XV Contents 38 Interactions of electrons and phonons in condensed matter 699 38.1 Fermi energy 69 38.2 Interacting electron gas 04 38.3 Phonons 708 38.4 Electron-phonon interaction 713 39 Superconductivity 719 39.1 Many-body system of half-integer spins 719 392 Normal states(△=0.G≠0) 724 39.3 39.4 BCS states(≠0) BCS condensate in Green's function formalism 39.5 Meissner effect 732 39.6 Problems 735 40 Bose-Einstein condensation and superfluidity 736 40.1 Many-body system of integer spins 736 40.2 Superfluidity 740 40.3 Problems 742 % Lagrangian formulation of classical fields 743 41.1 Basic structure 743 41.2 Noether's theorem 744 413 Examples 41.4 Maxwell's equations and consequences of gauge invariance Spontaneous symmetry breaking 7 42.1 BCS mechanism 75 42.2 Ferromagnetism 423 SSB for discrete symmetry in classical field theory 424 SSB for continuous 42.5 Nambu-Golds tone be symm osons 42.6 Higgs mechanism 101016 Basic quantum electrodynamics and Feynman diagrams 43.1 Perturbation theory 43.2 Feynman diagrams 43.3 T(HI (x1)H(x2))and Wick's theorem 777 434 Feynman rules 43.5 Cross-section for 1+23+4 43.6 Basic two-body scattering in QED 786 43.7 QED vs.nonrelativistic limit:electron-electron system 786 43.8 QED vs.nonrelativistic limit:electron-photon system 189 Radiative corrections 793 44.1 Radiative corrections and renormalization 793
xv Contents 38 Interactions of electrons and phonons in condensed matter 699 38.1 Fermi energy 699 38.2 Interacting electron gas 704 38.3 Phonons 708 38.4 Electron–phonon interaction 713 39 Superconductivity 719 39.1 Many-body system of half-integer spins 719 39.2 Normal states ( = 0, G = 0) 724 39.3 BCS states ( = 0) 725 39.4 BCS condensate in Green’s function formalism 727 39.5 Meissner effect 732 39.6 Problems 735 40 Bose–Einstein condensation and superfluidity 736 40.1 Many-body system of integer spins 736 40.2 Superfluidity 740 40.3 Problems 742 41 Lagrangian formulation of classical fields 743 41.1 Basic structure 743 41.2 Noether’s theorem 744 41.3 Examples 746 41.4 Maxwell’s equations and consequences of gauge invariance 750 42 Spontaneous symmetry breaking 755 42.1 BCS mechanism 755 42.2 Ferromagnetism 756 42.3 SSB for discrete symmetry in classical field theory 758 42.4 SSB for continuous symmetry 760 42.5 Nambu–Goldstone bosons 762 42.6 Higgs mechanism 765 43 Basic quantum electrodynamics and Feynman diagrams 770 43.1 Perturbation theory 770 43.2 Feynman diagrams 773 43.3 T(HI (x1) HI (x2)) and Wick’s theorem 777 43.4 Feynman rules 783 43.5 Cross-section for 1 + 2 →3 + 4 783 43.6 Basic two-body scattering in QED 786 43.7 QED vs. nonrelativistic limit: electron–electron system 786 43.8 QED vs. nonrelativistic limit: electron–photon system 789 44 Radiative corrections 793 44.1 Radiative corrections and renormalization 793
xvi Contents 44.2 Electron self-energy 79 44.3 Appendix to Chapter 44 99 45 Anomalous magnetic moment and Lamb shift 806 45.1 Calculating the divergent integrals 806 45.2 Vertex function and the magnetic moment 806 453 Calculation of the vertex function diagram 808 45.4 Divergent part of the vertex function 5 Radiative corrections to the photon propagator 456 Divergent part of the photon propagator 813 45.7 Modification of the photon propagator and photon wavefunction 814 45.8 Combination of all the divergent terms:basic renormalization 45.9 Convergent parts of the radiative corrections 817 45.10 Appendix to Chapter 45 821 Bibliography Index
xvi Contents 44.2 Electron self-energy 794 44.3 Appendix to Chapter 44 799 45 Anomalous magnetic moment and Lamb shift 806 45.1 Calculating the divergent integrals 806 45.2 Vertex function and the magnetic moment 806 45.3 Calculation of the vertex function diagram 808 45.4 Divergent part of the vertex function 810 45.5 Radiative corrections to the photon propagator 811 45.6 Divergent part of the photon propagator 813 45.7 Modification of the photon propagator and photon wavefunction 814 45.8 Combination of all the divergent terms: basic renormalization 816 45.9 Convergent parts of the radiative corrections 817 45.10 Appendix to Chapter 45 821 Bibliography 825 Index 828
