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Jaroslav Zamastil • Jakub Benda Translated with the assistance of Tereza Uhlifovä Quantum Mechanics and Electrodynamics Springer
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Quantum Mechanics and Electrodynamics - GBV

May 19, 2022

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Page 1: Quantum Mechanics and Electrodynamics - GBV

Jaroslav Zamastil • Jakub Benda Translated with the assistance of Tereza Uhlifovä

Quantum Mechanics and Electrodynamics

Springer

Page 2: Quantum Mechanics and Electrodynamics - GBV

Contents

1 Foundations of Quantum Mechanics 1 1.1 Basic Principles 1 1.2 Mathematical Scheine of the Quantum Theory 5

1.2.1 Stern-Gerlach Experiments 5 1.2.2 Operators 13 1.2.3 Time Evolution in Quantum Theory 14 1.2.4 Stationary States..... 15 1.2.5 Properties of Hermitian Operators 17 1.2.6 Ambiguity in the Determination of States 20 1.2.7 Rabi Method of Magnetic Moments 21

1.3 Systems with More Degrees of Freedom 23 1.3.1 Expected Values of Operators and Their Time Evolution ... 23 1.3.2 Canonical Quantization 25 1.3.3 Harmonie Oscillator 27 1.3.4 Abstract Solution 29 1.3.5 Matrix Representation 31 1.3.6 Dirac ̂ -Function 33 1.3.7 Coordinate Representation 34 1.3.8 Momentan Representation 37 1.3.9 Gaussian Packet and the Uncertainty Principle 39

1.4 FinalNotes 42 References 42

2 Approximate Methods in Quantum Mechanics 45 2.1 Variational Method 46

2.1.1 The Ritz Variational Principle 46 2.1.2 Optimization of Nonlinear Parameters 47 2.1.3 Optimization of Linear Parameters 48

xi

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xii Contents

2.2 Perturbation Method 52 2.2.1 Isolated Levels 52 2.2.2 Degenerate Levels 55 2.2.3 Note on the Error of the Perturbation Method 57

References 58

3 The Hydrogen Atom and Structure of Its Spectral Lines 59 3.1 A Particle in an Electromagnetic Field 60 3.2 The Gross Structure 60

3.2.1 The Problem of Two Particles 60 3.2.2 Electrostatic Potential 62 3.2.3 Units 63, 3.2.4 Spherical Coordinates 65 3.2.5 Solution for s-States 66 3.2.6 Comparison with Experiment 69

3.3 The Hyperfme Structure 70 3.3.1 Magnetic Field ofa Dipole 70 3.3.2 Hamiltonian of a Particle with Spin in an External

Electromagnetic Field 73 3.3.3 Hyperfme Splitting of the Hydrogen Ground State 76 3.3.4 Classification of States Using the Integrals of Motion 79

3.4 Orbital Angular Momentan 84 3.4.1 Significance of Angular Momentum 84 3.4.2 Angular Dependence of/»-States 87 3.4.3 Accidental Degeneracy 90

3.5 Fine Structure 90 3.5.1 Relativistic Corrections 90 3.5.2 Fine Splitting of the Energy Level n = 2 94 3.5.3 Classification of States Using the Integrals of Motion 97

3.6 Hamiltonian of Two Particles with Precision to a4 98 3.6.1 Magnetic Field of a Moving Charge 99 3.6.2 Hamiltonian of Two Particles in an External

Electromagnetic Field 102 3.6.3 Helium-Like Atoms 104 3.6.4 Hydrogen-Like Atoms 105 3.6.5 Final Notes 107

References 107

4 Treasures Hidden in Commutators 109 4.1 A General Solution To Angular Momentum.... 109 4.2 Addition of Angular Momenta 113 4.3 The Runge-Lenz Vector 120

4.3.1 The Runge-Lenz Vector in Classical Mechanics 120 4.3.2 The Runge-Lenz Vector in Quantum Mechanics 123

Page 4: Quantum Mechanics and Electrodynamics - GBV

Contents xiii

4.4 Matrix Elements of Vector Operators 124 4.4.1 Motivation 124 4.4.2 Commutation Relations 125 4.4.3 Selection Rules in m 126 4.4.4 Selection Rules in / 127 4.4.5 Nonzero Matrix Elements: Dependence on m 128 4.4.6 Generalization 130 4.4.7 The Zeeman Effect 132 4.4.8 Nonzero Matrix Elements: Dependence on / and n 135 4.4.9 Spherical Harmonics 135

