Quantum Mechanics for Pedestrians (part 2 - Applications and Extensions), 2nd Ed / Квантовая механика для прохожих, часть 2, 2я редакция
Год издания: 2018
Автор: Jochen Pade / Джошен Пэйд
Жанр или тематика: физика, астрофизика, квантовая механика
Издательство:
© Springer Nature Switzerland AG
ISBN:
978-3-319-00813-4
ISSN: 2192-4791
Серия:
Undergraduate Lecture Notes in Physics
Язык: Английский
Формат: PDF
Качество: Издательский макет или текст (eBook)
Интерактивное оглавление: Да
Количество страниц: 587
Описание:
The two-volume textbook Quantum Mechanics for Pedestrians provides an introduction to the basics of nonrelativistic quantum mechanics. Originally written as a course for students of science education, the book addresses all those science students and others who are looking for a reasonably simple, fresh and modern introduction to the field.
The basic principles of quantum mechanics are presented in the first volume. This second volume discusses applications and extensions to more complex problems. In addition to topics traditionally dealt with in quantum mechanics texts, such as symmetries or many-body problems, here also issues of current interest such as entanglement, Bell's inequalities, decoherence and various aspects of quantum information are treated in detail. Furthermore, questions of the basis of quantum mechanics and epistemological issues are discussed explicitly; these are relevant e.g. to the realism debate. A chapter on the interpretations of quantum mechanics completes this volume.
The necessary mathematical tools are introduced step by step; in the appendix, the most relevant mathematics is compiled in compact form. More advanced topics such as the Lenz vector, Hardy's experiment and Shor's algorithm are treated in more detail in the appendix. As an essential aid to learning and teaching, 130 exercises are included, most of them with their solutions.
Двухтомный учебник "Квантовая механика для пешеходов" представляет собой введение в основы нерелятивистской квантовой механики. Первоначально книга была написана как курс для студентов естественнонаучных специальностей, но адресована всем тем, кто изучает естественные науки, а также тем, кто ищет достаточно простое, свежее и современное введение в эту область.
Основные принципы квантовой механики представлены в первом томе. Во втором томе рассматриваются приложения и расширения к более сложным задачам. В дополнение к темам, традиционно рассматриваемым в текстах по квантовой механике, таким как симметрии или задачи о множестве тел, здесь также подробно рассматриваются актуальные вопросы, такие как запутанность, неравенства Белла, декогеренция и различные аспекты квантовой информации. Кроме того, подробно обсуждаются основы квантовой механики и эпистемологические проблемы; они имеют отношение, например, к дебатам о реализме. Завершает этот том глава, посвященная интерпретациям квантовой механики.
Необходимые математические инструменты представлены шаг за шагом; в приложении в компактной форме собраны наиболее актуальные математические данные. Более сложные темы, такие как вектор Ленца, эксперимент Харди и алгоритм Шора, более подробно рассматриваются в приложении. В качестве основного пособия для изучения и преподавания в приложение включено 130 упражнений, большинство из которых с решениями.
Примеры страниц (скриншоты)
Оглавление
Contents
Part II Applications and Extensions
15 One-Dimensional Piecewise-Constant Potentials ............... 3
15.1 General Remarks .................................. 4
15.2 Potential Steps .................................... 6
15.2.1 Potential Step, E\V0 ......................... 7
15.2.2 Potential Step, E [V0 ........................ 8
15.3 Finite Potential Well ................................ 11
15.3.1 Potential Well, E\0 ......................... 12
15.3.2 Potential Well, E [0 ......................... 15
15.4 Potential Barrier, Tunnel Effect ........................ 17
15.5 From the Finite to the Infinite Potential Well .............. 20
15.6 Wave Packets ..................................... 22
15.7 Exercises ........................................ 25
16 Angular Momentum .................................... 29
16.1 Orbital Angular Momentum Operator.................... 29
16.2 Generalized Angular Momentum, Spectrum ............... 30
16.3 Matrix Representation of Angular Momentum Operators ...... 34
16.4 Orbital Angular Momentum: Spatial Representation
of the Eigenfunctions ............................... 35
16.5 Addition of Angular Momenta ......................... 37
16.6 Exercises ........................................ 40
17 The Hydrogen Atom .................................... 43
17.1 Central Potential ................................... 44
17.2 The Hydrogen Atom ................................ 47
17.3 Complete System of Commuting Observables ............. 52
17.4 On Modelling ..................................... 53
17.5 Exercises ........................................ 54
ix
18 The Harmonic Oscillator ................................. 55