Preface While writing this book I was reminded at times of what Professor Francis Low used to your only other chance will be when you have to teach it."Knowing now what I know by having written this book,I would add that,if at that point one still does not understand the subjec t there will he vet anothe r opportunity when riting a book on it.That was certainly the case with me and this book For the last twenty years or so I have taught a one-year graduate course in quantum mechanics at the University of California,Riverside.I have used several books,including the text by Schff which als happens tobe the text Iused when I was taking my graduate courses at the University of California,Berkeley(along with my class notes from Professor Eyvind Wichmann who taught the quantum electrodynamics course).However,it became clear to me that I would need to expand the subject matter considerably if I wanted the book not only to be as thorough and up-to-date as possible but also organized so that one subject followed the other in a logical sequence.I hope I have succeeded Traditionally,books on graduate quantum mechanics go up to relativity and in some cases even cover the Dirac equation.But relativistic equations lead to the troublesome negative- energy solutions.It would be unsatisfactory then to just stop there and not go to second quantization,to show how the negative-energy states are reinterpreted as positive-energy states of antiparticles.It was,therefore,logical to cover elementary second quantization, which in a sense is many-body quantum mechanics with quantization conditions.And once great successes of many-body system tion.A logical concurrent step was to include also full relativistic quantum field theory,at least basic quantum electrodynamics(OED)and then finish on a triumphant note describing the stunning success of QED in explaining the anomalous magnetic moment and the Lamb shift.With the vast acreage that I wanted to cover,it seemed only appropriate to include a well the modern subject of spontaneous symmetry breaking,which has its applications both in condensed matter physics and in particle physics.This then was the rationale behind this book's content and or I have organized the book with small chapters in what I believe to be a logical order One can think of the layout of the chapters in terms of the following blocks,each with a common thread,with chapters arranged in an increasing degree of complexity within each block
Preface While writing this book I was reminded at times of what Professor Francis Low used to say when I took his class on undergraduate electromagnetism at the University of Illinois, Urbana-Champaign. “Be sure to understand the subject thoroughly,” he said, “otherwise, your only other chance will be when you have to teach it.” Knowing now what I know by having written this book, I would add that, if at that point one still does not understand the subject, there will be yet another opportunity when writing a book on it. That was certainly the case with me and this book. For the last twenty years or so I have taught a one-year graduate course in quantum mechanics at the University of California, Riverside. I have used several books, including the text by Schiff which also happens to be the text I used when I was taking my graduate courses at the University of California, Berkeley (along with my class notes from Professor Eyvind Wichmann who taught the quantum electrodynamics course). However, it became clear to me that I would need to expand the subject matter considerably if I wanted the book not only to be as thorough and up-to-date as possible but also organized so that one subject followed the other in a logical sequence. I hope I have succeeded. Traditionally, books on graduate quantum mechanics go up to relativity and in some cases even cover the Dirac equation. But relativistic equations lead to the troublesome negativeenergy solutions. It would be unsatisfactory then to just stop there and not go to second quantization, to show how the negative-energy states are reinterpreted as positive-energy states of antiparticles. It was, therefore, logical to cover elementary second quantization, which in a sense is many-body quantum mechanics with quantization conditions. And once this topic was addressed it would be unfair not to cover the great successes of many-body systems in condensed matter, in particular, superconductivity and Bose–Einstein condensation. A logical concurrent step was to include also full relativistic