Quantum Mechanics Using Computer Algebra: Includes Sample Programs in C++, Symbolicc++, Maxima, Maplend Mathematica (2nd Edition) Book

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