E-Book, Englisch, 484 Seiten
Zamastil / Benda Quantum Mechanics and Electrodynamics
1. Auflage 2017
ISBN: 978-3-319-65780-6
Verlag: Springer Nature Switzerland
Format: PDF
Kopierschutz: 1 - PDF Watermark
E-Book, Englisch, 484 Seiten
ISBN: 978-3-319-65780-6
Verlag: Springer Nature Switzerland
Format: PDF
Kopierschutz: 1 - PDF Watermark
This book highlights the power and elegance of algebraic methods of solving problems in quantum mechanics. It shows that symmetries not only provide elegant solutions to problems that can be solved exactly, but also substantially simplify problems that must be solved approximately. Furthermore, the book provides an elementary exposition of quantum electrodynamics and its application to low-energy physics, along with a thorough analysis of the role of relativistic, magnetic, and quantum electrodynamic effects in atomic spectroscopy. Included are essential derivations made clear through detailed, transparent calculations. The book's commitment to deriving advanced results with elementary techniques, as well as its inclusion of exercises will enamor it to advanced undergraduate and graduate students.
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.
Weitere Infos & Material
1;Preface;5
1.1;A Few Words of Explanation;5
1.2;Prerequisites;8
1.3;Acknowledgments;9
1.4;Errors;9
2;Contents;10
3;List of Exercises;17
4;Notation, Convention, Units, and Experimental Data;19
4.1;Notation;19
4.2;The Summation Convention;20
4.3;The Component Formalism;20
4.4;Units;21
4.5;Fundamental Constants;23
4.6;Experimental Data;23
4.7;References;24
5;1 Foundations of Quantum Mechanics;25
5.1;1.1 Basic Principles;25
5.2;1.2 Mathematical Scheme of the Quantum Theory;29
5.2.1;1.2.1 Stern-Gerlach Experiments;29
5.2.2;1.2.2 Operators;37
5.2.3;1.2.3 Time Evolution in Quantum Theory;38
5.2.4;1.2.4 Stationary States;39
5.2.5;1.2.5 Properties of Hermitian Operators;41
5.2.6;1.2.6 Ambiguity in the Determination of States;44
5.2.7;1.2.7 Rabi Method of Magnetic Moments;45
5.3;1.3 Systems with More Degrees of Freedom;47
5.3.1;1.3.1 Expected Values of Operators and Their Time Evolution;47
5.3.2;1.3.2 Canonical Quantization;49
5.3.3;1.3.3 Harmonic Oscillator;51
5.3.4;1.3.4 Abstract Solution;53
5.3.5;1.3.5 Matrix Representation;55
5.3.6;1.3.6 Dirac ?-Function;57
5.3.7;1.3.7 Coordinate Representation;58
5.3.8;1.3.8 Momentum Representation;61
5.3.9;1.3.9 Gaussian Packet and the Uncertainty Principle;63
5.4;1.4 Final Notes;66
5.5;References;66
6;2 Approximate Methods in Quantum Mechanics;68
6.1;2.1 Variational Method;69
6.1.1;2.1.1 The Ritz Variational Principle;69
6.1.2;2.1.2 Optimization of Nonlinear Parameters;70
6.1.3;2.1.3 Optimization of Linear Parameters;71
6.2;2.2 Perturbation Method;75
6.2.1;2.2.1 Isolated Levels;75
6.2.2;2.2.2 Degenerate Levels;78
6.2.3;2.2.3 Note on the Error of the Perturbation Method;80
6.3;References;81
7;3 The Hydrogen Atom and Structure of Its Spectral Lines;82
7.1;3.1 A Particle in an Electromagnetic Field;83
7.2;3.2 The Gross Structure;83
7.2.1;3.2.1 The Problem of Two Particles;83
7.2.2;3.2.2 Electrostatic Potential;85
7.2.3;3.2.3 Units;86
7.2.4;3.2.4 Spherical Coordinates;88