quantum field theory, at least basic quantum electrodynamics (QED) and then finish on a triumphant note describing the stunning success of QED in explaining the anomalous magnetic moment and the Lamb shift. With the vast acreage that I wanted to cover, it seemed only appropriate to include as well the modern subject of spontaneous symmetry breaking, which has its applications both in condensed matter physics and in particle physics. This then was the rationale behind this book’s content and organization. I have organized the book with small chapters in what I believe to be a logical order. One can think of the layout of the chapters in terms of the following blocks, each with a common thread, with chapters arranged in an increasing degree of complexity within each block
Preface Chs.1,2,3 Basic Formalism Chs.4,5,6,7 Free Particles Chs.89.10.1l,12 Exactly solvable bound state problems Chs.13,14,15 Two-Level Problems Chs.16,17,18 Perturbation Theory Ch.24 New approximation methods Ch.25 Lagrangian and fevnman integral formalisms Chs.19,20,21,22,23 Scattering Theory Chs.26,27,28,29,30 Symmetry,Rotations,and Angular Momentum Chs.31,32,33,34,35,36 Relativistic theory with Klein-Gordon,Dirac,and Maxwell's equations Chs.37,38,39,40 Second Quantization Condensed Matter Problems Chs.41,42 Classical Fields and Spontaneous Symmetry Breaking Chs.43,44,45 Quantum Electrodynamics and Radiative Corrections In the chapter on scattering theory,one may find an extra coverage in this book on the properties of the S-matrix especially with reference to its analytical properties.This is thanks to my thesis advisor at Berkeley,Professor Geoffrey Chew who emphasized the mportance of these properties to his students. quarters).The remaining chapters beginning with the Dirac equation could well be taugh in the first semester or first quarter of an advanced quantum mechanics course.since these opics cover quantum field theory applied to both particle physics and condensed matter physics,it could be taken by st spec ializ either subject Except at the beginning of each chapter,this book does not have as much narrative or as many long descriptive paragraphs as one normally finds in other textbooks.I have nstead spent extra space on deriving and solving the relevant equations.Ifeel that the extra narrative can alw ays be supplemente y the person teaching the course There are an adequate number of problems in this book.They are fairly straightforward.I suppose I still have scars left from the days when I took graduate quantum mechanics from Professor Edward Teller at Berkeley,who gave very inspiring lectures full of interesting and topical episodes while on the blackboard he usually wrote down just the basic form ulas But then he turned around and gave,as homework,a huge number of some of the toughest problems this side of the Atlantic!Those assignments routinely took care of our entire weekends. Ihave many people to thank,beginning with Dustin Urbanie c and Omar Moreno who did a good bit of the typing for me,and Barbara Simandl who did all the figures.I am also grateful to a number of graduate students from my Quantum mechanics course for pointing out errors in my write-up:in particular.i am thankful to eric barbagiovanni.for suggesting mpr I m nust also thank Dr.Steve Foulkes,a fo ner graduate studen UC Riverside.horembrof cptrnd.following my instructions not toshow any mercy in criticizing what he read,did exactly that!I also wish to thank my colleagues who critically read parts of the manuscript:Professors Robert Clare(who also directed me to Cambridge University Press).Leonid Pryadkov,G.Rajasekaran and Utpal Sarkar
xviii Preface Chs. 1, 2, 3 Basic Formalism Chs. 4, 5, 6, 7 Free Particles Chs. 8, 9, 10, 11, 12 Exactly Solvable Bound State Problems Chs. 13, 14, 15 Two-Level Problems Chs. 16, 17, 18 Perturbation Theory Ch. 24 New approximation methods Ch. 25 Lagrangian and Feynman integral formalisms Chs. 19, 20, 21, 22, 23 Scattering Theory Chs. 26, 27, 28, 29, 30 Symmetry, Rotations, and Angular Momentum Chs. 31, 32, 33, 34, 35, 36 Relativistic theory with Klein–Gordon, Dirac, and Maxwell’s equations Chs. 37, 38, 39, 40 Second Quantization, Condensed Matter Problems Chs. 41, 42 Classical Fields and Spontaneous Symmetry Breaking Chs. 43, 44, 45 Quantum Electrodynamics and Radiative Corrections