18.1 Algebraic Approach ................................ 56
18.1.1 Creation and Annihilation Operators .............. 56
18.1.2 Properties of the Occupation-Number Operator ...... 58
18.1.3 Derivation of the Spectrum ..................... 58
18.1.4 Spectrum of the Harmonic Oscillator .............. 61
18.2 Analytic Approach (Position Representation) .............. 61
18.3 Exercises ........................................ 63
19 Perturbation Theory .................................... 65
19.1 Stationary Perturbation Theory, Nondegenerate ............. 66
19.1.1 Calculation of the First-Order Energy Correction ..... 67
19.1.2 Calculation of the First-Order State Correction ....... 68
19.2 Stationary Perturbation Theory, Degenerate ............... 69
19.3 Hydrogen: Fine Structure ............................ 70
19.3.1 Relativistic Corrections to the Hamiltonian ......... 70
19.3.2 Results of Perturbation Theory .................. 72
19.3.3 Comparison with the Results of the Dirac Equation ... 73
19.4 Hydrogen: Lamb Shift and Hyperfine Structure ............ 74
19.5 Exercises ........................................ 76
20 Entanglement, EPR, Bell ................................. 79
20.1 Product Space ..................................... 79
20.2 Entangled States ................................... 80
20.2.1 Definition ................................. 81
20.2.2 Single Measurements on Entangled States .......... 83
20.2.3 Schrödinger’s Cat ............................ 85
20.2.4 A Misunderstanding .......................... 87
20.3 The EPR Paradox .................................. 88
20.4 Bell’s Inequality ................................... 91
20.4.1 Derivation of Bell’s Inequality .................. 91
20.4.2 EPR Photon Pairs............................ 92
20.4.3 EPR and Bell ............................... 93
20.5 Conclusions ...................................... 96
20.6 Exercises ........................................ 97
21 Symmetries and Conservation Laws ........................ 99
21.1 Continuous Symmetry Transformations .................. 101
21.1.1 General: Symmetries and Conservation Laws ........ 101
21.1.2 Time Translation ............................ 103
21.1.3 Spatial Translation ........................... 104
21.1.4 Spatial Rotation ............................. 106
21.1.5 Special Galilean Transformation ................. 109
x Contents
21.2 Discrete Symmetry Transformations..................... 109
21.2.1 Parity..................................... 109
21.2.2 Time Reversal .............................. 111
21.3 Exercises ........................................ 114
22 The Density Operator ................................... 117
22.1 Pure States ....................................... 117
22.2 Mixed States ..................................... 120
22.3 Reduced Density Operator ............................ 123
22.3.1 Example .................................. 125
22.3.2 Comparison ................................ 126
22.3.3 General Formulation .......................... 127
22.4 Exercises ........................................ 128
23 Identical Particles ...................................... 131
23.1 Distinguishable Particles ............................. 132
23.2 Identical Particles .................................. 133
23.2.1 A Simple Example ........................... 133
23.2.2 The General Case ............................ 134
23.3 The Pauli Exclusion Principle ......................... 137
23.4 The Helium Atom.................................. 138
23.4.1 Spectrum Without V1;2 ........................ 139
23.4.2 Spectrum with V1;2 (Perturbation Theory) .......... 141
23.5 The Ritz Method .................................. 143
23.6 How Far does the Pauli Principle Reach? ................. 145
23.6.1 Distinguishable Quantum Objects ................ 146
23.6.2 Identical Quantum Objects ..................... 146
23.7 Exercises ........................................ 147
24 Decoherence .......................................... 149
24.1 A Simple Example ................................. 150
24.2 Decoherence ...................................... 152
24.2.1 The Effect of the Environment I ................. 154
24.2.2 Simplified Description ........................ 156
24.2.3 The Effect of the Environment II................. 157
24.2.4 Interim Review ............................. 159
24.2.5 Formal Treatment ............................ 160
24.3 Time Scales, Universality ............................ 161
24.4 Decoherence-Free Subspaces, Basis ..................... 162
24.5 Historical Side Note ................................ 163
24.6 Conclusions ...................................... 164
24.7 Exercises ........................................ 166
Contents xi
25 Scattering ............................................ 169
25.1 Basic Idea; Scattering Cross Section .................... 170
25.1.1 Classical Mechanics .......................... 170
25.1.2 Quantum Mechanics .......................... 171
25.2 The Partial-Wave Method ............................ 173
