Quantum Mechanics in a Nutshell Book

Preface xi

Chapter 1: Introduction 1
1.1 Introduction 1
1.2 Schrödinger's Equation 2
1.3 Eigenfunctions 4
1.4 Measurement 8
1.5 Representations 8
1.5.1 Schrödinger Representation 9
1.5.2 Heisenberg Representation 10
1.6 Noncommuting Operators 11

Chapter 2: One Dimension 14
2.1 Square Well 14
2.2 Linear Potentials 26
2.3 Harmonic Oscillator 29
2.4 Raising and Lowering Operators 34
2.5 Exponential Potential 39
2.5.1 Bound State 40
2.5.2 Continuum State 42
2.6 Delta-Function Potential 45
2.7 Number of Solutions 48
2.8 Normalization 49
2.8.1 Bound States 49
2.8.2 Box Normalization 50
2.8.3 Delta-Function Normalization 51
2.8.4 The Limit of Infinite Volume 54
2.9 Wave Packets 56

Chapter 3: Approximate Methods 62
3.1 WKBJ 62
3.2 Bound States by WKBJ 68
3.2.1 Harmonic Oscillator 71
3.2.2 Morse Potential 71
3.2.3 Symmetric Ramp 73
3.2.4 Discontinuous Potentials 74
3.3 Electron Tunneling 76
3.4 Variational Theory 77
3.4.1 Half-Space Potential 80
3.4.2 Harmonic Oscillator in One Dimension 82

Chapter 4: Spin and Angular Momentum 87
4.1 Operators, Eigenvalues, and Eigenfunctions 87
4.1.1 Commutation Relations 88
4.1.2 Raising and Lowering Operators 89
4.1.3 Eigenfunctions and Eigenvalues 90
4.2 Representations 95
4.3 Rigid Rotations 100
4.4 The Addition of Angular Momentum 102

Chapter 5: Two and Three Dimensions 108
5.1 Plane Waves in Three Dimensions 108
5.2 Plane Waves in Two Dimensions 112
5.3 Central Potentials 114
5.3.1 Central Potentials in 3D 114
5.3.2 Central Potential in 2D 118
5.4 Coulomb Potentials 119
5.4.1 Bound States 119
5.4.2 Confluent Hypergeometric Functions 121
5.4.3 Hydrogen Eigenfunctions 121
5.4.4 Continuum States 125
5.5 WKBJ 126
5.5.1 Three Dimensions 126
5.5.2 3D Hydrogen Atom 127
5.5.3 Two Dimensions 128
5.6 Hydrogen-like Atoms 130
5.6.1 Quantum Defect 131
5.6.2 WKBJ Derivation 132
5.6.3 Expectation Values 134
5.7 Variational Theory 134
5.7.1 Hydrogen Atom: n 1 135
5.7.2 Hydrogen Atom: l 1 136
5.7.3 Helium Atom 137
5.8 Free Particles in a Magnetic Field 143
5.8.1 Gauges 143
5.8.2 Eigenfunctions and Eigenvalues 144
5.8.3 Density of States 146
5.8.4 Quantum Hall Effect 147
5.8.5 Flux Quantization 150

Chapter 6: Matrix Methods and Perturbation Theory 157
6.1 H and H0 157
6.2 Matrix Methods 158
6.2.1 2 x 2 160
6.2.2 Coupled Spins 160
6.2.3 Tight-Binding Model 163
6.3 The Stark Effect 166
6.4 Perturbation Theory 170
6.4.1 General Formulas 170
6.4.2 Harmonic Oscillator in Electric Field 174
6.4.3 Continuum States 176
6.4.4 Green's Function 180
6.5 The Polarizability 181
6.5.1 Quantum Definition 182
6.5.2 Polarizability of Hydrogen 183
6.6 Van der Waals Potential 188
6.7 Spin-Orbit Interaction 194
6.7.1 Spin-Orbit in Atoms 195
6.7.2 Alkali Valence Electron in Electric Field 199
6.8 Bound Particles in Magnetic Fields 202
6.8.1 Magnetic Susceptibility 204
6.8.2 Alkali Atom in Magnetic Field 205
6.8.3 Zeeman Effect 207
6.8.4 Paschen-Back Effect 209

