Contents
Part I Quantum Mechanics
1
Schrödinger Equation and Its Application . . . . . . . . . . . . . . . . . . .
3
1.1
Early-Stage Quantum Theory . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2
Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.3
Simple Applications of Schrödinger Equation . . . . . . . . . . . . . . 14
1.4
Quantum-Mechanical Operators and Matrices . . . . . . . . . . . . . . 21
1.5
Commutator and Canonical Commutation Relation . . . . . . . . . . 27
Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2
Quantum-Mechanical Harmonic Oscillator . . . . . . . . . . . . . . . . . . . 31
2.1
Classical Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2
Formulation Based on an Operator Method . . . . . . . . . . . . . . . . 33
2.3
Matrix Representation of Physical Quantities . . . . . . . . . . . . . . 41
2.4
Coordinate Representation of Schrödinger Equation . . . . . . . . . 44
2.5
Variance and Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . 51
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3
Hydrogen-Like Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.1
Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2
Constitution of Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.3
Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.4
Generalized Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . 72
3.5
Orbital Angular Momentum: Operator Approach . . . . . . . . . . . . 77
3.6
Orbital Angular Momentum: Analytic Approach . . . . . . . . . . . . 91
3.6.1
Spherical Surface Harmonics and Associated
Legendre Differential Equation . . . . . . . . . . . . . . . . . . 92
3.6.2
Orthogonality of Associated Legendre Functions . . . . . 103
3.7
Radial Wave Functions of Hydrogen-Like Atoms . . . . . . . . . . . 107
3.7.1
Operator Approach to Radial Wave Functions . . . . . . . 107
3.7.2
Normalization of Radial Wave Functions . . . . . . . . . . . 112
3.7.3
Associated Laguerre Polynomials . . . . . . . . . . . . . . . . 116
xiii
Part I Quantum Mechanics
1
Schrödinger Equation and Its Application . . . . . . . . . . . . . . . . . . .
3
1.1
Early-Stage Quantum Theory . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2
Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.3
Simple Applications of Schrödinger Equation . . . . . . . . . . . . . . 14
1.4
Quantum-Mechanical Operators and Matrices . . . . . . . . . . . . . . 21
1.5
Commutator and Canonical Commutation Relation . . . . . . . . . . 27
Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2
Quantum-Mechanical Harmonic Oscillator . . . . . . . . . . . . . . . . . . . 31
2.1
Classical Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2
Formulation Based on an Operator Method . . . . . . . . . . . . . . . . 33
2.3
Matrix Representation of Physical Quantities . . . . . . . . . . . . . . 41
2.4
Coordinate Representation of Schrödinger Equation . . . . . . . . . 44
2.5
Variance and Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . 51
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3
Hydrogen-Like Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.1
Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2
Constitution of Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.3
Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.4
Generalized Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . 72
3.5
Orbital Angular Momentum: Operator Approach . . . . . . . . . . . . 77
3.6
Orbital Angular Momentum: Analytic Approach . . . . . . . . . . . . 91
3.6.1
Spherical Surface Harmonics and Associated
Legendre Differential Equation . . . . . . . . . . . . . . . . . . 92
3.6.2
Orthogonality of Associated Legendre Functions . . . . . 103
3.7
Radial Wave Functions of Hydrogen-Like Atoms . . . . . . . . . . . 107
3.7.1
Operator Approach to Radial Wave Functions . . . . . . . 107
3.7.2
Normalization of Radial Wave Functions . . . . . . . . . . . 112
3.7.3
Associated Laguerre Polynomials . . . . . . . . . . . . . . . . 116
xiii