Table of Contents
Preface v
1 Introduction 1
2 Conservation Law and Schrödinger Equation 4
3 Wave Packet and Free Schrödinger Equation 6
4 Separation Ansatz and Schrödinger Equation 8
5 Matrix Representation in the Hilbert Space L2[-π,π] 10
6 One-Dimensional Potential and Trial Function 12
7 Heisenberg Equation of Motion 14
8 Variance 16
9 Unitary Operators 18
10 Unitary and Hermitian Operators 22
11 Magnus Expansion 24
12 Quantum Harmonic Oscillator 26
13 Harmonic Oscillator and Recursion Relation 28
14 Commutation Relations of p and q 30
15 Wigner Characteristic Functions 32
16 Anharmonic Oscillator 34
17 Morse Potential and Lie Algebra so(2,l) 36
18 One-Dimensional WKB-Solutions 38
19 Angular Momentum Operators I 40
20 Angular Momentum Operators II 42
21 Angular Momentum Operators III 44
22 Lie Algebra su(3) and Commutation Relations 46
23 Spin-1 Lie Algebra and Commutation Relations 48
24 Radial Symmetric Potential and Bound States 50
25 Wave Function of Hydrogen Atom I 54
26 Wave Function of Hydrogen Atom II 56
27 Two-Body Problem 58
28 Helium Atom and Trial Function 60
29 Stark Effect 64
30 Scattering in One-Dimension 68
31 Gauge Theory 70
32 Driven Two Level System 72
33 Berry Phase 74
34 Free Electron Spin Resonance 76
35 Two-Point Ising-Model with External Field 79
36 Two-Point Heisenberg Model 81
37 Spectra of Small Spin Clusters 83
38 Fermi Operators 86
39 Fermi Operators with Spin and the Hubbard Model 89
40 Bose Operators 96
41 Bose Operators and Number States 98
42 Matrix Representation of Bose Operators 100
43 Quartic Hamilton Operator and Bose Operators 102
44 Coherent States 104
45 Squeezed States 106
46 Bose-Fermi Systems 108
47 Dirac Equation and Dispersion Law 112
48 Perturbation Theory 115
49 Elastic Scattering 120
50 Entanglement I 123
51 Entanglement II 128
52 Teleportation 132
53 Exceptional Points 138
54 Expansion of exp(L)A exp(-L) 140
55 Expansion of (A - εB)-1 142
56 Heavyside Function and Delta Function 144
57 Legendre Polynomials 146
58 Associated Legendre Polynomials 148
59 Laguerre Polynomials 150
60 Hermite Polynomials 152
61 Chebyshev Polynomials 154
62 Airy Functions 156
63 Spherical Harmonics 158
64 Clebsch-Gordan Series 162
65 Hypergeometric Functions 164
66 Eigenvalues and Hypergeometric Differential Equation 167
67 Gamma Matrices and Spin Matrices 171
68 Hilbert Space and Fourier Expansion 173
69 Continuous Fourier Transform 175
70 Plancherel Theorem 178
71 Wavelets and Hilbert Space 180
72 Group Theory 183
73 Permutation Groups and Permutation Matrices 188
74 Reducible and Irreducible Representations 192
75 Pauli Group and Clifford Group 196
76 Lie Groups 198
77 Quantum Groups 200
78 Lie Algebras 203
79 Super-Lie Algebra 206
80 Casimir Operator and Lie Algebras 209
81 Gram-Schmidt Orthogonalisation Process 212
82 Soliton Theory and Quantum Mechanics 214
83 Padé Approximation 219
84 Cumulant Expansion 223
85 Kronecker and Tensor Product 225
Bibliography 229
Index 233