4.5 The Hydrogen Atom: A General Solution 139 4.5.1 Matrix Elements ofthe Runge-Lenz Vector 139 4.5.2 Energy Spectrum of the Hydrogen Atom 140 4.5.3 The Stark Effect 141 4.5.4 Radial Functions of the Hydrogen Atom 142 4.5.5 Parabolic Coordinates 144

4.6 Decomposition of a Plane Wave into Spherical Waves 145 4.7 Algebra of Radial Operators 148 4.8 FinalNotes 152 References 152

5 The Helium Atom 153 5.1 Symmetry in the Helium Atom 154

5.1.1 The Total Spin and the Antisymmetry of die Wave Function 154

5.1.2 Where Does the Indistinguishability Come From? 157 5.1.3 Additional Symmetries 157 5.1.4 Spectroscopic Notation 158

5.2 Variational Method with the Hartree-Fock Function 158 5.2.1 Multipole Expansion 160 5.2.2 A Note on the Legendre Polynomials 163 5.2.3 Calculation ofthe Integrals 164 5.2.4 Optimization of the Parameters 166

5.3 Variational Method: Configuration Interaction 169 5.3.1 Adaptation of the Basis to Symmetry 170 5.3.2 Angular Integration: The Wigner-Eckart Theorem 173 5.3.3 Angular Integration: Calculation of Reduced Matrix

Elements 176 5.3.4 Calculation of the One-Electron Matrix Elements 177 5.3.5 Radial Integrations 178 5.3.6 Convergence of the Variational Method 183 5.3.7 Comparison with the Experiment 184 5.3.8 A Note on the Parity 185 5.3.9 A Note on Complex Atoms 186

5.4 FinalNotes 187 References 188

Page 5: Quantum Mechanics and Electrodynamics - GBV

HV Contents

6 Dynamics: The Nonrelativistic Theory 189 6.1 Quantization of the Electromagnetic Field 190

6.1.1 WhyQuantize? 190 6.1.2 How to Quantize? 190 6.1.3 Classical Electrodynamics in Conventional Formalism 191 6.1.4 Gauge Invariance and Number of Degrees of Freedom 192 6.1.5 CoulombGauge 193 6.1.6 Hamiltonian of Free Electromagnetic Field 194 6.1.7 Classical Electrodynamics in Hamiltonian Formalism 195 6.1.8 Polarization 198 6.1.9 Quantized Electromagnetic Field 199 6.1.10 Transition to the Complex Basis 200 6.1.11 Transition to the Continuous Basis 202' 6.1.12 States of the Field 20$

6.2 Spontaneous Emission 204 6.2.1 Interaction Representation 205 6.2.2 Time-Dependent Perturbation Method and the Fermi

Golden Rule 206 6.2.3 Elimination of the Field Operators 208 6.2.4 Electric Dipole Radiation 209 6.2.5 Polarization and Angular Distribution of the Radiated

Photons 211 6.2.6 Lifetime of States 213 6.2.7 Circular States and Connection with Classical Theory 215 6.2.8 Forbidden Transitions 218 6.2.9 Radiation Associated with a Change of Spin 219

6.3 Photoelectric Effect 220 6.3.1 IntroductoryNotes 220 6.3.2 Parabolic Coordinates 225 6.3.3 Wave Functions of the Continuous Spectrum 226 6.3.4 Transition from the Discrete to Continuous Part of the

Spectrum I 230 6.3.5 Angular and Energy Distribution of Outgoing Electrons.... 233 6.3.6 Excitation of an Atom by an Electron Impact 236

6.4 Photon-Atom Scattering 240 6.4.1 Lippmann-Schwinger Equation 241 6.4.2 Elimination of Field Operators 243 6.4.3 Rayleigh, Raman, and Resonance Scattering 248 6.4.4 Averaging and Summing over Polarizations and Angles.... 252 6.4.5 Calculation of Expressions Containing a Function of

the Hamilton Operator 253 6.4.6 Transition from the Discrete to the Continuous Part

of the Spectrum II 255 6.4.7 Photon-Hydrogen Scattering 258 6.4.8 Thomson Scattering 261

Page 6: Quantum Mechanics and Electrodynamics - GBV

Contents xv

6.5 Virtual Processes 262 6.5.1 Introductory Notes 262 6.5.2 Lamb-Retherford Experiment 263 6.5.3 Self-energy: Bethe Estimate 264 6.5.4 Improved Bethe Estimate 269 6.5.5 One-Photon Exchange: Instantaneous Interaction . 271 6.5.6 One-Photon Exchange: Effect of Retardation 273 6.5.7 Two-Photon Exchange: Low Energies 277

