An Introduction to ADVANCED QUANTUM PHYSICS d0 Hans Paar WWILEY
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Contents Preface PART 1 Relativistic Quantum Physics 1 Electromagnetic Radiation and Matter 1.1 Hamiltonian and Vector Potential 1.2 Second Quantization 10 1.2.1 Commutation Relations 1.2) Energy 1.2. Momentum 127 1.2.4 Polarization and Spin 1.2.5 Hamiltonian 3 1.3 Time-Dependent Perturbation Theory 1.4 28 14.1 irst Order R 1.4. Dipole Tra 1.4.3 Higher Multipole Transition 1.5 Blackbody Radiation 36 1.6 Selection Rules Problems 2 Scattering 2.1 Scattering Amplitude and Cross Section 99 2.2 Born Approximation 2.2.1 Schrodinger Equation 2.2.2 Green's Function Formalism 52 2.2.3 Solution of the Schrodinger Equation 55
Contents Preface ix PART 1 Relativistic Quantum Physics 1 1 Electromagnetic Radiation and Matter 3 1.1 Hamiltonian and Vector Potential 3 1.2 Second Quantization 10 1.2.1 Commutation Relations 10 1.2.2 Energy 12 1.2.3 Momentum 17 1.2.4 Polarization and Spin 19 1.2.5 Hamiltonian 23 1.3 Time-Dependent Perturbation Theory 24 1.4 Spontaneous Emission 28 1.4.1 First Order Result 28 1.4.2 Dipole Transition 30 1.4.3 Higher Multipole Transition 32 1.5 Blackbody Radiation 36 1.6 Selection Rules 39 Problems 44 2 Scattering 49 2.1 Scattering Amplitude and Cross Section 49 2.2 Born Approximation 52 2.2.1 Schrodinger Equation ¨ 52 2.2.2 Green’s Function Formalism 52 2.2.3 Solution of the Schrodinger Equation ¨ 55
vi CONTENTS 2.2.4 Born Approximation 58 2.2.5 Electron-Atom Scattering 59 2.3 Photo-Electric Effect 2.4 Photon Scattering 61 2.4.1 Amplitudes 7 2.4.2 Cross Section 72 2.4.3 Rayleigh Scattering 2.4.4 Thomson Scattering 7 Problems 3 Symmetries and Conservation Laws 81 3.1 Symmetries and Conservation Laws 3.1.1 Symmetries 88 3.1.2 Conservation Laws 82 3.2 Continuous Symmetry Operators 84 3.2.1 Translations 84 3.2.2 Rotations 3.3 Discrete Symmetry Operators 83 3.4 Degeneracy 8 3.4.1 Example 3.4.2 Isospin 0 Problems 4 Relativistic Quantum Physics 93 4.1 Klein-Gordon Equation 93 4.2 Dirac Equation 4.2.1 Derivation of the Dirac Equation 4.2.2 Probability Density and Current 101 4.3 Solutions of the Dirac Equation,Anti-Particles 104 4.3.1 Solutions of the Dirac Equation 104 4.3.2 Anti-Particles 108 4.4 Spin,Non-Relativistic Limit and Magnetic Moment 111 4.4.1 Orbital Angular Momentum 111 4.4.2 Spin and Total Angular Momentum 112 4.4.3 Helicity 114 4.4.4 Non-Relativistic Limit 116 4.5 The Hydrogen Atom Re-Revisited 120 Problems 124 5 Special Topics 127 5.1 Introduction 127 5.2 Measurements in Quantum Physics 127 5.3 Einstein-Podolsky-Rosen Paradox 129 5.4 Schrodinger's Cat 133
vi CONTENTS 2.2.4 Born Approximation 58 2.2.5 Electron-Atom Scattering 59 2.3 Photo-Electric Effect 63 2.4 Photon Scattering 67 2.4.1 Amplitudes 67 2.4.2 Cross Section 72 2.4.3 Rayleigh Scattering 73 2.4.4 Thomson Scattering 75 Problems 78 3 Symmetries and Conservation Laws 81 3.1 Symmetries and Conservation Laws 81 3.1.1 Symmetries 81 3.1.2 Conservation Laws 82 3.2 Continuous Symmetry Operators 84 3.2.1 Translations 84 3.2.2 Rotations 86 3.3 Discrete Symmetry Operators 87 3.4 Degeneracy 89 3.4.1 Example 89 3.4.2 Isospin 90 Problems 91 4 Relativistic Quantum Physics 93 4.1 Klein-Gordon Equation 93 4.2 Dirac Equation 95 4.2.1 Derivation of the Dirac Equation 95 4.2.2 Probability Density and Current 101 4.3 Solutions of the Dirac Equation, Anti-Particles 104 4.3.1 Solutions of the Dirac Equation 104 4.3.2 Anti-Particles 108 4.4 Spin, Non-Relativistic Limit and Magnetic Moment 111 4.4.1 Orbital Angular Momentum 111 4.4.2 Spin and Total Angular Momentum 112 4.4.3 Helicity 114 4.4.4 Non-Relativistic Limit 116 4.5 The Hydrogen Atom Re-Revisited 120 Problems 124 5 Special Topics 127 5.1 Introduction 127 5.2 Measurements in Quantum Physics 127 5.3 Einstein-Podolsky-Rosen Paradox 129 5.4 Schrodinger’s Cat ¨ 133
