F.Schwabl Advanced Quantum Mechanics Third Edition Springer
Professor Dr.Franz Schwabl 7 Garching.Germany E-mail:schwabl@physik.tu-muenchen.de Title of the original German edition:Quantenmechanik fur Fortgeschrittene(QM I) Library of Congress Control Number:2005928641 ISBN-10 3-540-25901-5 3rd ed.Springer Berlin Heidelberg New York ISBN-13 978-3-540-25901-0 3rd ed.Springer Berlin Heidelberg New York ISBN 3-540-40152-0 and ed.Springer-Verlag Berlin Heidelberg New York ze in data banks.Duplication of Springer is a part of Springer Science+Business Media springeronline.com Springer-Verlag Berlin Heidelberg9, Printed in The Netherlands The use of general descriptive nam Printed on acid-free paper6//YL531
Professor Dr. Franz Schwabl Physik-Department Technische Universit¨at Munchen ¨ James-Franck-Strasse 85747 Garching, Germany E-mail: schwabl@physik.tu-muenchen.de Translator: Dr. Roginald Hilton Dr. Angela Lahee Title of the original German edition: Quantenmechanik für Fortgeschrittene (QM II) (Springer-Lehrbuch) ISBN 3-540-67730-5 © Springer-Verlag Berlin Heidelberg 2000 Library of Congress Control Number: 2005928641 ISBN-10 3-540-25901-5 3rd ed. Springer Berlin Heidelberg New York ISBN-13 978-3-540-25901-0 3rd ed. Springer Berlin Heidelberg New York ISBN 3-540-40152-0 2nd ed. Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 1999, 2004, 2005 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: A. Lahee and F. Herweg EDV Beratung using a Springer TEX macro package Production: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig Cover design: design & production GmbH, Heidelberg Printed on acid-free paper 56/3141/YL 5 4 3 2 1 0
The true physics is that which will,one day achieve the inclusion of man in his whole in a coherent picture of the world. Pierre Teilhard de Chardin To my daughter Birgitta
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.Fig res have been redra wn and the lay out has been improved.in all additions I have intended to cha ct c 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.Tauber for occasional advice.Special thanks go to them and to Mrs.Jorg-Muiller 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
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 problems. 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. Wollenweber. 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
Preface to the First Edition This textbook deals with advanced topics in the field of quantum mechanics mater whichissnda ondpi tum mechanic. into three parts:I -Body Systems,II.Relat vistic Wave Equations,and III.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 prior knowledge,except in the field of basic quantum mechanics.The inclusion of all mathematical steps and full presentation of intermediate calculations ensures ease of understanding.A number of problems are included at the end of each chapter Sections or parts thereof that can be omitted in a first emarked with a star,a nd subsidia emarks sma pri It is not necessary have read P works 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 encourage further reading.A list of other textbooks is included at the end of each of the three parts. In contrast to Quantum Mechanics I.the present book treats relativistic phenomena,and classical and relativistic quantum fields. Part Iin oduces the forr of s ems that nd quntization and a applies this t can be e described using s mple methods These incl tron ga sand excit in weakly interacting Bose gases.The basic properties of the correlation and response functions of many-particle systems are also treated here. The second part deals with the Klein-Gordon and Dirac equations.Im- portant aspects,such as motion in a Coulomb potential are discussed,and particular attention is paid to symmetry properties. The third part presents Noether's theorem,the quantization of the Klein- Gordon Dira and adiatio n fields,and the tatistics the n.The final interac rix theory, heorem,Feynman rules,a few simple proces scattering and scattering,and basic aspects of radiative corrections are discussed
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 quantum mechanics. The book, which comprises a total of 15 chapters, is divided into three parts: I. Many-Body Systems, II. Relativistic Wave Equations, and III. Relativistic Fields. The text is written in such a way as to attach importance to a rigorous presentation while, at the same time, requiring no prior knowledge, except in the field of basic quantum mechanics. The inclusion of all mathematical steps and full presentation of intermediate calculations ensures ease of understanding. A number of problems are included at the end of each chapter. Sections or parts thereof that can be omitted in a first reading are marked with a star, and subsidiary calculations and remarks not essential for comprehension are given in small print. It is not necessary to have read Part I in order to understand Parts II and III. References to other works in the literature are given whenever it is felt they serve a useful purpose. These are by no means complete and are simply intended to encourage further reading. A list of other textbooks is included at the end of each of the three parts. In contrast to Quantum Mechanics I, the present book treats relativistic phenomena, 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 weakly interacting Bose gases. The basic properties of the correlation and response functions of many-particle systems are also treated here. The second part deals with the Klein–Gordon and Dirac equations. Important aspects, such as motion in a Coulomb potential are discussed, and particular attention is paid to symmetry properties. The third part presents Noether’s theorem, the quantization of the Klein– Gordon, Dirac, and radiation fields, and the spin-statistics theorem. The final chapter treats interacting fields using the example of quantum electrodynamics: S-matrix theory, Wick’s theorem, Feynman rules, a few simple processes such as Mott scattering and electron–electron scattering, and basic aspects of radiative corrections are discussed
