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Quantum Mechanics with Basic Field Theory

Students and instructors alike will find this organized and detailed approach to quantummechanics ideal for a two-semester graduate course on the subject.

This textbook covers, step-by-step, important topics in quantum mechanics, from tra-ditional subjects like bound states, perturbation theory and scattering, to more currenttopics such as coherent states, quantum Hall effect, spontaneous symmetry breaking, super-conductivity, and basic quantum electrodynamics with radiative corrections. The largenumber of diverse topics are covered in concise, highly focused chapters, and are explainedin simple but mathematically rigorous ways. Derivations of results and formulas are carriedout from beginning to end, without leaving students to complete them.

With over 200 exercises to aid understanding of the subject, this textbook provides athorough grounding for students planning to enter research in physics. Several exercisesare solved in the text, and password-protected solutions for remaining exercises are availableto instructors at www.cambridge.org/9780521877602.

Bipin R. Desai is a Professor of Physics at the University of California, Riverside, where hedoes research in elementary particle theory. He obtained his Ph.D. in Physics from theUniversity of California, Berkeley. He was a visiting Fellow at Clare Hall, CambridgeUniversity, UK, and has held research positions at CERN, Geneva, Switzerland, and CENSaclay, France. He is a Fellow of the American Physical Society.

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Quantum Mechanics withBasic Field Theory

Bipin R. DesaiUniversity of California at Riverside






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To Ba, Bapuji, and Blaire






Contents

Preface page xviiPhysical constants xx

1 Basic formalism 11.1 State vectors 11.2 Operators and physical observables 31.3 Eigenstates 41.4 Hermitian conjugation and Hermitian operators 51.5 Hermitian operators: their eigenstates and eigenvalues 61.6 Superposition principle 71.7 Completeness relation 81.8 Unitary operators 91.9 Unitary operators as transformation operators 101.10 Matrix formalism 121.11 Eigenstates and diagonalization of matrices 161.12 Density operator 181.13 Measurement 201.14 Problems 21

2 Fundamental commutator and time evolution of state vectorsand operators 242.1 Continuous variables: X and P operators 242.2 Canonical commutator [X , P] 262.3 P as a derivative operator: another way 292.4 X and P as Hermitian operators 302.5 Uncertainty principle 322.6 Some interesting applications of uncertainty relations 352.7 Space displacement operator 362.8 Time evolution operator 412.9 Appendix to Chapter 2 442.10 Problems 52

3 Dynamical equations 553.1 Schrödinger picture 553.2 Heisenberg picture 57






viii Contents

3.3 Interaction picture 593.4 Superposition of time-dependent states and energy–time

uncertainty relation 633.5 Time dependence of the density operator 663.6 Probability conservation 673.7 Ehrenfest’s theorem 683.8 Problems 70

4 Free particles 734.1 Free particle in one dimension 734.2 Normalization 754.3 Momentum eigenfunctions and Fourier transforms 784.4 Minimum uncertainty wave packet 794.5 Group velocity of a superposition of plane waves 834.6 Three dimensions – Cartesian coordinates 844.7 Three dimensions – spherical coordinates 874.8 The radial wave equation 914.9 Properties of Ylm(θ , φ) 924.10 Angular momentum 944.11 Determining L2 from the angular variables 974.12 Commutator

[Li, Lj

]and

[L2, Lj

]98

4.13 Ladder operators 1004.14 Problems 102

5 Particles with spin ½ 1035.1 Spin ½ system 1035.2 Pauli matrices 1045.3 The spin ½ eigenstates 1055.4 Matrix representation of σ x and σ y 1065.5 Eigenstates of σ x and σ y 1085.6 Eigenstates of spin in an arbitrary direction 1095.7 Some important relations for σ i 1105.8 Arbitrary 2 × 2 matrices in terms of Pauli matrices 1115.9 Projection operator for spin ½ systems 1125.10 Density matrix for spin ½ states and the ensemble average 1145.11 Complete wavefunction 1165.12 Pauli exclusion principle and Fermi energy 1165.13 Problems 118

6 Gauge invariance, angular momentum, and spin 1206.1 Gauge invariance 1206.2 Quantum mechanics 1216.3 Canonical and kinematic momenta 1236.4 Probability conservation 124






ix Contents

6.5 Interaction with the orbital angular momentum 1256.6 Interaction with spin: intrinsic magnetic moment 1266.7 Spin–orbit interaction 1286.8 Aharonov–Bohm effect 1296.9 Problems 131