CONTENTS vii 5.5 The Watched Pot 135 5.6 Hidden Variables and Bell's Theorem 137 Problems 140 PART 2 Introduction to Quantum Field Theory 143 6 Second Quantization of Spin 1/2 and Spin 1 Fields 145 6.1 Second Quantization of Spin Fields 145 6.1.1 Plane Wave Solutions 145 6.1.2 Normalization of Spinors 146 6.1.3 Energy 148 6.1.4 Momentum 151 6.1.5 Creation and Annihilation Operators 151 6.2 Second Quantization of Spin 1 Fields 155 Problems 159 7 Covariant Perturbation Theory and Applications 161 7.1 Covariant Perturbation Theory 161 7.1.1 Hamiltonian Density 161 7.1.2 Interaction Representation 165 7.1.3 Covariant Perturbation Theory 168 7.2 W and Z Boson Decays 171 7.2.1 Amplitude 171 7.2.2 Decay Rate 173 7.2.3 Summation over Spin 174 7.2.4 Integration over Phase Space 179 7.2.5 Interpretation 181 7.3 Feynman Graphs 183 7.4 Second Order Processes and Propagators 185 7.4.1 Annihilation and Scattering 185 7.4.2 Time-Ordered Product 187 Problems 193 8 Quantum Electrodynamics 195 8.1 Electron-Positron Annihilation 195 8.2 Electron-Muon Scattering 201 Problems 204 Index 207
CONTENTS vii 5.5 The Watched Pot 135 5.6 Hidden Variables and Bell’s Theorem 137 Problems 140 PART 2 Introduction to Quantum Field Theory 143 6 Second Quantization of Spin 1/2 and Spin 1 Fields 145 6.1 Second Quantization of Spin 1 2 Fields 145 6.1.1 Plane Wave Solutions 145 6.1.2 Normalization of Spinors 146 6.1.3 Energy 148 6.1.4 Momentum 151 6.1.5 Creation and Annihilation Operators 151 6.2 Second Quantization of Spin 1 Fields 155 Problems 159 7 Covariant Perturbation Theory and Applications 161 7.1 Covariant Perturbation Theory 161 7.1.1 Hamiltonian Density 161 7.1.2 Interaction Representation 165 7.1.3 Covariant Perturbation Theory 168 7.2 W and Z Boson Decays 171 7.2.1 Amplitude 171 7.2.2 Decay Rate 173 7.2.3 Summation over Spin 174 7.2.4 Integration over Phase Space 179 7.2.5 Interpretation 181 7.3 Feynman Graphs 183 7.4 Second Order Processes and Propagators 185 7.4.1 Annihilation and Scattering 185 7.4.2 Time-Ordered Product 187 Problems 193 8 Quantum Electrodynamics 195 8.1 Electron-Positron Annihilation 195 8.2 Electron-Muon Scattering 201 Problems 204 Index 207
Preface Over the years,material that used to be taught in graduate school made its 选 ts in extension to Quantum Field Theory are a case in point.This book is intended to facilitate this process.It is written to support a second course in Quantum Physics and attempts to present the material in such a way that it is accessible to adv nced ur dergraduates and starting graduate students in Part 1,comprising Chapters 1 through 5,contains the material for a second course in Quantum Physics.This is where concepts from classical mechanics,electricity and magnetism,statistical physics,and quantum ysics are pulled together n sio of the interaction of radiation aws,scattering, relativistic quantum physics,questions related to the validity of quantum physics,and more.This is material that is suitable to be taught as part of an undergraduate quantum physics course for physics and electrical ngnering majors.Surprisingly,there isno undergraduate textbook that reatsthis material at the undergradu ate level,although it is (or ought to be) taught at many institutions. In Part 2,comprising Chapters 6 through 8,we present elementary Quan- tum Field Theory.That this material should be studied by undergraduates is controversial but I expect it will become accepted practice in the future. This material is intended for undergraduates that are interested in the topics discussed and need it,for exa in a course on elementa ry particle physics or condensed matter. Traditionally such a course is taught in the beginning of graduate school.When teaching particle physics to advanced undergrad- uate students I felt that the time was ripe for an elementary introduction to quantum field theory,concentrating on only those topics that have an
