Quantum Mechanics and Electrodynamics (eBook)

Jaroslav Zamastil is an Associate Professor in the Department of Chemical Physics and Optics at Charles University in Prague. He has published a number of articles in the fields of mathematical and atomic physics.Jakub Benda is a PhD student of theoretical physics at Charles University in Prague.

Preface 5
A Few Words of Explanation 5
Prerequisites 8
Acknowledgments 9
Errors 9
Contents 10
List of Exercises 17
Notation, Convention, Units, and Experimental Data 19
Notation 19
The Summation Convention 20
The Component Formalism 20
Units 21
Fundamental Constants 23
Experimental Data 23
References 24
1 Foundations of Quantum Mechanics 25
1.1 Basic Principles 25
1.2 Mathematical Scheme of the Quantum Theory 29
1.2.1 Stern-Gerlach Experiments 29
1.2.2 Operators 37
1.2.3 Time Evolution in Quantum Theory 38
1.2.4 Stationary States 39
1.2.5 Properties of Hermitian Operators 41
1.2.6 Ambiguity in the Determination of States 44
1.2.7 Rabi Method of Magnetic Moments 45
1.3 Systems with More Degrees of Freedom 47
1.3.1 Expected Values of Operators and Their Time Evolution 47
1.3.2 Canonical Quantization 49
1.3.3 Harmonic Oscillator 51
1.3.4 Abstract Solution 53
1.3.5 Matrix Representation 55
1.3.6 Dirac ?-Function 57
1.3.7 Coordinate Representation 58
1.3.8 Momentum Representation 61
1.3.9 Gaussian Packet and the Uncertainty Principle 63
1.4 Final Notes 66
References 66
2 Approximate Methods in Quantum Mechanics 68
2.1 Variational Method 69
2.1.1 The Ritz Variational Principle 69
2.1.2 Optimization of Nonlinear Parameters 70
2.1.3 Optimization of Linear Parameters 71
2.2 Perturbation Method 75
2.2.1 Isolated Levels 75
2.2.2 Degenerate Levels 78
2.2.3 Note on the Error of the Perturbation Method 80
References 81
3 The Hydrogen Atom and Structure of Its Spectral Lines 82
3.1 A Particle in an Electromagnetic Field 83
3.2 The Gross Structure 83
3.2.1 The Problem of Two Particles 83
3.2.2 Electrostatic Potential 85
3.2.3 Units 86
3.2.4 Spherical Coordinates 88
3.2.5 Solution for s-States 89
3.2.6 Comparison with Experiment 92
3.3 The Hyperfine Structure 93
3.3.1 Magnetic Field of a Dipole 93
3.3.2 Hamiltonian of a Particle with Spin in an External Electromagnetic Field 96
3.3.3 Hyperfine Splitting of the Hydrogen Ground State 99
3.3.4 Classification of States Using the Integrals of Motion 102
3.4 Orbital Angular Momentum 107
3.4.1 Significance of Angular Momentum 107
3.4.2 Angular Dependence of p-States 110
3.4.3 Accidental Degeneracy 113
3.5 Fine Structure 113
3.5.1 Relativistic Corrections 113
3.5.2 Fine Splitting of the Energy Level n = 2 117
3.5.3 Classification of States Using the Integrals of Motion 120
3.6 Hamiltonian of Two Particles with Precision to ?4 121
3.6.1 Magnetic Field of a Moving Charge 122
3.6.2 Hamiltonian of Two Particles in an External Electromagnetic Field 125
3.6.3 Helium-Like Atoms 127
3.6.4 Hydrogen-Like Atoms 128
3.6.5 Final Notes 130
References 130
4 Treasures Hidden in Commutators 131
4.1 A General Solution To Angular Momentum 131
4.2 Addition of Angular Momenta 135
4.3 The Runge-Lenz Vector 142
4.3.1 The Runge-Lenz Vector in Classical Mechanics 142
4.3.2 The Runge-Lenz Vector in Quantum Mechanics 145
4.4 Matrix Elements of Vector Operators 146
4.4.1 Motivation 146
4.4.2 Commutation Relations 147
4.4.3 Selection Rules in m 148
4.4.4 Selection Rules in l 149
4.4.5 Nonzero Matrix Elements: Dependence on m 150
4.4.6 Generalization 152
