Chapter 1 Quantum states and physical quantities 1
1.1 Quantum states as linear vectors in Hilbert space 1
1.1.1 Quantum states as linear vectors 1
1.1.2 Axiom of quantum mechanics concerning quantum states 2
1.2 Physical quantities as operators in Hilbert space 3
1.2.1 Eigenvalue and eigenvector of Hermitian operator 3
1.2.2 Orthogonal-normalization condition of eigenvectors 6
1.2.3 Axiom of quantum mechanics concerning physical quantities 7
1.2.4 Completeness condition of eigenvector system 8
1.2.5 Projection operator 9
1.2.6 Density operator Pure state and mixed state 10
1.3 Representation 13
1.3.1 Definition of representation 13
1.3.2 Representation transformation 15
1.3.3 Enter in and escape from a representation 16
1.3.4 Invariances of representation transformation 16
1.4 Operators as non-commuting quantities 17
1.4.1 Simultaneous measurability of physical quantities 17
1.4.2 Commutator algebra 17
1.4.3 Function of operator 19
1.5 Unitary transformation Generator of continuous transformation 20
1.5.1 Similar transformation and unitary transformation 20
1.5.2 Relation between unitary operator and Hermitian operator 22
1.5.3 Continuous transformation Generator 23
1.5.4 Space displacement Momentum 25
1.5.5 Space rotation Angular momentum 26
Chapter 2 Time evolution of microscopic system Schr？dinger equation and propagator 29
2.1 Time evolution in coordinate representation 29
2.2 Time evolution operator in Hilbert space 32
2.3 Picture in quantum mechanics 33
2.3.1 Definition of picture and picture-transform in quantum mechanics 33
2.3.2 Schr？dinger picture and Heisenberg picture 35
2.3.3 Interaction picture 36
2.4 Time evolution in the form of path integral 40
2.4.1 Functional integral formula for propagator 40
2.4.2 Functional integral as limit of multiple-integral 42
2.4.3 Functional integral as sum over path 43
2.4.4 Path-integral of Gaussian type 47
2.4.5 From path integral to Schr？dinger equation 48
2.5 Diffraction phenomena in quantum mechanics 49
2.5.1 The role of phase in quantum mechanics 49
2.5.2 Double-slot diffraction 50
2.5.3 Aharonov-Bohm effect Magnetic-flux quantum 52
2.6 Symmetry of microscopic system and conservation of physical quantity 54
2.6.1 Symmetry of microscopic system 54
2.6.2 Conservation of physical quantity 55
2.6.3 Parity 55
2.6.4 Time reversal 56
Chapter 3 Angular momentum 59
3.1 General solution of the eigenvalue problem of angular momentum 59
3.1.1 The procedure for solving eigenvalue problem directly in Hilbert space 60
3.1.2 Solution of the eigenvalue problem of angular momentum 60
3.2 Two kinds of angular momentum 63
3.2.1 Orbital angular momentum 63
3.2.2 Spin 64
3.3 Spin 1/2 67
3.3.1 Properties of Pauli matrix 68
3.3.2 Density matrix of spin 1/2 state Polarization vector 72
3.4 Representation of angular momentum 76
3.4.1 Reducible and irreducible representations 76
3.4.2 Irreducible tensor operator 77
3.4.3 Property of D function 78
3.5 Addition of angular momentum Clebsch-Gordan coefficients 79
3.5.1 Addition of angular momentum 79
3.5.2 Clebsch-Gordan coefficients 85
3.5.3 Addition of D function 87
Chapter 4 Multi-particle system 89
4.1 Axiom on indistinguishability of identical particles 89
4.2 Fock representation of multi-particle state——Discrete spectrum 90
4.2.1 The representation basis 90
4.2.2 Basic operators 91
4.2.3 Action of basic operators on representation basis 95
4.2.4 Representation-and canonical-transformations of annihilation and creation operators 96
4.2.5 Operators of physical quantities in multi-particle system 98