7 Stern–Gerlach experiments 1337.1 Experimental set-up and electron’s magnetic moment 1337.2 Discussion of the results 1347.3 Problems 136

8 Some exactly solvable bound-state problems 1378.1 Simple one-dimensional systems 1378.2 Delta-function potential 1458.3 Properties of a symmetric potential 1478.4 The ammonia molecule 1488.5 Periodic potentials 1518.6 Problems in three dimensions 1568.7 Simple systems 1608.8 Hydrogen-like atom 1648.9 Problems 170

9 Harmonic oscillator 1749.1 Harmonic oscillator in one dimension 1749.2 Problems 184

10 Coherent states 18710.1 Eigenstates of the lowering operator 18710.2 Coherent states and semiclassical description 19210.3 Interaction of a harmonic oscillator with an electric field 19410.4 Appendix to Chapter 10 19910.5 Problems 200

11 Two-dimensional isotropic harmonic oscillator 20311.1 The two-dimensional Hamiltonian 20311.2 Problems 207

12 Landau levels and quantum Hall effect 20812.1 Landau levels in symmetric gauge 20812.2 Wavefunctions for the LLL 21212.3 Landau levels in Landau gauge 21412.4 Quantum Hall effect 21612.5 Wavefunction for filled LLLs in a Fermi system 22012.6 Problems 221






x Contents

13 Two-level problems 22313.1 Time-independent problems 22313.2 Time-dependent problems 23413.3 Problems 246

14 Spin ½ systems in the presence of magnetic fields 25114.1 Constant magnetic field 25114.2 Spin precession 25414.3 Time-dependent magnetic field: spin magnetic resonance 25514.4 Problems 258

15 Oscillation and regeneration in neutrinos and neutral K-mesonsas two-level systems 26015.1 Neutrinos 26015.2 The solar neutrino puzzle 26015.3 Neutrino oscillations 26315.4 Decay and regeneration 26515.5 Oscillation and regeneration of stable and unstable systems 26915.6 Neutral K-mesons 27315.7 Problems 276

16 Time-independent perturbation for bound states 27716.1 Basic formalism 27716.2 Harmonic oscillator: perturbative vs. exact results 28116.3 Second-order Stark effect 28416.4 Degenerate states 28716.5 Linear Stark effect 28916.6 Problems 290

17 Time-dependent perturbation 29317.1 Basic formalism 29317.2 Harmonic perturbation and Fermi’s golden rule 29617.3 Transitions into a group of states and scattering cross-section 29917.4 Resonance and decay 30317.5 Appendix to Chapter 17 31017.6 Problems 315

18 Interaction of charged particles and radiation in perturbation theory 31818.1 Electron in an electromagnetic field: the absorption cross-section 31818.2 Photoelectric effect 32318.3 Coulomb excitations of an atom 32518.4 Ionization 32818.5 Thomson, Rayleigh, and Raman scattering in second-order

perturbation 33118.6 Problems 339






xi Contents

19 Scattering in one dimension 34219.1 Reflection and transmission coefficients 34219.2 Infinite barrier 34419.3 Finite barrier with infinite range 34519.4 Rigid wall preceded by a potential well 34819.5 Square-well potential and resonances 35119.6 Tunneling 35419.7 Problems 356

20 Scattering in three dimensions – a formal theory 35820.1 Formal solutions in terms of Green’s function 35820.2 Lippmann–Schwinger equation 36020.3 Born approximation 36320.4 Scattering from a Yukawa potential 36420.5 Rutherford scattering 36520.6 Charge distribution 36620.7 Probability conservation and the optical theorem 36720.8 Absorption 37020.9 Relation between the T -matrix and the scattering amplitude 37220.10 The S-matrix 37420.11 Unitarity of the S-matrix and the relation between S and T 37820.12 Properties of the T -matrix and the optical theorem (again) 38220.13 Appendix to Chapter 20 38320.14 Problems 384

21 Partial wave amplitudes and phase shifts 38621.1 Scattering amplitude in terms of phase shifts 38621.2 χ l , Kl , and Tl 39221.3 Integral relations for χ l , Kl , and Tl 39321.4 Wronskian 39521.5 Calculation of phase shifts: some examples 40021.6 Problems 405

22 Analytic structure of the S-matrix 40722.1 S-matrix poles 40722.2 Jost function formalism 41322.3 Levinson’s theorem 42022.4 Explicit calculation of the Jost function for l = 0 42122.5 Integral representation of F0(k) 42422.6 Problems 426

