particle, bosons and fermions. These have states that are, respectively, completely symmetric and completely an- tisymmetric. Fermions possess half-integer spin values, whereas boson spins have integer values. This connection between spin and symmetry (statistics) is proved within relativistic quantum field theory (the spin-statistics theo- rem). An important consequence in many-particle physics is the existence of Fermi–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 Mechanics^1. For the later sections, only the first part, Sect. 1.1.1, is essential.

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

H = H(1, 2 ,... , N ) (1.1.1)

is symmetric in the variables 1, 2 ,... , N. Here 1 ≡ x 1 , σ 1 denotes the position and spin degrees of freedom of particle 1 and correspondingly for the other particles. Similarly, we write a wave function in the form

ψ = ψ(1, 2 ,... , N ). (1.1.2)

The permutation operator Pij , which interchanges i and j, has the following effect on an arbitrary N -particle wave function

(^1) F. Schwabl, Quantum Mechanics, 3rd (^) ed., Springer, Berlin Heidelberg, 2002; in subsequent citations this book will be referred to as QM I.

4 1. Second Quantization

Pij ψ(... , i,... , j,... ) = ψ(... , j,... , i,... ). (1.1.3)

We remind the reader of a few important properties of this operator. Since P (^) ij^2 = 1, the eigenvalues of Pij are ±1. Due to the symmetry of the Hamilto- nian, one has for every element P of the permutation group

P H = HP. (1.1.4)

The permutation group SN which consists of all permutations of N objects has N! elements. Every permutation P can be represented as a product of transpositions Pij. An element is said to be even (odd) when the number of Pij ’s is even (odd).^2

A few properties:

(i) If ψ(1,... , N ) is an eigenfunction of H with eigenvalue E, then the same also holds true for P ψ(1,... , N ). Proof. Hψ = Eψ ⇒ HP ψ = P Hψ = EP ψ. (ii) For every permutation one has

〈ϕ|ψ〉 = 〈P ϕ|P ψ〉 , (1.1.5) as follows by renaming the integration variables. (iii) The adjoint permutation operator P †^ is defined as usual by

〈ϕ|P ψ〉 =

P †ϕ|ψ

It follows from this that 〈ϕ|P ψ〉 =

P −^1 ϕ|P −^1 P ψ

P −^1 ϕ|ψ

⇒ P †^ = P −^1

and thus P is unitary P †P = P P †^ = 1. (1.1.6)

(iv) For every symmetric operator S(1,... , N ) we have

[P, S] = 0 (1.1.7) and 〈P ψi| S |P ψj 〉 = 〈ψi| P †SP |ψj 〉 = 〈ψi| P †P S |ψj 〉 = 〈ψi| S |ψj 〉. (1.1.8) This proves that the matrix elements of symmetric operators are the same in the states ψi and in the permutated states P ψi. (^2) It is well known that every permutation can be represented as a product of cycles that have no element in common, e.g., (124)(35). Every cycle can be written as a product of transpositions, e.g. (12) odd P 124 ≡ (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.

6 1. Second Quantization

as parasymmetric states.^4. The fictitious particles that are described by these 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 P 12 leads to P 12 ψ(1, 2) = ψ(2, 1). From these two wave functions one can form

ψs = ψ(1, 2) + ψ(2, 1) symmetric ψa = ψ(1, 2) − ψ(2, 1) antisymmetric (1.1.12)

under the operation P 12. For two particles, the symmetric and antisymmetric states exhaust all possibilities.

(ii) Three particles We consider the example of a wave function that is a function only of the spatial coordinates

ψ(1, 2 , 3) = ψ(x 1 , x 2 , x 3 ).

Application of the permutation P 123 yields

P 123 ψ(x 1 , x 2 , x 3 ) = ψ(x 2 , x 3 , x 1 ),

i.e., particle 1 is replaced by particle 2, particle 2 by particle 3, and parti- cle 3 by particle 1, e.g., ψ(1, 2 , 3) = e−x

(^21) (x (^22) −x (^23) ) 2 , P 12 ψ(1, 2 , 3) = e−x

(^22) (x (^21) −x (^23) ) 2 , P 123 ψ(1, 2 , 3) = e−x (^22) (x (^23) −x (^21) ) 2

. We consider

P 13 P 12 ψ(1, 2 , 3) = P 13 ψ(2, 1 , 3) = ψ(2, 3 , 1) = P 123 ψ(1, 2 , 3) P 12 P 13 ψ(1, 2 , 3) = P 12 ψ(3, 2 , 1) = ψ(3, 1 , 2) = P 132 ψ(1, 2 , 3) (P 123 )^2 ψ(1, 2 , 3) = P 123 ψ(2, 3 , 1) = ψ(3, 1 , 2) = P 132 ψ(1, 2 , 3).