7.2.5;3.2.5 Solution for s-States;89
7.2.6;3.2.6 Comparison with Experiment;92
7.3;3.3 The Hyperfine Structure;93
7.3.1;3.3.1 Magnetic Field of a Dipole;93
7.3.2;3.3.2 Hamiltonian of a Particle with Spin in an External Electromagnetic Field;96
7.3.3;3.3.3 Hyperfine Splitting of the Hydrogen Ground State;99
7.3.4;3.3.4 Classification of States Using the Integrals of Motion;102
7.4;3.4 Orbital Angular Momentum;107
7.4.1;3.4.1 Significance of Angular Momentum;107
7.4.2;3.4.2 Angular Dependence of p-States;110
7.4.3;3.4.3 Accidental Degeneracy;113
7.5;3.5 Fine Structure;113
7.5.1;3.5.1 Relativistic Corrections;113
7.5.2;3.5.2 Fine Splitting of the Energy Level n = 2;117
7.5.3;3.5.3 Classification of States Using the Integrals of Motion;120
7.6;3.6 Hamiltonian of Two Particles with Precision to ?4;121
7.6.1;3.6.1 Magnetic Field of a Moving Charge;122
7.6.2;3.6.2 Hamiltonian of Two Particles in an External Electromagnetic Field;125
7.6.3;3.6.3 Helium-Like Atoms;127
7.6.4;3.6.4 Hydrogen-Like Atoms;128
7.6.5;3.6.5 Final Notes;130
7.7;References;130
8;4 Treasures Hidden in Commutators;131
8.1;4.1 A General Solution To Angular Momentum;131
8.2;4.2 Addition of Angular Momenta;135
8.3;4.3 The Runge-Lenz Vector;142
8.3.1;4.3.1 The Runge-Lenz Vector in Classical Mechanics;142
8.3.2;4.3.2 The Runge-Lenz Vector in Quantum Mechanics;145
8.4;4.4 Matrix Elements of Vector Operators;146
8.4.1;4.4.1 Motivation;146
8.4.2;4.4.2 Commutation Relations;147
8.4.3;4.4.3 Selection Rules in m;148
8.4.4;4.4.4 Selection Rules in l;149
8.4.5;4.4.5 Nonzero Matrix Elements: Dependence on m;150
8.4.6;4.4.6 Generalization;152
8.4.7;4.4.7 The Zeeman Effect;154
8.4.8;4.4.8 Nonzero Matrix Elements: Dependence on l and n;157
8.4.9;4.4.9 Spherical Harmonics;157
8.5;4.5 The Hydrogen Atom: A General Solution;161
8.5.1;4.5.1 Matrix Elements of the Runge-Lenz Vector;161
8.5.2;4.5.2 Energy Spectrum of the Hydrogen Atom;162
8.5.3;4.5.3 The Stark Effect;163
8.5.4;4.5.4 Radial Functions of the Hydrogen Atom;164
8.5.5;4.5.5 Parabolic Coordinates;166
8.6;4.6 Decomposition of a Plane Wave into Spherical Waves;167
8.7;4.7 Algebra of Radial Operators;170
8.8;4.8 Final Notes;174
8.9;References;174
9;5 The Helium Atom;175
9.1;5.1 Symmetry in the Helium Atom;176
9.1.1;5.1.1 The Total Spin and the Antisymmetry of the Wave Function;176
9.1.2;5.1.2 Where Does the Indistinguishability Come From?;179
9.1.3;5.1.3 Additional Symmetries;179
9.1.4;5.1.4 Spectroscopic Notation;180
9.2;5.2 Variational Method with the Hartree-Fock Function;180
9.2.1;5.2.1 Multipole Expansion;182
9.2.2;5.2.2 A Note on the Legendre Polynomials;185
9.2.3;5.2.3 Calculation of the Integrals;186
9.2.4;5.2.4 Optimization of the Parameters;188
9.3;5.3 Variational Method: Configuration Interaction;191
9.3.1;5.3.1 Adaptation of the Basis to Symmetry;192
9.3.2;5.3.2 Angular Integration: The Wigner-Eckart Theorem;195
9.3.3;5.3.3 Angular Integration: Calculation of Reduced Matrix Elements;198
9.3.4;5.3.4 Calculation of the One-Electron Matrix Elements;199
9.3.5;5.3.5 Radial Integrations;200
9.3.6;5.3.6 Convergence of the Variational Method;205
9.3.7;5.3.7 Comparison with the Experiment;206
9.3.8;5.3.8 A Note on the Parity;207