In the chapters on scattering theory, one may find an extra coverage in this book on the properties of the S-matrix especially with reference to its analytical properties. This is thanks to my thesis advisor at Berkeley, Professor Geoffrey Chew who emphasized the importance of these properties to his students. I believe it is feasible to complete the first 32 chapters in one year (two semesters or three quarters). The remaining chapters beginning with the Dirac equation could well be taught in the first semester or first quarter of an advanced quantum mechanics course. Since these topics cover quantum field theory applied to both particle physics and condensed matter physics, it could be taken by students specializing in either subject. Except at the beginning of each chapter, this book does not have as much narrative or as many long descriptive paragraphs as one normally finds in other textbooks. I have instead spent extra space on deriving and solving the relevant equations. I feel that the extra narrative can always be supplemented by the person teaching the course. There are an adequate number of problems in this book. They are fairly straightforward. I suppose I still have scars left from the days when I took graduate quantum mechanics from Professor Edward Teller at Berkeley, who gave very inspiring lectures full of interesting and topical episodes while on the blackboard he usually wrote down just the basic formulas. But then he turned around and gave, as homework, a huge number of some of the toughest problems this side of the Atlantic! Those assignments routinely took care of our entire weekends. I have many people to thank, beginning with Dustin Urbaniec and Omar Moreno who did a good bit of the typing for me, and Barbara Simandl who did all the figures. I am also grateful to a number of graduate students from my Quantum Mechanics course for pointing out errors in my write-up; in particular, I am thankful to Eric Barbagiovanni, for suggesting a number of improvements. I must also thank Dr. Steve Foulkes, a former graduate student at UC Riverside, who read a number of chapters and, following my instructions not to show any mercy in criticizing what he read, did exactly that! I also wish to thank my colleagues who critically read parts of the manuscript: Professors Robert Clare (who also directed me to Cambridge University Press), Leonid Pryadkov, G. Rajasekaran and Utpal Sarkar
Preface At Cambridge University Press,my special thanks to Simon Mitton,with whom I cor- responded in the early years.for his kind support and encouragement;to John Fowler and Lindsay Barnes for their constant help and,more importantly,for their patience with this long project. There is one individual,Alex Vaucher,whom I must single out,without whose help this book would neither have bee started nor completed.After finishing my graduate cours onQuntum Mechanics at UC Riverside somye he stroyodmeto write this book.He supplied the necessary software and,knowing how computer-ignorant I was,continued to provide me with technical instructions during all phases of this project. Initially the two ofu vere planning to collaborate on this book but,because of his full time position with the Physics and Astronomy department at the University of Califomia, Los Angeles,he was not able to participate.My deepest gratitude to him
xix Preface At Cambridge University Press, my special thanks to Simon Mitton, with whom I corresponded in the early years, for his kind support and encouragement; to John Fowler and Lindsay Barnes for their constant help and, more importantly, for their patience with this long project. There is one individual, Alex Vaucher, whom I must single out, without whose help this book would neither have been started nor completed. After finishing my graduate course on Quantum Mechanics at UC Riverside some years ago, he strongly encouraged me to write this book. He supplied the necessary software and, knowing how computer-ignorant I was, continued to provide me with technical instructions during all phases of this project. Initially the two of us were planning to collaborate on this book but, because of his full time position with the Physics and Astronomy department at the University of California, Los Angeles, he was not able to participate. My deepest gratitude to him