4.3 Fock representation for continuous spectrum 100
4.3.1 Second quantization 102
4.4 Time evolution in Fock representation 104
4.5 Theory of superconductivity 104
4.5.1 The BCS theory of superconductivity 106
4.6 Two-state system 111
4.6.1 Two-electron system 111
4.6.2 Quantum bit Bell inequality 113
Chapter 5 Uncertainty relation Coherent state 119
5.1 Uncertainty relation Minimum-uncertainty state 121
5.1.1 Uncertainty relation 121
5.1.2 Minimum-uncertainty state 122
5.1.3 Schwartz inequality 124
5.2 Harmonic oscillator 125
5.2.1 Solution of eigenvalue problem 125
5.2.2 Time evolution of harmonic oscillator in Heisenberg picture 127
5.3 Coherent state 127
5.3.1 Definition of coherent state 127
5.3.2 Properties of coherent state 128
5.3.3 Representation with coherent states as basis 131
5.3.4 Production of coherent state 134
5.3.5 Coherent state in coordinate representation 138
5.3.6 Displacement,rotation and squeeze operators for coherent states 139
Chapter 6 One-dimensional problem Bound states and resonances 145
6.1 Three-dimensional scattering and one-dimensional transmission and reflection 145
6.2 Piecewise constant potential 148
6.2.1 Potential box 149
6.2.2 Rectangular potential well 150
6.3 Poles of scattering amplitude in the complex Eplane 154
6.3.1 Scattering resonance 154
6.3.2 Poles of scattering amplitude on negative E axis Discrete energy levels 155
6.3.3 Poles of scattering amplitude in complex E plane Breit-Wigner formula 156
6.4 Space-time evolution of one-dimensional scattering 157
6.4.1 Wave packet 158
6.4.2 One-dimensional wave packet scattering 159
6.4.3 Energy-time uncertainty relation 161
6.5 Slowly-varying potential Semi-classical(WKB)approximation 162
6.5.1 Canonical and path-integral quantizations 162
6.5.2 The semi-classical approximation(WKB approximation) 163
6.5.3 Application of semi-classical approximation 170
6.6 The connection formulae for WKB approximation 173
Chapter 7 Three-dimensional Scattering 181
7.1 Asymptotic form of scattering Differential cross section 181
7.2 Formal theory of scattering 182
7.2.1 Lippmann-Schwinger Equation 182
7.2.2 An alternative derivation of L-S equation 185
7.2.3 L-S equation in coordinate representation 188
7.2.4 Properties of scattering state 190
7.2.5 Scattering matrix 192
7.3 Pattial wave phase shift 195
7.3.1 Wave function of free particle in spherical coordinate 195
7.3.2 Spherical Hankel and spherical Neumann functions 197
7.3.3 Partial wave expansion of scattering amplitude 198
7.3.4 Three dimensional square well 200
7.4 Elastic and inelastic scattering 202
7.4.1 Pure elastic scattering 203
7.4.2 Elastic scattering in the presence of inelastic processes 203
7.4.3 Optical theorem 204
7.5 Approximation methods for high energy scattering 205
7.5.1 Born approximation 205
7.5.2 Eikonal approximation 208
7.6 Wigner-Eckart theorem 213
Chapter 8 Relativistic quantum mechanics 217
8.1 Klein-Gordan equation and the difficulty of negative probability 217
8.2 Dirac equation 219
8.3 The spin of Dirac particle 221
8.4 Free particle solution of Dirac equation 222
8.5 Dirac equation in electro-magnetic field Magnetic moment of electron 226
Hints to Selected Exercises 229
H.1 Quantum states and Physical quantities 229
H.2 Time evolution of microscopic system 231
H.3 Angular momentum 235
H.4 Multi-particle system 240
H.5 Uncertainty relation Coherent state 243
H.6 One-dimensional problem Bound states and resonances 246
H.7 Three-dimensional Scattering 250
H.8 Relativistic quantum mechanics 252
Index 254