Quantum Mechanics - J. Norbury, Angielskie techniczne

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QUANTUM MECHANICS
Professor John W. Norbury
Physics Department
University of Wisconsin-Milwaukee
P.O. Box 413
Milwaukee, WI 53201
November 20, 2000
Contents
1 WAVE FUNCTION 7
1.1 Probability Theory ........................ 8
1.1.1 Mean, Average, Expectation Value ........... 8
1.1.2 Average of a Function .................. 10
1.1.3 Mean, Median, Mode .................. 10
1.1.4 Standard Deviation and Uncertainty .......... 11
1.1.5 Probability Density ................... 14
1.2 Postulates of Quantum Mechanics ............... 14
1.3 Conservation of Probability (Continuity Equation) ...... 19
1.3.1 Conservation of Charge ................. 19
1.3.2 Conservation of Probability ............... 22
1.4 Interpretation of the Wave Function .............. 23
1.5 Expectation Value in Quantum Mechanics ........... 24
1.6 Operators ............................. 24
1.7 Commutation Relations ..................... 27
1.8 Problems ............................. 32
1.9 Answers .............................. 33
2 DIFFERENTIAL EQUATIONS 35
2.1 Ordinary Di®erential Equations ................. 35
2.1.1 Second Order, Homogeneous, Linear, Ordinary Di®er-
ential Equations with Constant Coe±cients ...... 36
2.1.2 Inhomogeneous Equation ................ 39
2.2 Partial Di®erential Equations .................. 42
2.3 Properties of Separable Solutions ................ 44
2.3.1 General Solutions ..................... 44
2.3.2 Stationary States ..................... 44
2.3.3 De¯nite Total Energy .................. 45
1
2
CONTENTS
2.3.4 Alternating Parity .................... 46
2.3.5 Nodes ........................... 46
2.3.6 Complete Orthonormal Sets of Functions ....... 46
2.3.7 Time-dependent Coe±cients .............. 49
2.4 Problems ............................. 50
2.5 Answers .............................. 51
3 INFINITE 1-DIMENSIONAL BOX 53
3.1 Energy Levels ........................... 54
3.2 Wave Function .......................... 57
3.3 Problems ............................. 63
3.4 Answers .............................. 64
4 POSTULATES OF QUANTUM MECHANICS 65
4.1 Mathematical Preliminaries ................... 65
4.1.1 Hermitian Operators ................... 65
4.1.2 Eigenvalue Equations .................. 66
4.2 Postulate 4 ............................ 67
4.3 Expansion Postulate ....................... 68
4.4 Measurement Postulate ..................... 69
4.5 Reduction Postulate ....................... 70
4.6 Summary of Postulates of Quantum Mechanics (Simple Version) 71
4.7 Problems ............................. 74
4.8 Answers .............................. 75
I 1-DIMENSIONAL PROBLEMS
77
5 Bound States 79
5.1 Boundary Conditions ....................... 80
5.2 Finite 1-dimensional Well .................... 81
5.2.1 Regions I and III With Real Wave Number ...... 82
5.2.2 Region II ......................... 83
5.2.3 Matching Boundary Conditions ............. 84
5.2.4 Energy Levels ....................... 87
5.2.5 Strong and Weak Potentials ............... 88
5.3 Power Series Solution of ODEs ................. 89
5.3.1 Use of Recurrence Relation ............... 91
5.4 Harmonic Oscillator ....................... 92
CONTENTS
3
5.5 Algebraic Solution for Harmonic Oscillator .......... 100
5.5.1 Further Algebraic Results for Harmonic Oscillator . . 108
6 SCATTERING STATES 113
6.1 Free Particle ........................... 113
6.1.1 Group Velocity and Phase Velocity ........... 117
6.2 Transmission and Re°ection ................... 119
6.2.1 Alternative Approach .................. 120
6.3 Step Potential ........................... 121
6.4 Finite Potential Barrier ..................... 124
6.5 Quantum Description of a Colliding Particle .......... 126
6.5.1 Expansion Coe±cients .................. 128
6.5.2 Time Dependence .................... 129
6.5.3 Moving Particle ...................... 130
6.5.4 Wave Packet Uncertainty ................ 131
7 FEW-BODY BOUND STATE PROBLEM 133
7.1 2-Body Problem ......................... 133
7.1.1 Classical 2-Body Problem ................ 134
7.1.2 Quantum 2-Body Problem ................ 137
7.2 3-Body Problem ......................... 139
II 3-DIMENSIONAL PROBLEMS
141
8 3-DIMENSIONAL SCHR ODINGER EQUATION 143
8.1 Angular Equations ........................ 144
8.2 Radial Equation ......................... 147
8.3 Bessel's Di®erential Equation .................. 148
8.3.1 Hankel Functions ..................... 150
9 HYDROGEN-LIKE ATOMS 153
9.1 Laguerre Associated Di®erential Equation ........... 153
9.2 Degeneracy ............................ 157
10 ANGULAR MOMENTUM 159
10.1 Orbital Angular Momentum ................... 159
10.1.1 Uncertainty Principle .................. 162
10.2 Zeeman E®ect ........................... 163
10.3 Algebraic Method ......................... 164
4
CONTENTS
2
........................... 166
10.4.2 Spin-Orbit Coupling ................... 167
10.5 Addition of Angular Momentum ................ 169
10.5.1 Wave Functions for Singlet and Triplet Spin States . . 171
10.5.2 Clebsch-Gordon Coe±cients ............... 172
10.6 Total Angular Momentum .................... 172
10.6.1
LS
and
jj
Coupling ................... 173
11 SHELL MODELS 177
11.1 Atomic Shell Model ....................... 177
11.1.1 Degenerate Shell Model ................. 177
11.1.2 Non-Degenerate Shell Model .............. 178
11.1.3 Non-Degenerate Model with Surface E®ects ...... 178
11.1.4 Spectra .......................... 179
11.2 Hartree-Fock Self Consistent Field Method .......... 180
11.3 Nuclear Shell Model ....................... 181
11.3.1 Nuclear Spin ....................... 181
11.4 Quark Shell Model ........................ 182
12 DIRAC NOTATION 183
12.1 Finite Vector Spaces ....................... 183
12.1.1 Real Vector Space .................... 183
12.1.2 Complex Vector Space .................. 185
12.1.3 Matrix Representation of Vectors ............ 188
12.1.4 One-Forms ........................ 188
12.2 In¯nite Vector Spaces ...................... 189
12.3 Operators and Matrices ..................... 191
12.3.1 Matrix Elements ..................... 191
12.3.2 Hermitian Conjugate ................... 194
12.3.3 Hermitian Operators ................... 195
12.3.4 Expectation Values and Transition Amplitudes .... 197
12.4 Postulates of Quantum Mechanics (Fancy Version) ...... 198
12.5 Uncertainty Principle ....................... 198
13 TIME-INDEPENDENT PERTURBATION THEORY, HY-
DROGEN ATOM, POSITRONIUM, STRUCTURE OF HADRONS201
13.1 Non-degenerate Perturbation Theory .............. 204
13.2 Degenerate Perturbation Theory ................ 208
1
10.4 Spin ................................ 165
10.4.1 Spin
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