Contents
Preface 1
Ⅰ.Brief review of historical development 1
1.Black body radiation 1
2.Photoelectric effect 2
3.Specific heat of solids 3
4.Compton effect 4
Problems 4
Ⅱ.Uncertainty and complementarily 6
1.Einstein relations and Bohr complimentarily principle 6
2.Wave-particle duality and quantum behavior 9
3.De Broglie relation and Heisenberg uncertainty relation 10
4.Further remarks on the uncertainty principle 11
Problems 14
Ⅲ.The Schrodinger wave equation 15
1.Postulates of quantum mechanics 15
2.The Schrodinger equation for free particles 15
3.Probability distributions 17
4.Operators and expectation values 20
5.Motion of a free wave packet 21
6.Schrodinger equation for a particle in external fields 23
7.Schrodinger equation for a system of interacting particles 25
Problems 28
Ⅳ.Heisenberg equation of motion and commutators 31
1.Heisenberg equation of motion 31
2.Commutation relations 33
Problems 36
Ⅴ.Symmetry properties and conservation laws 38
1.Uniformity of time 38
2.Uniformity of space 39
3.Isotropy of space 40
4.Discrete transformation 42
5.Reduction of the two-body problem 43
Problems 44
Ⅵ.Eigen functions and Eigen values 46
1.Stationary states 46
2.Spectrum of the Hamiltonian 48
3.Dirac’s δ-function 49
4.Orthonormality and completeness 52
5.Density of states 56
6.Linear vector space 58
7.Simultaneous Eigen functions and compatible observables 59
8.Probability amplitudes 62
Problems 64
Ⅶ.The classical limit and WKB method 66
1.Ehrenfest theorem 66
2.Classical limit of Schrodinger equation 69
3.The semi-classical approximation for stationary states 70
4.The quantization rule of Bohr and Sommerfield 74
Problems 77
Ⅷ.Illustrative examples in one dimension 78
1.Square well potential 78
2.Scattering from the square well—resonances and virtual states 80
3.Periodic potential——Koenig-Penney model 83
4.The d function potential 85
5.Linear harmonic oscillator 88
Problems 92
Ⅸ.Illustrative examples in three dimensional space 95
1.The wave equation in spherical coordinates 95
2.Symmetry properties of the central field problem 97
3.Angular momentum eigenstates 98
4.Free particle motion with a definite I 101
5.Isotropic square well 104
6.Hydrogen atom 106
7.Degeneracy of hydrogen energy levels 108
8.Electron in magnetic fields——a cylindrical field problem 111
9.Examples in the confined and low-dimensional space 112
Problems 118
Ⅹ.Angular momentum 122
1.Matrix representation of angular momentum operators 122
2.Spin eigenvectors 124
*3.Coupling of two angular momentum vectors 125
*4.Rotation matrices 130
*5.Arbitrary rotation of a rigid body 133
Problems 135
XI.Unitary transformation 138
*1.States and operators 138
*2.Unitary operators and unitary transformations 141
*3.Observables in different representations 143
*4.Schrodinger, Heisenberg and interaction pictures 146
*5.The interaction picture 148
*6.Pure and mixed states 150
Problems 157
Ⅻ.Approximation methods 158
1.Variation method 158
2.Perturbation theory for non-degenerate stationary states 162
3.Validity of the perturbation method 166
4.Perturbation theory for degenerate states 167
5.Fine structure of hydrogen atomic spectra 168
6.Atoms in external magnetic fields 172
Problems 177
XIII.Many-electron systems 180
1.Indistinguishability and Pauii principle 180
2.Symmetrization and anti-symmetrization of wave functions 182
3.Ground state of helium atom 186
*4.Excited states of helium atom 189
*5.Slater determinant for many-electronatoms 192
*6.Hartree-Fock method 194
*7.Statistical model of Fermi-Thomas 198
Problems 201
XIV.Theory of time-dependent perturbation 203
1.Time-dependent perturbation 203
2.Transition probability per unit time 206
3.Adiabatic and sudden approximation 208
4.Induced emission, absorption and spontaneous emission 209
*5.Multipole radiation and selection rules 214
*6.Lifetime and width of excited levels 216
*7.Photoelectric effect 219
*8.Magnetic resonance 223
*9.Oscillating strength 229
Problems 230
XV.Theory of scattering 233
1.General theory of elastic scattering 233
2.Green function for a free particle 235
3.The Born approximation 238
4.Validity criteria for the Born approximation 240
*5.Method of partial waves 241
*6.Eikonal approximation 247
*7.Elastic scattering from the Coulomb potential 249
*8.General theory of inelastic scattering—The Lipman-Schwingwer formulation 251
*9.Scattering from complex targets 255
Problems 257
References 260
Index 262