Preface Over the years, material that used to be taught in graduate school made its way into the undergraduate curriculum to create room in graduate courses for new material that must be included because of new developments in various fields of physics. Advanced Quantum Physics and its relativistic extension to Quantum Field Theory are a case in point. This book is intended to facilitate this process. It is written to support a second course in Quantum Physics and attempts to present the material in such a way that it is accessible to advanced undergraduates and starting graduate students in Physics or Electrical Engineering. This book consists of two parts. Part 1, comprising Chapters 1 through 5, contains the material for a second course in Quantum Physics. This is where concepts from classical mechanics, electricity and magnetism, statistical physics, and quantum physics are pulled together in a discussion of the interaction of radiation and matter, selection rules, symmetries and conservation laws, scattering, relativistic quantum physics, questions related to the validity of quantum physics, and more. This is material that is suitable to be taught as part of an undergraduate quantum physics course for physics and electrical engineering majors. Surprisingly, there is no undergraduate textbook that treats this material at the undergraduate level, although it is (or ought to be) taught at many institutions. In Part 2, comprising Chapters 6 through 8, we present elementary Quantum Field Theory. That this material should be studied by undergraduates is controversial but I expect it will become accepted practice in the future. This material is intended for undergraduates that are interested in the topics discussed and need it, for example, in a course on elementary particle physics or condensed matter. Traditionally such a course is taught in the beginning of graduate school. When teaching particle physics to advanced undergraduate students I felt that the time was ripe for an elementary introduction to quantum field theory, concentrating on only those topics that have an
PREfAce application in particle physics at that level.I have also taught the materia in Chapters 6 through 8 to undergraduates and have found that they had no problem in understanding the material and doing the homework. It is hoped that the presentation of the material is such that any good undergraduate studen hysics or electrical engineering can follow that sucha stude ill be motivated to co e the study of quantum field theory beyond its present scope.Additionally,beginning graduate students may also find it of use. Please communicate suggestions,criticisms and errors to the author at hpaar@ucsd.edu. Hans P.Paar January 2010
x PREFACE application in particle physics at that level. I have also taught the material in Chapters 6 through 8 to undergraduates and have found that they had no problem in understanding the material and doing the homework. It is hoped that the presentation of the material is such that any good undergraduate student in physics or electrical engineering can follow it, and that such a student will be motivated to continue the study of quantum field theory beyond its present scope. Additionally, beginning graduate students may also find it of use. Please communicate suggestions, criticisms and errors to the author at hpaar@ucsd.edu. Hans P. Paar January 2010
PREFACE 5 UNITS AND METRIC It is customary in advanced quantum physics to use natural units.These are cgs units with Planck's constant divided by 2 and the velocity of light set to unity.Thus we set =c=1. In natural units we have for example that the Bohr radius of the hydrogen atom is 1/am with a=/(4x)th fine-structure constant and m the mass of the electron.Likewise we have that the energy of the hydrogen atom is am/(2n2)and the classical radius of the electron is a/m. When results of calculations have to be compared with experiment, we must introduce powers of h and c.This can be done easily with dimensional analysis u 9GeV-mb g,for example,that the product he=0.197 GeVfm and(hc2=0.3 In summations I use the Einstein convention that requires one to sum over repeated indices,1 to 3 for Latin and 1 to 4 for Greek letter indices. Three-vectors are written bold faced such as x for the coordinate vector. We use the 'East-coast metric',introduced by Minkovski and made popular by Pauli in special relativity.Its name is obviously US-cen tric.In it ,four vectors have an com nent The alte mative is to use ativity. s is called th West-co me made popular by the textbook on Field Theory by Bjorken and Drell,but is overkill for our purposes.Thus=(x,it),pu=(p,iE),and k=(k,iw) with k the wave vector and the angular frequ ency.a traveling wave can be written as e mple 2 expi(kxx+ y+k: )Squaring