9.3.9;5.3.9 A Note on Complex Atoms;208
9.4;5.4 Final Notes;209
9.5;References;210
10;6 Dynamics: The Nonrelativistic Theory;211
10.1;6.1 Quantization of the Electromagnetic Field;212
10.1.1;6.1.1 Why Quantize?;212
10.1.2;6.1.2 How to Quantize?;212
10.1.3;6.1.3 Classical Electrodynamics in Conventional Formalism;213
10.1.4;6.1.4 Gauge Invariance and Number of Degrees of Freedom;214
10.1.5;6.1.5 Coulomb Gauge;215
10.1.6;6.1.6 Hamiltonian of Free Electromagnetic Field;216
10.1.7;6.1.7 Classical Electrodynamics in Hamiltonian Formalism;217
10.1.8;6.1.8 Polarization;220
10.1.9;6.1.9 Quantized Electromagnetic Field;221
10.1.10;6.1.10 Transition to the Complex Basis;222
10.1.11;6.1.11 Transition to the Continuous Basis;224
10.1.12;6.1.12 States of the Field;225
10.2;6.2 Spontaneous Emission;226
10.2.1;6.2.1 Interaction Representation;227
10.2.2;6.2.2 Time-Dependent Perturbation Method and the Fermi Golden Rule;228
10.2.3;6.2.3 Elimination of the Field Operators;230
10.2.4;6.2.4 Electric Dipole Radiation;231
10.2.5;6.2.5 Polarization and Angular Distribution of the Radiated Photons;233
10.2.6;6.2.6 Lifetime of States;235
10.2.7;6.2.7 Circular States and Connection with Classical Theory;237
10.2.8;6.2.8 Forbidden Transitions;240
10.2.9;6.2.9 Radiation Associated with a Change of Spin;241
10.3;6.3 Photoelectric Effect;242
10.3.1;6.3.1 Introductory Notes;242
10.3.2;6.3.2 Parabolic Coordinates;247
10.3.3;6.3.3 Wave Functions of the Continuous Spectrum;248
10.3.4;6.3.4 Transition from the Discrete to Continuous Part of the Spectrum I;252
10.3.5;6.3.5 Angular and Energy Distribution of Outgoing Electrons;255
10.3.6;6.3.6 Excitation of an Atom by an Electron Impact;258
10.4;6.4 Photon-Atom Scattering;262
10.4.1;6.4.1 Lippmann-Schwinger Equation;263
10.4.2;6.4.2 Elimination of Field Operators;265
10.4.3;6.4.3 Rayleigh, Raman, and Resonance Scattering;270
10.4.4;6.4.4 Averaging and Summing over Polarizations and Angles;274
10.4.5;6.4.5 Calculation of Expressions Containing a Function of the Hamilton Operator;275
10.4.6;6.4.6 Transition from the Discrete to the Continuous Part of the Spectrum II;277
10.4.7;6.4.7 Photon-Hydrogen Scattering;280
10.4.8;6.4.8 Thomson Scattering;283
10.5;6.5 Virtual Processes;284
10.5.1;6.5.1 Introductory Notes;284
10.5.2;6.5.2 Lamb-Retherford Experiment;285
10.5.3;6.5.3 Self-energy: Bethe Estimate;286
10.5.4;6.5.4 Improved Bethe Estimate;291
10.5.5;6.5.5 One-Photon Exchange: Instantaneous Interaction;293
10.5.6;6.5.6 One-Photon Exchange: Effect of Retardation;295
10.5.7;6.5.7 Two-Photon Exchange: Low Energies;299
10.6;6.6 Formalism of the Second Quantization;302
10.6.1;6.6.1 Quantization of Free Fields;302
10.6.2;6.6.2 States of a Free Electron Field;306
10.6.3;6.6.3 Self-interacting Electron Field;307
10.7;6.7 Final Notes;310
10.8;References;311
11;7 Dynamics: The Relativistic Theory;312
11.1;7.1 Relativistic Equation for an Electron;313
11.1.1;7.1.1 Relativistic Notation;313
11.1.2;7.1.2 Klein-Gordon Equation;316
11.1.3;7.1.3 Dirac Equation;317
11.1.4;7.1.4 External EM Field;318
11.1.5;7.1.5 Difficulties Associated with the Interpretation of the Dirac Equation and Their Resolution;322
