Non-Relativistic Quantum Theory: Dynamics, Symmetry, and Geometry (Hardcover)

Kai S. Lam

  • 出版商: World Scientific Pub
  • 出版日期: 2009-08-25
  • 售價: $1,450
  • 貴賓價: 9.8$1,421
  • 語言: 英文
  • 頁數: 441
  • 裝訂: Hardcover
  • ISBN: 9814271799
  • ISBN-13: 9789814271790
  • 相關分類: 量子 Quantum
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商品描述

This textbook is mainly for physics students at the advanced undergraduate and beginning graduate levels, especially those with a theoretical inclination. Its chief purpose is to give a systematic introduction to the main ingredients of the fundamentals of quantum theory, with special emphasis on those aspects of group theory (spacetime and permutational symmetries and group representations) and differential geometry (geometrical phases, topological quantum numbers, and Chern–Simons Theory) that are relevant in modern developments of the subject. It will provide students with an overview of key elements of the theory, as well as a solid preparation in calculational techniques.

Contents

Preface vii
1 HowDid Schr‥odinger Get His Equation? 1
2 Heisenberg’s Matrix Mechanics and Dirac’s Re-creation of it 11
3 Dirac’s Derivation of the Quantum Conditions 17
4 The Equivalence between Matrix Mechanics
and Wave Mechanics 21
5 The Dirac Delta Function 27
6 Why Do We Need Hilbert Space? 33
7 The Dirac Bra Ket Notation and the Riesz Theorem 37
8 Self-Adjoint Operators in Hilbert Space 45
9 The Spectral Theorem, Discrete and Continuous Spectra 53
10 Coordinate and Momentum Representations of Quantum
States, Fourier Transforms 59
11 The Uncertainty Principle 63
12 Commutator Algebra 73
13 Ehrenfest’s Theorem 77
14 The Simple Harmonic Oscillator 81
15 Complete Set of Commuting Observables 95
16 Solving Schr‥odinger’s Equation 99
17 Symmetry, Invariance, and Conservation
in Quantum Mechanics 113
18 Why is Group Theory Useful in Quantum Mechanics? 125
19 SO(3) and SU(2) 131
20 The Spectrum of the Angular Momentum Operators 145
21 Whence the Spherical Harmonics? 151
22 Irreducible Representations of SU(2) and SO(3),
Rotation Matrices 159
23 Direct Product Representations,
Clebsch-Gordon Coefficients 169
24 Transformations of Wave Functions and
Vector Operators under SO(3) 173
25 Irreducible Tensor Operators and
the Wigner-Eckart Theorem 179
26 Reduction of Direct Product Representations of SO(3):
The Addition of Angular Momenta 183
27 The Calculation of Clebsch-Gordon Coefficients:
The 3-j Symbols 189
28 Applications of the Wigner-Eckart Theorem 199
29 The Symmetric Groups 209
30 The Lie Algebra of SO(4) and the Hydrogen Atom 233
31 Stationary Perturbations 243
32 The Fine Structure of Hydrogen:
Application of Degenerate Perturbation Theory 255
33 Time-Dependent Perturbation Theory 263
34 Interaction of Matter with the Classical Radiation Field:
Application of Time-Dependent Perturbation Theory 275
35 Potential Scattering Theory 285
36 Analytic Properties of the S-Matrix:
Bound States and Resonances 307
37 Non-Perturbative Bound-State and Scattering-State
Solutions: Radiation-Induced Bound-Continuum
Interactions 317
38 Geometric Phases: The Aharonov-Bohm Effect
and the Magnetic Monopole 333
39 The Berry Phase in Molecular Dynamics 339
40 The Dynamic Phase: Riemann Surfaces in the
Semiclassical Theory of Non-Adiabatic Collisions;
Homotopy and Homology 349
41 “The Connection is the Gauge Field and the
Curvature is the Force”: Some Differential Geometry 367
42 Topological Quantum (Chern) Numbers:
The Integer Quantum Hall Effect 385
43 de Rham Cohomology and Chern Classes:
Some More Differential Geometry 405
44 Chern-Simons Forms: The Fractional Quantum Hall
Effect, Anyons and Knots 413
References 429
Index 433

商品描述(中文翻譯)

這本教科書主要針對高年級本科生和初級研究生的物理學生,特別是那些對理論傾向的學生。它的主要目的是系統地介紹量子理論基礎的主要要素,特別強調在該主題的現代發展中相關的群論(時空和置換對稱性以及群表示)和微分幾何(幾何相位,拓撲量子數和Chern-Simons理論)。它將為學生提供理論的關鍵要素概述,以及計算技巧的扎實準備。

目錄:
前言
1. 薛定谔如何得到他的方程式?
2. 海森堡的矩陣力學和狄拉克的重建
3. 狄拉克對量子條件的推導
4. 矩陣力學和波動力學的等價性
5. 狄拉克δ函數
6. 為什麼我們需要希爾伯特空間?
7. 狄拉克布拉凱特符號和Riesz定理
8. 希爾伯特空間中的自伴算子
9. 譜定理,離散和連續譜
10. 量子狀態的坐標和動量表示,傅立葉變換
11. 不確定性原理
12. 反交換代數
13. Ehrenfest定理
14. 簡谐振子
15. 完全的可交換觀測量集合
16. 解薛定谔方程
17. 量子力學中的對稱性,不變性和守恆
18. 為什麼群論在量子力學中有用?
19. SO(3)和SU(2)
20. 角動量算子的譜
21. 球面調和函數的由來
22. SU(2)和SO(3)的不可約表示,旋轉矩陣
23. 直積表示,Clebsch-Gordon系數
24. 波函數和向量算子在SO(3)下的變換
25. 不可約張量算子和Wigner-Eckart定理
26. SO(3)的直積表示的縮減:角動量的相加
27. Clebsch-Gordon系數的計算:3-j符號
28. Wigner-Eckart定理的應用
29. 對稱群
30. SO(4)的李代數和氫原子
31. 定常微擾
32. 氫的精細結構:退化微擾理論的應用
33. 時間相依微擾理論
34. 物質與經典輻射場的相互作用:時間相依微擾理論的應用
35. 势散射理論
36. S-矩陣的解析性質:束縛態和共振
37. 非微擾束縛態和散射態解:輻射誘導的束縛-連續相互作用
38. 幾何相位:阿哈羅諾夫-波姆效應和磁單極子
39. 分子動力學中的貝里相位
40. 動態相位:非絕熱碰撞的半經典理論中的黎曼曲面;同伦和同调
41. "連接是規範場,曲率是力量":一些微分幾何
42. 拓撲量子(Chern)數:整數量子霍爾效應
43. de Rham上同调和Chern类:更多微分幾何
44. Chern-Simons形式:分數量子霍爾效應,任意子和結

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