Clearly, P 13 P 12 = P 12 P 13. S 3 , the permutation group for three objects, consists of the following 3! = 6 ele- ments:

S 3 = { 1 , P 12 , P 23 , P 31 , P 123 , P 132 = (P 123 )^2 }. (1.1.13)

We now consider the effect of a permutation P on a ket vector. Thus far we have only allowed P to act on spatial wave functions or inside scalar products which lead to integrals over products of spatial wave functions. Let us assume that we have the state

|ψ〉 =

X

x 1 ,x 2 ,x 3

direct product z }| { |x 1 〉 1 |x 2 〉 2 |x 3 〉 3 ψ(x 1 , x 2 , x 3 ) (1.1.14)

(^4) A.M.L. Messiah and O.W. Greenberg, Phys. Rev. B 136, 248 (1964), B 138, 1155 (1965).

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

with ψ(x 1 , x 2 , x 3 ) = 〈x 1 | 1 〈x 2 | 2 〈x 3 | 3 |ψ〉. In |xi〉j the particle is labeled by the num- ber j and the spatial coordinate is xi. The effect of P 123 , for example, is defined as follows:

P 123 |ψ〉 =

X x 1 ,x 2 ,x 3

|x 1 〉 2 |x 2 〉 3 |x 3 〉 1 ψ(x 1 , x 2 , x 3 ).

=

X

x 1 ,x 2 ,x 3

|x 3 〉 1 |x 1 〉 2 |x 2 〉 3 ψ(x 1 , x 2 , x 3 )

In the second line the basis vectors of the three particles in the direct product are once more written in the usual order, 1,2,3. We can now rename the summation variables according to (x 1 , x 2 , x 3 ) → P 123 (x 1 , x 2 , x 3 ) = (x 2 , x 3 , x 1 ). From this, it follows that

P 123 |ψ〉 =

X

x 1 ,x 2 ,x 3

|x 1 〉 1 |x 2 〉 2 |x 3 〉 3 ψ(x 2 , x 3 , x 1 ).

If the state |ψ〉 has the wave function ψ(x 1 , x 2 , x 3 ), then P |ψ〉 has the wave function P ψ(x 1 , x 2 , x 3 ). 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 S 3 , we get the six states

|α〉 |β〉 |γ〉 P 12 |α〉 |β〉 |γ〉 = |β〉 |α〉 |γ〉 , P 23 |α〉 |β〉 |γ〉 = |α〉 |γ〉 |β〉 , P 31 |α〉 |β〉 |γ〉 = |γ〉 |β〉 |α〉 , P 123 |α〉 1 |β〉 2 |γ〉 3 = |α〉 2 |β〉 3 |γ〉 1 = |γ〉 |α〉 |β〉 , P 132 |α〉 |β〉 |γ〉 = |β〉 |γ〉 |α〉.

(1.1.15)

Except in the fourth line, the indices for the particle number are not written out, but are determined by the position within the product (particle 1 is the first factor, etc.). It is the particles that are permutated, not the arguments of the 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 to yield invariant subspaces 5 :

Invariant subspaces:

Basis 1 (symmetric basis):

1 √ 6

(|α〉 |β〉 |γ〉 + |β〉 |α〉 |γ〉 + |α〉 |γ〉 |β〉 + |γ〉 |β〉 |α〉 + |γ〉 |α〉 |β〉 + |β〉 |γ〉 |α〉) (1.1.16a)

Basis 2 (antisymmetric basis):

1 √ 6

(|α〉 |β〉 |γ〉 − |β〉 |α〉 |γ〉 − |α〉 |γ〉 |β〉 − |γ〉 |β〉 |α〉 + |γ〉 |α〉 |β〉 + |β〉 |γ〉 |α〉) (1.1.16b)

(^5) An invariant subspace is a subspace of states which transforms into itself on application of the group elements.