4.4.7 The Zeeman Effect 154
4.4.8 Nonzero Matrix Elements: Dependence on l and n 157
4.4.9 Spherical Harmonics 157
4.5 The Hydrogen Atom: A General Solution 161
4.5.1 Matrix Elements of the Runge-Lenz Vector 161
4.5.2 Energy Spectrum of the Hydrogen Atom 162
4.5.3 The Stark Effect 163
4.5.4 Radial Functions of the Hydrogen Atom 164
4.5.5 Parabolic Coordinates 166
4.6 Decomposition of a Plane Wave into Spherical Waves 167
4.7 Algebra of Radial Operators 170
4.8 Final Notes 174
References 174
5 The Helium Atom 175
5.1 Symmetry in the Helium Atom 176
5.1.1 The Total Spin and the Antisymmetry of the Wave Function 176
5.1.2 Where Does the Indistinguishability Come From? 179
5.1.3 Additional Symmetries 179
5.1.4 Spectroscopic Notation 180
5.2 Variational Method with the Hartree-Fock Function 180
5.2.1 Multipole Expansion 182
5.2.2 A Note on the Legendre Polynomials 185
5.2.3 Calculation of the Integrals 186
5.2.4 Optimization of the Parameters 188
5.3 Variational Method: Configuration Interaction 191
5.3.1 Adaptation of the Basis to Symmetry 192
5.3.2 Angular Integration: The Wigner-Eckart Theorem 195
5.3.3 Angular Integration: Calculation of Reduced Matrix Elements 198
5.3.4 Calculation of the One-Electron Matrix Elements 199
5.3.5 Radial Integrations 200
5.3.6 Convergence of the Variational Method 205
5.3.7 Comparison with the Experiment 206
5.3.8 A Note on the Parity 207
5.3.9 A Note on Complex Atoms 208
5.4 Final Notes 209
References 210
6 Dynamics: The Nonrelativistic Theory 211
6.1 Quantization of the Electromagnetic Field 212
6.1.1 Why Quantize? 212
6.1.2 How to Quantize? 212
6.1.3 Classical Electrodynamics in Conventional Formalism 213
6.1.4 Gauge Invariance and Number of Degrees of Freedom 214
6.1.5 Coulomb Gauge 215
6.1.6 Hamiltonian of Free Electromagnetic Field 216
6.1.7 Classical Electrodynamics in Hamiltonian Formalism 217
6.1.8 Polarization 220
6.1.9 Quantized Electromagnetic Field 221
6.1.10 Transition to the Complex Basis 222
6.1.11 Transition to the Continuous Basis 224
6.1.12 States of the Field 225
6.2 Spontaneous Emission 226
6.2.1 Interaction Representation 227
6.2.2 Time-Dependent Perturbation Method and the Fermi Golden Rule 228
6.2.3 Elimination of the Field Operators 230
6.2.4 Electric Dipole Radiation 231
6.2.5 Polarization and Angular Distribution of the Radiated Photons 233
6.2.6 Lifetime of States 235
6.2.7 Circular States and Connection with Classical Theory 237
6.2.8 Forbidden Transitions 240
6.2.9 Radiation Associated with a Change of Spin 241
6.3 Photoelectric Effect 242
6.3.1 Introductory Notes 242
6.3.2 Parabolic Coordinates 247
6.3.3 Wave Functions of the Continuous Spectrum 248
6.3.4 Transition from the Discrete to Continuous Part of the Spectrum I 252
6.3.5 Angular and Energy Distribution of Outgoing Electrons 255
6.3.6 Excitation of an Atom by an Electron Impact 258
6.4 Photon-Atom Scattering 262
6.4.1 Lippmann-Schwinger Equation 263
6.4.2 Elimination of Field Operators 265
6.4.3 Rayleigh, Raman, and Resonance Scattering 270
6.4.4 Averaging and Summing over Polarizations and Angles 274
6.4.5 Calculation of Expressions Containing a Function of the Hamilton Operator 275
6.4.6 Transition from the Discrete to the Continuous Part of the Spectrum II 277
6.4.7 Photon-Hydrogen Scattering 280
6.4.8 Thomson Scattering 283
6.5 Virtual Processes 284
6.5.1 Introductory Notes 284
6.5.2 Lamb-Retherford Experiment 285
6.5.3 Self-energy: Bethe Estimate 286