Chapter 7: Time-Dependent Perturbations 213
7.1 Time-Dependent Hamiltonians 213
7.2 Sudden Approximation 215
7.2.1 Shake-up and Shake-off 216
7.2.2 Spin Precession 218
7.3 Adiabatic Approximation 220
7.4 Transition Rates: The Golden Rule 222
7.5 Atomic Excitation by a Charged Particle 226
7.6 Born Approximation to Scattering 231
7.6.1 Cross Section 232
7.6.2 Rutherford Scattering 235
7.6.3 Electron Scattering from Hydrogen 236
7.7 Particle Decay 237

Chapter 8: Electromagnetic Radiation 244
8.1 Quantization of the Field 245
8.1.1 Gauges 246
8.1.2 Lagrangian 250
8.1.3 Hamiltonian 253
8.1.4 Casimir Force 256
8.2 Optical Absorption by a Gas 258
8.2.1 Entangled Photons 268
8.3 Oscillator Strength 269
8.4 Polarizability 273
8.5 Rayleigh and Raman Scattering 278
8.6 Compton Scattering 283

Chapter 9: Many-Particle Systems 288
9.1 Introduction 288
9.2 Fermions and Bosons 289
9.2.1 Two Identical Particles 290
9.3 Exchange Energy 291
9.3.1 Two-Electron Systems 291
9.3.2 Parahelium and Orthohelium 293
9.3.3 Hund's Rules 293
9.4 Many-Electron Systems 295
9.4.1 Evaluating Determinants 295
9.4.2 Ground-State Energy 297
9.4.3 Hartree-Fock Equations 299
9.4.4 Free Electrons 301
9.4.5 Pair Distribution Function 303
9.4.6 Correlation Energy 304
9.4.7 Thomas-Fermi Theory 304
9.4.8 Density Functional Theory 307
9.5 Second Quantization 309
9.5.1 Bosons 309
9.5.2 Fermions 312
9.6 Bose-Einstein Condensation 313
9.6.1 Noninteracting Particles 313
9.6.2 Off-Diagonal Long-Range Order 314

Chapter 10: Scattering Theory 320
10.1 Elastic Scattering 320
10.1.1 Partial Wave Analysis 323
10.1.2 Scattering in Two Dimensions 326
10.1.3 Hard-Sphere Scattering 328
10.1.4 Ramsauer-Townsend Effect 330
10.1.5 Born Approximation 332
10.2 Scattering of Identical Particles 333
10.2.1 Two Free Particles 333
10.2.2 Electron Scattering from Hydrogen 335
10.3 T-Matrices 337
10.4 Distorted Wave Scattering 340
10.5 Scattering from Many Particles 343
10.5.1 Bragg Scattering 343
10.5.2 Scattering by Fluids 344
10.5.3 Strong Scattering 346
10.6 Wave Packets 347
10.6.1 Three Dimensions 347
10.6.2 Scattering by Wave Packets 348

Chapter 11: Relativistic Quantum Mechanics 352
11.1 Four-Vectors 352
11.2 Klein-Gordon Equation 354
11.2.1 Derivation 354
11.2.2 Free Particle 354
11.2.3 Currents and Densities 355
11.2.4 Step Potential 356
11.2.5 Nonrelativistic Limit 356
11.2.6 ?-Mesonic Atoms 357
11.3 Dirac Equation 360
11.3.1 Derivation 361
11.3.2 Current and Charge Density 364
11.3.3 Gamma-Matrices 364
11.3.4 Free-Particle Solutions 366
11.3.5 Spin-Projection Operators 369
11.3.6 Scattering of Dirac Particles 371
11.4 Antiparticles and Negative Energy States 374
11.5 Spin Averages 377
11.6 Nonrelativistic Limit 379
11.6.1 First Approximation 379
11.6.2 Second Approximation 380
11.6.3 Relativistic Corrections for Hydrogenic States 382
11.7 Relativistic Interactions 384
11.7.1 Photon Green's Function 384
11.7.2 Electron Green's Function 387
11.7.3 Boson Exchange 387
11.8 Scattering of Electron and Muon 388

Index 397