25.3 Integral Equations, Born Approximation.................. 177
25.4 Exercises ........................................ 180
26 Quantum Information ................................... 183
26.1 No-Cloning Theorem (Quantum Copier) ................. 183
26.2 Quantum Cryptography .............................. 185
26.3 Quantum Teleportation .............................. 185
26.4 The Quantum Computer ............................. 188
26.4.1 Qubits, Registers (Basic Concepts) ............... 188
26.4.2 Quantum Gates and Quantum Computers .......... 190
26.4.3 The Basic Idea of the Quantum Computer .......... 194
26.4.4 The Deutsch Algorithm ....................... 194
26.4.5 Grover’s Search Algorithm ..................... 196
26.4.6 Shor’s Algorithm ............................ 198
26.4.7 On The Construction of Real Quantum Computers.... 199
26.5 Exercises ........................................ 201
27 Is Quantum Mechanics Complete? ......................... 203
27.1 The Kochen–Specker Theorem ........................ 204
27.1.1 Value Function.............................. 205
27.1.2 From the Value Function to Coloring ............. 206
27.1.3 Coloring .................................. 207
27.1.4 Interim Review: The Kochen–Specker Theorem ...... 209
27.2 GHZ States ...................................... 210
27.3 Discussion and Outlook ............................. 214
27.4 Exercises ........................................ 216
28 Interpretations of Quantum Mechanics...................... 219
28.1 Preliminary Remarks................................ 221
28.1.1 Problematic Issues ........................... 221
28.1.2 Difficulties in the Representation of Interpretations .... 224
28.2 Some Interpretations in Short Form ..................... 225
28.2.1 Copenhagen Interpretation(s) .................... 225
28.2.2 Ensemble Interpretation ....................... 227
28.2.3 Bohm’s Interpretation ......................... 228
28.2.4 Many-Worlds Interpretation .................... 228
28.2.5 Consistent-Histories Interpretation ................ 230
xii Contents
28.2.6 Collapse Theories............................ 230
28.2.7 Other Interpretations .......................... 231
28.3 Conclusion ....................................... 232
Appendix A: Abbreviations and Notations ........................ 235
Appendix B: Special Functions ................................. 237
Appendix C: Tensor Product ................................... 247
Appendix D: Wave Packets .................................... 253
Appendix E: Laboratory System, Center-of-Mass System ............ 263
Appendix F: Analytic Treatment of the Hydrogen Atom............. 267
Appendix G: The Lenz Vector .................................. 279
Appendix H: Perturbative Calculation of the Hydrogen Atom ........ 293
Appendix I: The Production of Entangled Photons ................. 297
Appendix J: The Hardy Experiment ............................. 301
Appendix K: Set-Theoretical Derivation of the Bell Inequality ........ 309
Appendix L: The Special Galilei Transformation................... 311
Appendix M: Kramers’ Theorem ............................... 323
Appendix N: Coulomb Energy and Exchange Energy in the
Helium Atom .................................... 325
Appendix O: The Scattering of Identical Particles .................. 329
Appendix P: The Hadamard Transformation...................... 333
Appendix Q: From the Interferometer to the Computer ............. 339
Appendix R: The Grover Algorithm, Algebraically ................. 345
Appendix S: Shor Algorithm ................................... 351
Appendix T: The Gleason Theorem ............................. 367
Appendix U: What is Real? Some Quotations ..................... 369
Appendix V: Remarks on Some Interpretations of
Quantum Mechanics............................... 375
Appendix W: Elements of Quantum Field Theory .................. 387
W.1 Foreword ........................................... 387
W.2 Quantizing a Field - A Toy Example ..................... 388
W.3 Quantization of Free Fields, Introduction ................. 396
W.4 Quantization of Free Fields, Klein–Gordon................ 397
W.5 Quantization of Free Fields, Dirac ....................... 405
Contents xiii
W.6 Quantization of Free Fields, Photons ..................... 418
W.7 Operator Ordering ................................... 423
W.8 Interacting Fields, Quantum Electrodynamics.............. 431
W.9 S-Matrix, First Order ................................. 436
W.10 Contraction, Propagator, Wick’s Theorem ............... 447
W.11 S-Matrix, 2. Order, General ........................... 458
W.12 S-Matrix, 2. Order, 4 Lepton Scattering ................. 462
W.13 High Precision and Infinities........................... 476
Appendix X: Exercises and Solutions ............................ 485
Further Reading ............................................. 577
Index of Volume 1 ........................................... 579
Index of Volume 2 ........................................... 583