four-vecto we get for 2 2 unfortunate negative)and -0 for electromagn etic radiation When integratin 1g four-dimensional space we use dx which one might think is equal to dxidt but this is not so.By dx we mean d3xdt(the West-coast metric does not have this inconsistency,sorry).With this convention the gamma matrices in the Dirac equation are all Hermitian. We also find that e can write the mmutation relations of p and x on the one hand and E and t on th e other in relativistic ly covariant form as [pa,=v/i where is the Kronecker delta.Thus it is seen that the (at first sight odd)difference in sign of the two original commutation relations is required by relativistic invariance. The partial derivatives a/ax will often be abbreviated to a.For example the Lor ntz condition VA+/at=0can be written as aA=0,shov ving that the covarian Furtherm re we have that a ==(V-0-/0t-)o,an expression that is useful in writing down a Lorentz invariant wave equation for the function
PREFACE xi UNITS AND METRIC It is customary in advanced quantum physics to use natural units. These are cgs units with Planck’s constant divided by 2π and the velocity of light set to unity. Thus we set = c = 1. In natural units we have for example that the Bohr radius of the hydrogen atom is 1/αm with α = e2/(4π) the fine-structure constant and m the mass of the electron. Likewise we have that the energy of the hydrogen atom is αm/(2n2) and the classical radius of the electron is α/m. When results of calculations have to be compared with experiment, we must introduce powers of and c. This can be done easily with dimensional analysis using, for example, that the product c = 0.197 GeVfm and (c) 2 = 0.389 GeV2mb. In summations I use the Einstein convention that requires one to sum over repeated indices, 1 to 3 for Latin and 1 to 4 for Greek letter indices. Three-vectors are written bold faced such as x for the coordinate vector. We use the ‘East-coast metric’, introduced by Minkovski and made popular by Pauli in special relativity. Its name is obviously US-centric. In it, fourvectors have an imaginary fourth component. The alternative is to use a metric tensor as in General Relativity. This is called the ‘West-coast metric’, made popular by the textbook on Field Theory by Bjorken and Drell, but is overkill for our purposes. Thus xµ = (x, it), pµ = (p, iE), and kµ = (k, iω) with k the wave vector and ω the angular frequency. A traveling wave can be written as exp ikx = exp i(kxx + kyy + kzz − ωt). Squaring four-vectors, we get for example p2 = pµpµ = p2 − E2 which is −m2 (unfortunately negative) and k2 = 0 for electromagnetic radiation. When integrating in four-dimensional space we use d4x which one might think is equal to d3xidt but this is not so. By d4x we mean d3xdt (the West-coast metric does not have this inconsistency, sorry). With this convention the gamma matrices in the Dirac equation are all Hermitian. We also find that we can write the commutation relations of p and x on the one hand and E and t on the other in relativistically covariant form as [pµ, xν ] = δµν /i where δµν is the Kronecker delta. Thus it is seen that the (at first sight odd) difference in sign of the two original commutation relations is required by relativistic invariance. The partial derivatives ∂/∂xµ will often be abbreviated to ∂µ. For example, the Lorentz condition ∇A + ∂φ/∂t = 0 can be written as ∂µAµ = 0, showing that the Lorentz condition is relativistically covariant. Furthermore we have that ∂2φ = ∂µ∂µφ = (∇2 − ∂2/∂t 2)φ, an expression that is useful in writing down a Lorentz invariant wave equation for the function φ
PREFACE some equations the notation h.c.appears.This differs from the usual perators to which it is applied.So of AB is AtBt e的T0
xii PREFACE In some equations the notation h.c. appears. This differs from the usual h.c. in that h.c. preserves the order of operators to which it is applied. So the h.c. of AB is B†A† while the h.c. of AB is A†B†. We do not follow the convention of some textbooks in which e stands for the absolute value of the electron charge; we use e = −1.602 × 10−19 C
Part I Relativistic Quantum Physics
Part I Relativistic Quantum Physics