11.2;7.2 Hamiltonian of Relativistic Quantum Electrodynamics;324
11.2.1;7.2.1 Quantization of the Electron-Positron Field;324
11.2.2;7.2.2 Interaction Hamiltonian;326
11.2.3;7.2.3 Note on Charge Symmetry;329
11.2.4;7.2.4 Note on Gauge Invariance;332
11.3;7.3 Ordinary Perturbation Method;333
11.3.1;7.3.1 Interaction of a Bound Electron with Fluctuations of Fields;335
11.3.2;7.3.2 Positronium I;340
11.4;7.4 Feynman Space-Time Approach;351
11.4.1;7.4.1 Electron in an External EM Field;351
11.4.2;7.4.2 Electron Interacting with Its Own EM Field;358
11.4.3;7.4.3 Photon Propagator and Time Ordered Operator Product;360
11.4.4;7.4.4 Electron Self-energy via Green Functions;362
11.4.5;7.4.5 Integration over k0;364
11.4.6;7.4.6 Electron Self-energy: Cancellation of the Non-covariant Terms;366
11.4.7;7.4.7 Vacuum Polarization: Covariant Formulation;369
11.4.8;7.4.8 Discussion of the Lorentz Invariance;370
11.4.9;7.4.9 What View of Positrons Is the Correct One?;372
11.4.10;7.4.10 Note on the Feynman Diagrams and Feynman Rules;374
11.5;7.5 Electron Self-energy: Calculation;376
11.5.1;7.5.1 Regularization;377
11.5.2;7.5.2 Integration over the Four-Momenta of the Virtual Photon;378
11.5.3;7.5.3 Mass Renormalization;384
11.5.4;7.5.4 Calculation of the Observable Part of the Effect;388
11.5.5;7.5.5 Low-Energy Part of the Effect;394
11.5.6;7.5.6 High-Energy Part of the Effect;396
11.5.7;7.5.7 Electron Anomalous Magnetic Moment;397
11.5.8;7.5.8 Lamb Shift;399
11.5.9;7.5.9 Nuclear Motion Effect;400
11.6;7.6 Vacuum Polarization: Calculation;401
11.6.1;7.6.1 Propagator Expansion;401
11.6.2;7.6.2 Gauge Invariance and Degree of Divergence;406
11.6.3;7.6.3 Note on a Massive Vector Field;408
11.6.4;7.6.4 Charge Renormalization;409
11.6.5;7.6.5 Calculation of the Observable Part of the Effect;412
11.6.6;7.6.6 Comparison with Experiment;413
11.7;7.7 Two-Photon Exchange at High Energies;416
11.7.1;7.7.1 Longitudinal Photons;417
11.7.2;7.7.2 Two-Photon Exchange in Feynman Approach;417
11.7.3;7.7.3 Photon Propagator and Time Ordered Operator Product;418
11.7.4;7.7.4 Note on Gauge Invariance;422
11.7.5;7.7.5 Longitudinal Part of the Interaction;423
11.7.6;7.7.6 The Remaining Part of the Interaction;427
11.7.7;7.7.7 Comparison with Experiment;428
11.8;7.8 Positronium II;429
11.8.1;7.8.1 Virtual Positronium Annihilation in Feynman Approach;430
11.8.2;7.8.2 Vacuum Polarization Correction;432
11.8.3;7.8.3 Photon Exchange Correction;433
11.8.4;7.8.4 Virtual Two-Photon Annihilation;446
11.8.5;7.8.5 Comparison with Experiment;447
11.9;7.9 Final Notes;450
11.10;References;450
12;Closing Remarks;452
13;Epilogue: Electrodynamics as a Part of a Greater Structure;454
13.1;?-decay and Its Problems;454
13.2;Fermi Theory;456
13.3;Weyl Representation;457
13.4;Feynman – Gell-Mann Theory;460
13.5;Conserved Lepton Number and Generalization of Electrodynamics;463
13.6;Glashow Theory of Electroweak Interactions;465
13.7;Extension to Quarks;468
13.8;Extension to Nucleons;470
13.9;Effective Interactions at Low Energies;471
13.10;Masses of Intermediate Bosons;472
13.11;Electroweak Neutral Currents in Atoms;473
13.12;Final Notes;475
13.13;References;475
14;Index;477