6.6 Formalism of the Second Quantization 280 6.6.1 Quantization of Free Fields 280 6.6.2 Statesofa Free Electron Field 284 6.6.3 Self-interacting Electron Field 285

6.7 FinalNotes 288 References 289

7 Dynamics: The Relativistic Theory 291 7.1 Relativistic Equation for an Electron 292

7.1.1 Relativistic Notation 292 7.1.2 Klein-Gordon Equation 295 7.1.3 Dirac Equation 296 7.1.4 External EM Field 297 7.1.5 Difficulties Associated with the Interpretation of the

Dirac Equation and Their Resolution 301 7.2 Hamiltonian of Relativistic Quantum Electrodynamics 303

7.2.1 Quantization of the Electron-Positron Field 303 7.2.2 Interaction Hamiltonian 305 7.2.3 Note on Charge Symmetry 308 7.2.4 Note on Gauge Invariance 311

7.3 Ordinary Perturbation Method 312 7.3.1 Interaction of a Bound Electron with Fluctuations

of Fields 314 7.3.2 PositroniumI 319

7.4 Feynman Space-Time Approach 330 7.4.1 Electron in an External EM Field 330 7.4.2 Electron Interacting with Its Own EM Field 337 7.4.3 Photon Propagator and Time Ordered Operator Product.... 339 7.4.4 Electron Self-energy via Green Functions 341 7.4.5 Integration over ko 343 7.4.6 Electron Self-energy: Cancellation of the

Non-covariant Terms 345 7.4.7 Vacuum Polarization: Covariant Formulation 348 7.4.8 Discussion of the Lorentz Invariance 349 7.4.9 What View of Positrons Is the Correct One? 351 7.4.10 Note on the Feynman Diagrams and Feynman Rules 353

Page 7: Quantum Mechanics and Electrodynamics - GBV

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7.5 Electron Self-energy: Calculation 355 7.5.1 Regularization 356 7.5.2 Integration over the Four-Momenta of the Virtual Photon .. 357 7.5.3 Mass Renormalization 363 7.5.4 Calculation of the Observable Part of the Effect 367 7.5.5 Low-Energy Part of the Effect 373 7.5.6 High-Energy Part of the Effect 375 7.5.7 Electron Anomalous Magnetic Moment 376 7.5.8 LambShift 378 7.5.9 Nuclear Motion Effect 379

7.6 Vacuum Polarization: Calculation 380 7.6.1 Propagator Expansion 380 7.6.2 Gauge Invariance and Degree of Divergence 38*5 7.6.3 Note ona Massive VectorField 387 7.6.4 Charge Renormalization 388 7.6.5 Calculation of the Observable Part of the Effect 391 7.6.6 Comparison with Experiment 392

7.7 Two-Photon Exchange at High Energies 395 7.7.1 Longitudinal Photons 396 7.7.2 Two-Photon Exchange in Feynman Approach 396 7.7.3 Photon Propagator and Time Ordered Operator Product.... 397 7.7.4 Note on Gauge Invariance 401 7.7.5 Longitudinal Part of the Interaction 402 7.7.6 The Remaining Part of the Interaction 406 7.7.7 Comparison with Experiment. 407

7.8 Positronium II 408 7.8.1 Virtual Positronium Annihilation in Feynman Approach ... 409 7.8.2 Vacuum Polarization Correction 411 7.8.3 Photon Exchange Correction 412 7.8.4 Virtual Two-Photon Annihilation 425 7.8.5 Comparison with Experiment 426

7.9 FinalNotes 429 References 429

Closing Remarks 431

Epilogue: Electrodynamics as a Part of a Greater Structure 433 /J-decay and Its Problems 433 FermiTheory 435 Weyl Representation 436 Feynman - Gell-Mann Theory 439 Conserved Lepton Number and Generalization of Electrodynamics 442 Glashow Theory of Electroweak Interactions 444 Extension to Quarks 447 Extension to Nucleons 449

Page 8: Quantum Mechanics and Electrodynamics - GBV

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Contents xvii

Effective Interactions at Low Energies 450 Masses of Intermediate Bosons 451 Electroweak Neutral Currents in Atoms 452 Final Notes 454 References 454

Index 457