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 Uni rsity Munich Many and co assisted in the t:Ms.I Wefers,Ms.E.Jorg-Mille Schwier Schenk,M. .alM.nd Wb.Maier. Wefers 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 fi nal rendition Toall those mentioned here,and to the ues who gave their】 Hans-Jiirgen Kolsch of Munich,March 1999 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 Technical University Munich. Many colleagues and coworkers assisted in the production 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 Part I.Nonrelativistic Many-Particle Systems 1.Second Quantization. 3 1.1 Identical Particles.Many-Particle States. and Permutation Symmetry... 1.1.1 States and Observables of Identical Particles......... 1.1.2 Examples 6 1.Completely Symmetric and States 1.3Bo 10 1.3.1 States,Fock Space .Creation lation Operators 10 1.3.2 The Particle-Number Operator. 1.3.3 General Single-and Manv-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......... 10 1.5 Field Op rmations Between Different Basis Systems.... 20 Field 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 2.1.1 The Fermi Sphere,Excitations 2.1 Noninteracting Fermions 33 23 2.1.2Si ngle-Particle Co lation Function 35 Distribution Function......................... 36 Pair Distribution Function, Density Correlation Functions,and Structure Factor..39
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
Table of Contents 2.2 Ground State Energy and Elementary Theory of the Electron Gas.....................................41 2.2.1 Hamiltonian. …41 2.2.2 Ground State Energy 42 2.2.3 Modificatio of Elec ergy Levels due to the Coulom nteraction.................... 2.3 Hartree-Fock Equations for Atoms....................... 4 Problems.…52 Bo. n 55 3.1 Free Bosons. 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 Bogoliuboy Theory of the Weakly Interacting Bose Gas 62 *393 Superftuidity..................................... problems ................................................. 72 4.Correlation Functions,Scattering,and Response 75 Scatt d Re spons 4.4.4。。。。”””””””0”4”4.44.4.4.4 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. 93 100 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.. 4.9.2 Application to the Excitations in He II..............108 Problems.· ...109 Bibliography for Part I... 。。。 ..111
XII Table of Contents 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 5.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 110 100 5.3 Dirac E uation.. 120 5.3. 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 195 Problems .130 6.Lorentz Transformations and Covariance of the Dirac Equation ,.131 6.1 Lorentz Transformations 121 6.2 entz Covariance of the Dirac Equation 135 0 Lorentz Co e and Transformation of Spinors.... 135 6.2.2 Determination of the Representation S(A)..........13 6.2.3 Further Properties of S...........................142 6.2.4 Transformation of Bilinear Forms................... 144 6.2.5 Properties of the y Matrices.... 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 1A0 6.3.3 Projection Operators 151 Problems................................. 152 7.Orbital Angular Mom 155 ve and Acti orma 15 Rotations and Angular Momentum...................... Problems.................................................. 159 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.........162 8.2 Dirac Equation for the Coulomb Potential.................168 Problems.. ..179
Table of Contents XIII Part II. Relativistic Wave Equations 5. 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
XIV Table of Contents 9.The Foldy-Wouthuysen Transformation and Relativistic Corrections..............................181 9.1 The Foldy-Wouthuysen Transformation......... .181 9.1.1 Description of the Problem 1只1 91 2 Transfo mation for Free Particles 1只0 0121m ction with the Elec etic Field 1只3 9.2 Relativist ra trom ions and the Lamb Shift.. 187 9.2 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 l0.1 Wave Packets and“Zitterbewegung”. 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 Re tatic 200 10.1.4 Potential Steps and the Klein Paradox 202 10.2 The Hole Theory...............20 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 )12 11.2.4 Spatial Relection (Parity Transformation) 912 11.3 Charge Conjugatic 214 11.4 Time Reversal (Motion Reversal) 21 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 226 11.6 Zero-Mass Fermions (Neutrinos)..................... )20 Problem 244 Bibliography for Part II …245
XIV Table of Contents 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
