Advanced Quantum Mechanics www.com Franz Schwabl Advanced Quantum Mechanics Translated by Roginald Hilton and Angela Lahee Fourth Edition With 79 Figures, 4 Tables, and 104 Problems 13 www.com Professor Dr. Franz Schwabl Physik-Department Technische Universität München James-Franck-Str. 2 85748 Garching, Germany schwabl@physik.de Translators: Dr. Roginald Hilton Dr.
Angela Lahee Title of the original German edition: Quantenmechanik für Fortgeschrittene (QM II) (Springer-Lehrbuch) ISBN 978-3-540-85075-5 © Springer-Verlag Berlin Heidelberg 2008 ISBN 978-3-540-85061-8 e-ISBN 978-3-540-85062-5 DOI 10.1007/978-3-540-85062-5 Library of Congress Control Number: 2008933497 © 2008, 2005, 2004, 1999 Springer-Verlag Berlin Heidelberg 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, broad- casting, 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 to prosecution under the German Copyright Law.
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 and production: le-tex publishing services oHG, Leipzig, Germany Cover design: eStudio Calamar, Girona/Spain Printed on acid-free paper 987654321 springer.com 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 www.com Preface to the Fourth Edition In this latest edition new material has been added, which includes many additional clarifying remarks and cross references.
The design of all figures has been reworked, the layout has been improved and unified to enhance the didactic appeal of the book, however, in the course of these changes I have attempted to keep intact its underlying compact nature. I am grateful to many colleagues for their help with this substantial revision. Again, special thanks go to Uwe Täuber and Roger Hilton for discussions, comments and many constructive suggestions. I should like to thank Dr.
Herbert Müller for his generous help in all computer problems. Concerning the graphics, I am very grateful to Mr Wenzel Schürmann for essential support and to Ms Christina Di Stefano and Mr Benjamin Sánchez who undertook the graphical design of the diagrams. It is my pleasure to thank Dr. Thorsten Schneider and Mrs Jacqueline Lenz of Springer for the excellent co-operation, as well as the le-tex setting team for their careful incorporation of the amendments for this new edition.
Finally, I should like to thank all colleagues and students who, over the years, have made suggestions to improve the usefulness of this book.com 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 divided into three parts: I. Many-Body Systems, II. Relativistic Wave Equations, and III.
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 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 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 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. 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, Dirac, and radiation fields, and the spin-statistics theorem. The final chapter treats interacting fields using the example of quantum electrodynam- ics: 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.com 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. Jörg-Müller, Ms. The problems were conceived with the help of E.
Gasser also read through the entire manuscript and made many valuable suggestions. I am indebted to Dr. 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-Jürgen Kölsch of Springer-Verlag, I wish to express my sincere gratitude.com Table of Contents Part I. Nonrelativistic Many-Particle Systems 1.1 Identical Particles, Many-Particle States, and Permutation Symmetry .1 States and Observables of Identical Particles .2 Completely Symmetric and Antisymmetric States .1 States, Fock Space, Creation and Annihilation Operators .2 The Particle-Number Operator .3 General Single- and Many-Particle Operators .1 States, Fock Space, Creation and Annihilation Operators .2 Single- and Many-Particle Operators .1 Transformations Between Different Basis Systems .1 Momentum Eigenfunctions and the Hamiltonian .2 Fourier Transformation of the Density .3 The Inclusion of Spin .1 The Fermi Sphere, Excitations .2 Single-Particle Correlation Function .3 Pair Distribution Function .4 Pair Distribution Function, Density Correlation Functions, and Structure Factor .com XII Table of Contents 2.2 Ground State Energy and Elementary Theory of the Electron Gas .2 Ground State Energy in the Hartree–Fock Approximation .3 Modification of Electron Energy Levels due to the Coulomb Interaction .3 Hartree–Fock Equations for Atoms .1 Pair Distribution Function for Free Bosons .2 Two-Particle States of Bosons .2 Weakly Interacting, Dilute Bose Gas .1 Quantum Fluids and Bose–Einstein Condensation .2 Bogoliubov Theory of the Weakly Interacting Bose Gas. Correlation Functions, Scattering, and Response .1 Scattering and Response .2 Density Matrix, Correlation Functions .6 Fluctuation–Dissipation Theorem .7 Examples of Applications .1 General Symmetry Relations .2 Symmetry Properties of the Response Function for Hermitian Operators .1 General Structure of Sum Rules .2 Application to the Excitations in He II.