23 Poles of the Green’s function and composite systems 42723.1 Relation between the time-evolution operator and the

Green’s function 42723.2 Stable and unstable states 429






xii Contents

23.3 Scattering amplitude and resonance 43023.4 Complex poles 43123.5 Two types of resonances 43123.6 The reaction matrix 43223.7 Composite systems 44223.8 Appendix to Chapter 23 447

24 Approximation methods for bound states and scattering 45024.1 WKB approximation 45024.2 Variational method 45824.3 Eikonal approximation 46124.4 Problems 466

25 Lagrangian method and Feynman path integrals 46925.1 Euler–Lagrange equations 46925.2 N oscillators and the continuum limit 47125.3 Feynman path integrals 47325.4 Problems 478

26 Rotations and angular momentum 47926.1 Rotation of coordinate axes 47926.2 Scalar functions and orbital angular momentum 48326.3 State vectors 48526.4 Transformation of matrix elements and representations of the

rotation operator 48726.5 Generators of infinitesimal rotations: their eigenstates

and eigenvalues 48926.6 Representations of J 2 and Ji for j = 1

2 and j = 1 49426.7 Spherical harmonics 49526.8 Problems 501

27 Symmetry in quantum mechanics and symmetry groups 50227.1 Rotational symmetry 50227.2 Parity transformation 50527.3 Time reversal 50727.4 Symmetry groups 51127.5 Dj(R) for j = 1

2 and j = 1: examples of SO(3) and SU (2) groups 51427.6 Problems 516

28 Addition of angular momenta 51828.1 Combining eigenstates: simple examples 51828.2 Clebsch–Gordan coefficients and their recursion relations 52228.3 Combining spin ½ and orbital angular momentum l 52428.4 Appendix to Chapter 28 52728.5 Problems 528






xiii Contents

29 Irreducible tensors and Wigner–Eckart theorem 52929.1 Irreducible spherical tensors and their properties 52929.2 The irreducible tensors: Ylm(θ , φ) and Dj(χ) 53329.3 Wigner–Eckart theorem 53629.4 Applications of the Wigner–Eckart theorem 53829.5 Appendix to Chapter 29: SO(3), SU (2) groups and Young’s tableau 54129.6 Problems 548

30 Entangled states 54930.1 Definition of an entangled state 54930.2 The singlet state 55130.3 Differentiating the two approaches 55230.4 Bell’s inequality 55330.5 Problems 555

31 Special theory of relativity: Klein–Gordon and Maxwell’s equations 55631.1 Lorentz transformation 55631.2 Contravariant and covariant vectors 55731.3 An example of a covariant vector 56031.4 Generalization to arbitrary tensors 56131.5 Relativistically invariant equations 56331.6 Appendix to Chapter 31 56931.7 Problems 572

32 Klein–Gordon and Maxwell’s equations 57532.1 Covariant equations in quantum mechanics 57532.2 Klein–Gordon equations: free particles 57632.3 Normalization of matrix elements 57832.4 Maxwell’s equations 57932.5 Propagators 58132.6 Virtual particles 58632.7 Static approximation 58632.8 Interaction potential in nonrelativistic processes 58732.9 Scattering interpreted as an exchange of virtual particles 58932.10 Appendix to Chapter 32 593

33 The Dirac equation 59733.1 Basic formalism 59733.2 Standard representation and spinor solutions 60033.3 Large and small components of u(p) 60133.4 Probability conservation 60533.5 Spin ½ for the Dirac particle 607

34 Dirac equation in the presence of spherically symmetric potentials 61134.1 Spin–orbit coupling 611






xiv Contents

34.2 K-operator for the spherically symmetric potentials 61334.3 Hydrogen atom 61634.4 Radial Dirac equation 61834.5 Hydrogen atom states 62334.6 Hydrogen atom wavefunction 62434.7 Appendix to Chapter 34 626

35 Dirac equation in a relativistically invariant form 63135.1 Covariant Dirac equation 63135.2 Properties of the γ -matrices 63235.3 Charge–current conservation in a covariant form 63335.4 Spinor solutions: ur(p) and vr(p) 63535.5 Normalization and completeness condition for ur(p) and vr(p) 63635.6 Gordon decomposition 64035.7 Lorentz transformation of the Dirac equation 64235.8 Appendix to Chapter 35 644