6.5.4 Improved Bethe Estimate 291
6.5.5 One-Photon Exchange: Instantaneous Interaction 293
6.5.6 One-Photon Exchange: Effect of Retardation 295
6.5.7 Two-Photon Exchange: Low Energies 299
6.6 Formalism of the Second Quantization 302
6.6.1 Quantization of Free Fields 302
6.6.2 States of a Free Electron Field 306
6.6.3 Self-interacting Electron Field 307
6.7 Final Notes 310
References 311
7 Dynamics: The Relativistic Theory 312
7.1 Relativistic Equation for an Electron 313
7.1.1 Relativistic Notation 313
7.1.2 Klein-Gordon Equation 316
7.1.3 Dirac Equation 317
7.1.4 External EM Field 318
7.1.5 Difficulties Associated with the Interpretation of the Dirac Equation and Their Resolution 322
7.2 Hamiltonian of Relativistic Quantum Electrodynamics 324
7.2.1 Quantization of the Electron-Positron Field 324
7.2.2 Interaction Hamiltonian 326
7.2.3 Note on Charge Symmetry 329
7.2.4 Note on Gauge Invariance 332
7.3 Ordinary Perturbation Method 333
7.3.1 Interaction of a Bound Electron with Fluctuations of Fields 335
7.3.2 Positronium I 340
7.4 Feynman Space-Time Approach 351
7.4.1 Electron in an External EM Field 351
7.4.2 Electron Interacting with Its Own EM Field 358
7.4.3 Photon Propagator and Time Ordered Operator Product 360
7.4.4 Electron Self-energy via Green Functions 362
7.4.5 Integration over k0 364
7.4.6 Electron Self-energy: Cancellation of the Non-covariant Terms 366
7.4.7 Vacuum Polarization: Covariant Formulation 369
7.4.8 Discussion of the Lorentz Invariance 370
7.4.9 What View of Positrons Is the Correct One? 372
7.4.10 Note on the Feynman Diagrams and Feynman Rules 374
7.5 Electron Self-energy: Calculation 376
7.5.1 Regularization 377
7.5.2 Integration over the Four-Momenta of the Virtual Photon 378
7.5.3 Mass Renormalization 384
7.5.4 Calculation of the Observable Part of the Effect 388
7.5.5 Low-Energy Part of the Effect 394
7.5.6 High-Energy Part of the Effect 396
7.5.7 Electron Anomalous Magnetic Moment 397
7.5.8 Lamb Shift 399
7.5.9 Nuclear Motion Effect 400
7.6 Vacuum Polarization: Calculation 401
7.6.1 Propagator Expansion 401
7.6.2 Gauge Invariance and Degree of Divergence 406
7.6.3 Note on a Massive Vector Field 408
7.6.4 Charge Renormalization 409
7.6.5 Calculation of the Observable Part of the Effect 412
7.6.6 Comparison with Experiment 413
7.7 Two-Photon Exchange at High Energies 416
7.7.1 Longitudinal Photons 417
7.7.2 Two-Photon Exchange in Feynman Approach 417
7.7.3 Photon Propagator and Time Ordered Operator Product 418
7.7.4 Note on Gauge Invariance 422
7.7.5 Longitudinal Part of the Interaction 423
7.7.6 The Remaining Part of the Interaction 427
7.7.7 Comparison with Experiment 428
7.8 Positronium II 429
7.8.1 Virtual Positronium Annihilation in Feynman Approach 430
7.8.2 Vacuum Polarization Correction 432
7.8.3 Photon Exchange Correction 433
7.8.4 Virtual Two-Photon Annihilation 446
7.8.5 Comparison with Experiment 447
7.9 Final Notes 450
References 450
Closing Remarks 452
Epilogue: Electrodynamics as a Part of a Greater Structure 454
?-decay and Its Problems 454
Fermi Theory 456
Weyl Representation 457
Feynman – Gell-Mann Theory 460
Conserved Lepton Number and Generalization of Electrodynamics 463
Glashow Theory of Electroweak Interactions 465
Extension to Quarks 468
Extension to Nucleons 470
Effective Interactions at Low Energies 471
Masses of Intermediate Bosons 472
Electroweak Neutral Currents in Atoms 473
Final Notes 475
References 475
Index 477