109 Bibliography for Part I .com Table of Contents XIII Part II. Relativistic Wave Equations 5. Relativistic Wave Equations and their Derivation .2 The Klein–Gordon Equation .1 Derivation by Means of the Correspondence Principle .2 The Continuity Equation .3 Free Solutions of the Klein–Gordon Equation .1 Derivation of the Dirac Equation .2 The Continuity Equation .3 Properties of the Dirac Matrices .4 The Dirac Equation in Covariant Form .5 Nonrelativistic Limit and Coupling to the Electromagnetic Field. Lorentz Transformations and Covariance of the Dirac Equation .2 Lorentz Covariance of the Dirac Equation .1 Lorentz Covariance and Transformation of Spinors .2 Determination of the Representation S(Λ) .3 Further Properties of S .4 Transformation of Bilinear Forms .5 Properties of the γ Matrices .3 Solutions of the Dirac Equation for Free Particles .1 Spinors with Finite Momentum .2 Orthogonality Relations and Density.
Orbital Angular Momentum and Spin .1 Passive and Active Transformations .2 Rotations and Angular Momentum. The Coulomb Potential .1 Klein–Gordon Equation with Electromagnetic Field .1 Coupling to the Electromagnetic Field .2 Klein–Gordon Equation in a Coulomb Field .2 Dirac Equation for the Coulomb Potential .com XIV Table of Contents 9. The Foldy–Wouthuysen Transformation and Relativistic Corrections .1 The Foldy–Wouthuysen Transformation .1 Description of the Problem .2 Transformation for Free Particles .3 Interaction with the Electromagnetic Field .2 Relativistic Corrections and the Lamb Shift .2 Estimate of the Lamb Shift. Physical Interpretation of the Solutions to the Dirac Equation .1 Wave Packets and “Zitterbewegung” .1 Superposition of Positive Energy States .2 The General Wave Packet .3 General Solution of the Free Dirac Equation in the Heisenberg Representation .4 Potential Steps and the Klein Paradox .2 The Hole Theory.
Symmetries and Further Properties of the Dirac Equation .1 Active and Passive Transformations, Transformations of Vectors .2 Invariance and Conservation Laws .1 The General Transformation .4 Spatial Reflection (Parity Transformation) .1 Reversal of Motion in Classical Physics .2 Time Reversal in Quantum Mechanics .3 Time-Reversal Invariance of the Dirac Equation .4 Racah Time Reflection .6 Zero-Mass Fermions (Neutrinos). 244 Bibliography for Part II .com Table of Contents XV Part III. Quantization of Relativistic Fields .1 Coupled Oscillators, the Linear Chain, Lattice Vibrations .1 Linear Chain of Coupled Oscillators .2 Continuum Limit, Vibrating String .3 Generalization to Three Dimensions, Relationship to the Klein–Gordon Field .2 Classical Field Theory .1 Lagrangian and Euler–Lagrange Equations of Motion .4 Symmetries and Conservation Laws, Noether’s Theorem .1 The Energy–Momentum Tensor, Continuity Equations, and Conservation Laws .2 Derivation from Noether’s Theorem of the Conservation Laws for Four-Momentum, Angular Momentum, and Charge .1 The Real Klein–Gordon Field .1 The Lagrangian Density, Commutation Relations, and the Hamiltonian .2 The Complex Klein–Gordon Field .3 Quantization of the Dirac Field .5 The Infinite-Volume Limit .4 The Spin Statistics Theorem .1 Propagators and the Spin Statistics Theorem .2 Further Properties of Anticommutators and Propagators of the Dirac Field. Quantization of the Radiation Field .2 The Coulomb Gauge .3 The Lagrangian Density for the Electromagnetic Field .4 The Free Electromagnatic Field and its Quantization .com XVI Table of Contents 14.5 Calculation of the Photon Propagator.
Interacting Fields, Quantum Electrodynamics .1 Lagrangians, Interacting Fields .2 Fermions in an External Field .3 Interaction of Electrons with the Radiation Field: Quantum Electrodynamics (QED) .2 The Interaction Representation, Perturbation Theory .1 The Interaction Representation (Dirac Representation) 324 15.5 Simple Scattering Processes, Feynman Diagrams .1 The First-Order Term .3 Second-Order Processes .4 Feynman Rules of Quantum Electrodynamics .1 The Self-Energy of the Electron .2 Self-Energy of the Photon, Vacuum Polarization .4 The Ward Identity and Charge Renormalization .5 Anomalous Magnetic Moment of the Electron. 373 Bibliography for Part III. 377 A Alternative Derivation of the Dirac Equation. 380 C Projection Operators for the Spin .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 .1 Quantization and the Feynman Propagator .com Table of Contents XVII E.2 The Physical Significance of Longitudinal and Scalar Photons .3 The Feynman Photon Propagator. 393 F Coupling of Charged Scalar Mesons to the Electromagnetic Field .com Part I Nonrelativistic Many-Particle Systems www. Second Quantization In this first chapter, 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 quantization. Nature has given us two types of particle, bosons and fermions. These have states that are, respectively, completely symmetric and completely antisym- metric.