36 Interaction of a Dirac particle with an electromagnetic field 64736.1 Charged particle Hamiltonian 64736.2 Deriving the equation another way 65036.3 Gordon decomposition and electromagnetic current 65136.4 Dirac equation with EM field and comparison with the

Klein–Gordon equation 65336.5 Propagators: the Dirac propagator 65536.6 Scattering 65736.7 Appendix to Chapter 36 661

37 Multiparticle systems and second quantization 66337.1 Wavefunctions for identical particles 66337.2 Occupation number space and ladder operators 66437.3 Creation and destruction operators 66637.4 Writing single-particle relations in multiparticle language: the

operators, N , H , and P 67037.5 Matrix elements of a potential 67137.6 Free fields and continuous variables 67237.7 Klein–Gordon/scalar field 67437.8 Complex scalar field 67837.9 Dirac field 68037.10 Maxwell field 68337.11 Lorentz covariance for Maxwell field 68737.12 Propagators and time-ordered products 68837.13 Canonical quantization 69037.14 Casimir effect 69337.15 Problems 697






xv Contents

38 Interactions of electrons and phonons in condensed matter 69938.1 Fermi energy 69938.2 Interacting electron gas 70438.3 Phonons 70838.4 Electron–phonon interaction 713

39 Superconductivity 71939.1 Many-body system of half-integer spins 71939.2 Normal states (� = 0, G �= 0) 72439.3 BCS states (� �= 0) 72539.4 BCS condensate in Green’s function formalism 72739.5 Meissner effect 73239.6 Problems 735

40 Bose–Einstein condensation and superfluidity 73640.1 Many-body system of integer spins 73640.2 Superfluidity 74040.3 Problems 742

41 Lagrangian formulation of classical fields 74341.1 Basic structure 74341.2 Noether’s theorem 74441.3 Examples 74641.4 Maxwell’s equations and consequences of gauge invariance 750

42 Spontaneous symmetry breaking 75542.1 BCS mechanism 75542.2 Ferromagnetism 75642.3 SSB for discrete symmetry in classical field theory 75842.4 SSB for continuous symmetry 76042.5 Nambu–Goldstone bosons 76242.6 Higgs mechanism 765

43 Basic quantum electrodynamics and Feynman diagrams 77043.1 Perturbation theory 77043.2 Feynman diagrams 77343.3 T (HI (x1) HI (x2)) and Wick’s theorem 77743.4 Feynman rules 78343.5 Cross-section for 1 + 2 →3 + 4 78343.6 Basic two-body scattering in QED 78643.7 QED vs. nonrelativistic limit: electron–electron system 78643.8 QED vs. nonrelativistic limit: electron–photon system 789

44 Radiative corrections 79344.1 Radiative corrections and renormalization 793






xvi Contents

44.2 Electron self-energy 79444.3 Appendix to Chapter 44 799

45 Anomalous magnetic moment and Lamb shift 80645.1 Calculating the divergent integrals 80645.2 Vertex function and the magnetic moment 80645.3 Calculation of the vertex function diagram 80845.4 Divergent part of the vertex function 81045.5 Radiative corrections to the photon propagator 81145.6 Divergent part of the photon propagator 81345.7 Modification of the photon propagator and photon wavefunction 81445.8 Combination of all the divergent terms: basic renormalization 81645.9 Convergent parts of the radiative corrections 81745.10 Appendix to Chapter 45 821

Bibliography 825Index 828






Preface

While writing this book I was reminded at times of what Professor Francis Low used tosay 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 byhaving written this book, I would add that, if at that point one still does not understand thesubject, there will be yet another opportunity when writing a book on it. That was certainlythe case with me and this book.

For the last twenty years or so I have taught a one-year graduate course in quantummechanics at the University of California, Riverside. I have used several books, includingthe text by Schiff which also happens to be the text I used when I was taking my graduatecourses at the University of California, Berkeley (along with my class notes from ProfessorEyvind Wichmann who taught the quantum electrodynamics course). However, it becameclear to me that I would need to expand the subject matter considerably if I wanted the booknot only to be as thorough and up-to-date as possible but also organized so that one subjectfollowed the other in a logical sequence. I hope I have succeeded.

