An Introduction to Advanced Quantum Physics

Professor Hans Paar has been on the faculty of the University of California San Diego since 1986. He has taught Quantum Physics in all four quarters it is offered at the undergraduate level and has developed and taught the third and fourth quarter of this course (which corresponds to the content of the book) almost each of those 22 years. He has been vice-chair education in UCSD's Physics Department from 1992-1999 and again from 2008-present. Professor Paar is an experimental particle-astrophysicist and has spent nearly 20 years in experimental high energy physics (particle physics by another name) and nearly three years in Cosmology (the Big-Bang and the particle physics of that era). I am an author on 938 publications.

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 Schrödinger Equation 52

2.2.2 Green’s Function Formalism 52

2.2.3 Solution of the Schrödinger Equation 55

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 Schrödinger’s Cat 133

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