Quantum Mechanics Book

xi

1 Genesis 1

1.1 Introduction 1

1.2 Correspondence Principle 1

1.3 Birth of Quantum Mechanics 2

1.4 Heisenberg's Uncertainty Principle 4

1.5 de Broglie's Matter Waves 6

1.6 Differential Equation for Matter Waves 7

2 Foundations 10

2.1 Ground Rules of Quantum Mechanics 10

2.2 Uncertainty Relations 13

2.3 Hidden Variable Theory 14

2.4 Schrödinger, Heisenberg and Dirac Pictures 15

3 Symmetry and Conservation Laws 17

3.1 Introduction 17

3.2 Conservation of Energy 18

3.3 Conservation of Momentum 19

3.4 Conservation of Angular Momentum 19

3.5 Conservation of Parity 20

3.6 Conservation of Probability 20

4 Energy, Momentum and Angular Momentum 24

4.1 Energy 24

4.2 Momentum 25

4.3 Angular Momentum 25

5 Quantum Mechanics of Photon and Neutrino 34

5.1 Introduction 34

5.2 Schrödinger Equation 34

5.3 Photon with Definite Momentum 36

5.4 Photon with Definite Angular Momentum 37

5.5 Photon Polarisation 41

5.6 Neutrino with Definite Momentum 43

5.7 Neutrino with Definite Angular Momentum 44

6 Passage from Quantum to Classical Mechanics 48

6.1 Introduction 48

6.2 Geometrical Optics from Wave Optics 48

6.3 Classical Mechanics from Wave Mechanics 49

6.4 WKB Approximation 50

7 Solution of Schrödinger Equation 55

7.1 Free Particle with Definite Momentum 55

7.2 Free Particle with Definite Angular Momentum 55

7.3 Particle on a Line Segment 57

7.4 Particle on a Circle 58

7.5 Particle Confined to a Circular Plane 59

7.6 Particle Confined Inside a Sphere 60

7.7 Simple Harmonic Oscillator (Simple Pendulum) 62

7.8 Plane Pendulum 67

7.9 Particle in a Cylindrical Well 70

7.10 Particle in a Spherical Well 71

7.11 Particle in a Coulomb Potential in Two Dimensions 73

7.12 Harmonic Oscillator in Two Dimensions 75

7.13 Electron in a Magnetic Field 76

8 The Hydrogen Atom 81

8.1 Two-Body Schrödinger Equation 81

8.2 Solutions in Spherical Polar Coordinates 82

8.3 Solutions in Parabolic Coordinates 86

8.4 Origin of Coulomb Degeneracy 89

9 Action in Quantum Mechanics 92

9.1 Introduction 92

9.2 Azimuthal Action Operator 92

9.3 Action Operator for Spin Angular Momentum 95

9.4 Principal Action Operator for H-atom 96

9.5 Solution of Eigenvalue Equation for Action for H-atom 99

10 Perturbation Theory 102

10.1 Introduction 102

10.2 Time-Independent Perturbation for Non-Degenerate States 102

10.3 Time-Independent Perturbation for Degenerate States 105

10.4 Time-Dependent Perturbation 112

11 Electron Spin and Hydrogen Fine Structure 117

11.1 Electron Spin 117

11.2 Hydrogen Fine Structure (Pauli Theory) 119

11.3 Pauli Eigenfunction 121

12 Identical Particles 124

12.1 Introduction 124

12.2 Pauli Exclusion and Bose Aggregation Principle 125

12.3 Exchange Interaction 127

12.4 Fock Space (Second) Quantization 128

12.5 Quantization of the Electromagnetic Field 131

12.6 Parabosons and Parafermions 133

13 The Helium Atom 137

13.1 Ground State 137

13.2 First Excited State 139

14 Many-Electron Atoms 142

14.1 Self-Consistent Field Method 142

14.2 Statistical Method 145

15 Emission and Absorption of Photons by Atoms 148

15.1 Introduction 148

15.2 Spontaneous Emission of Photons of Definite Momentum 149

15.3 Spontaneous Emission of Photons of Definite Angular Momentum 154

15.4 Photoelectric Effect 158

15.5 Radiative Electron Capture by Proton 159

15.6 Stimulated (Induced) Emission and Absorption 160

15.7 Thermodynamic Equilibrium between Atom and Radiation 161

16 Scattering of Photons by Atoms 163

16.1 Introduction 163

16.2 Scattering of Photon by Charged Particle 164

16.3 Scattering of Photon by Bound Electron 165

17 Lamb Shift 167

17.1 Introduction 167

17.2 Bethe Method 167

17.3 Welton Method 170

18 Theory of Scattering 173

18.1 Introduction 173

18.2 Schematic Presentation of Scattering Experiment 173

18.3 Scattering Amplitude from Wave Equation 174

18.4 Heisenberg's S-Matrix 177

18.5 Unitarity of S-Matrix 178

18.6 Reciprocity of S-Matrix 179

18.7 Lippmann—Schwinger Formalism 180

18.8 Method of Partial Waves for Elastic Scattering 182

18.9 Method of Partial Waves for Inelastic Scattering 185

18.10 Coulomb Scattering 187

18.11 Analytic Structure of S-Matrix in Complex Momentum Plane 190

18.12 Analytic Structure of S-Matrix in Complex Angular Momentum Plane 197

19 Phase of the Wavefunction 203

19.1 Introduction 203

19.2 Charged Particle in a Faraday Cage 203

19.3 Charged Particle in the Magnetic Field of a Solenoid 204

19.4 Magnetic Dipole in the Field of a Line Charge 206

19.5 Phase of the Spin Wavefunction 206

19.6 Neutral Particle in Earth's Gravitational Field 207

19.7 Phase of Neutron Wavefunction due to Earth's Rotation 208

19.8 Charged Particle in the Field of a Magnetic Charge 210

19.9 Phase of Coherent Macroscopic Quantum System 211

19.10 Topological Phase 213

20 Lagrangian Formulation of Quantum Mechanics 216

20.1 Dirac's Transformation Theory Approach 216

20.2 Feynman's Path Integral Approach 219

20.3 Dirac-Feynman Action Principle 221

20.4 Equivalence of Feynman and Schrödinger Equations 221

20.5 Concluding Remarks 223

21 Paradoxes in Quantum Mechanics 225

21.1 Schrödinger's Cat Paradox 225

21.2 Einstein-Podolosky-Rosen (EPR) Paradox 227

21.3 Bell Inequality for Hidden Variable Theory 228

Appendix-A 231

Appendix-B 233

Subject Index 235

Author Index 23