Traditionally, books on graduate quantum mechanics go up to relativity and in some caseseven 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 secondquantization, to show how the negative-energy states are reinterpreted as positive-energystates of antiparticles. It was, therefore, logical to cover elementary second quantization,which in a sense is many-body quantum mechanics with quantization conditions. And oncethis topic was addressed it would be unfair not to cover the great successes of many-bodysystems in condensed matter, in particular, superconductivity and Bose–Einstein condensa-tion. A logical concurrent step was to include also full relativistic quantum field theory, atleast basic quantum electrodynamics (QED) and then finish on a triumphant note describingthe stunning success of QED in explaining the anomalous magnetic moment and the Lambshift. With the vast acreage that I wanted to cover, it seemed only appropriate to include aswell the modern subject of spontaneous symmetry breaking, which has its applications bothin condensed matter physics and in particle physics. This then was the rationale behind thisbook’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 acommon thread, with chapters arranged in an increasing degree of complexity within eachblock






xviii Preface

Chs. 1, 2, 3 Basic FormalismChs. 4, 5, 6, 7 Free ParticlesChs. 8, 9, 10, 11, 12 Exactly Solvable Bound State ProblemsChs. 13, 14, 15 Two-Level ProblemsChs. 16, 17, 18 Perturbation TheoryCh. 24 New approximation methodsCh. 25 Lagrangian and Feynman integral formalismsChs. 19, 20, 21, 22, 23 Scattering TheoryChs. 26, 27, 28, 29, 30 Symmetry, Rotations, and Angular MomentumChs. 31, 32, 33, 34, 35, 36 Relativistic theory with Klein–Gordon, Dirac, and

Maxwell’s equationsChs. 37, 38, 39, 40 Second Quantization, Condensed Matter ProblemsChs. 41, 42 Classical Fields and Spontaneous Symmetry BreakingChs. 43, 44, 45 Quantum Electrodynamics and Radiative Corrections

In the chapters on scattering theory, one may find an extra coverage in this book onthe properties of the S-matrix especially with reference to its analytical properties. Thisis thanks to my thesis advisor at Berkeley, Professor Geoffrey Chew who emphasized theimportance of these properties to his students.

I believe it is feasible to complete the first 32 chapters in one year (two semesters or threequarters). The remaining chapters beginning with the Dirac equation could well be taughtin the first semester or first quarter of an advanced quantum mechanics course. Since thesetopics cover quantum field theory applied to both particle physics and condensed matterphysics, it could be taken by students specializing in either subject.

Except at the beginning of each chapter, this book does not have as much narrativeor as many long descriptive paragraphs as one normally finds in other textbooks. I haveinstead spent extra space on deriving and solving the relevant equations. I feel that the extranarrative can always be supplemented by the person teaching the course.

There are an adequate number of problems in this book. They are fairly straightforward. Isuppose I still have scars left from the days when I took graduate quantum mechanics fromProfessor Edward Teller at Berkeley, who gave very inspiring lectures full of interestingand 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 toughestproblems this side of the Atlantic! Those assignments routinely took care of our entireweekends.

I have many people to thank, beginning with Dustin Urbaniec and Omar Moreno whodid a good bit of the typing for me, and Barbara Simandl who did all the figures. I am alsograteful to a number of graduate students from my Quantum Mechanics course for pointingout errors in my write-up; in particular, I am thankful to Eric Barbagiovanni, for suggestinga number of improvements. I must also thank Dr. Steve Foulkes, a former graduate studentat UC Riverside, who read a number of chapters and, following my instructions not to showany mercy in criticizing what he read, did exactly that! I also wish to thank my colleagueswho critically read parts of the manuscript: Professors Robert Clare (who also directed meto Cambridge University Press), Leonid Pryadkov, G. Rajasekaran and Utpal Sarkar.






xix 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 andLindsay Barnes for their constant help and, more importantly, for their patience with thislong project.

There is one individual, Alex Vaucher, whom I must single out, without whose help thisbook would neither have been started nor completed. After finishing my graduate courseon Quantum Mechanics at UC Riverside some years ago, he strongly encouraged me towrite this book. He supplied the necessary software and, knowing how computer-ignorantI 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 fulltime 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.






Physical constants

Planck’s constant � 6.581 × 10−16 eV sVelocity of light in vacuum c 2.9979 × 1010 cm/sFine structure constant α = e2/�c 1/137.04Rest mass of the electron mc2 0.511 MeVMass of the proton Mc2 938.28 MeVBohr radius �

2/me2 5.2918 ×10−9 cmBohr magneton e�/2mc 0.58 × 10−8 eV/gaussBoltzmann constant k 8.62 × 10−5 eV/K1 eV 1.6 × 10−12 erg






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