Центральный Дом Знаний - Norbury J.W. Undergraduate quantum mechanics (lecture notes, Wisconsin, 2000)

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Norbury J.W. Undergraduate quantum mechanics (lecture notes, Wisconsin, 2000)

Norbury J.W. 

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 Differential Equations................. 35
2.1.1 Second Order, Homogeneous, Linear, Ordinary Differential Equations with Constant Coefficients...... 36
2.1.2 Inhomogeneous Equation ................ 39
2.2 Partial Differential Equations.................. 42
2.3 Properties of Separable Solutions ................ 44
2.3.1 General Solutions ..................... 44
2.3.2 Stationary States ..................... 44
2.3.3 Definite Total Energy.................. 45
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 Coefficients .............. 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
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 Reflection................... 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 Coefficients.................. 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 SCHRODINGER EQUATION 143
8.1 Angular Equations ........................ 144
8.2 Radial Equation ......................... 147
8.3 Bessel's Differential Equation.................. 148
8.3.1   Hankel Functions..................... 150
9 HYDROGEN-LIKE ATOMS 153
9.1 Laguerre Associated Differential 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 Effect........................... 163
10.3 Algebraic Method ......................... 164
10.4 Spin ................................ 165
10.4.1 Spin 1........................... 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 Coefficients............... 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 Effects...... 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 Infinite 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, HYDROGEN ATOM, POSITRONIUM, STRUCTURE OF HADRONS201
13.1 Non-degenerate Perturbation Theory .............. 204
13.2 Degenerate Perturbation Theory ................ 208
13.2.1 Two-fold Degeneracy ................... 209
13.2.2 Another Approach .................... 211
13.2.3 Higher Order Degeneracies ............... 212
13.3 Fine Structure of Hydrogen ................... 212
13.3.1 1-Body Relativistic Correction ............. 212
13.3.2 Two-Body Relativistic Correction ........... 216
13.3.3 Spin-Orbit Coupling ................... 217
13.4 Zeeman effect........................... 220
13.5 Stark effect............................ 221
13.6 Hyperfine splitting........................ 221
13.7 Lamb shift ............................. 221
13.8 Positronium and Muonium ................... 221
13.9 Quark Model of Hadrons..................... 221
14 VARIATIONAL PRINCIPLE, HELIUM ATOM, MOLECULES223
14.1 Variational Principle ....................... 223
14.2 Helium Atom ........................... 223
14.3 Molecules ............................. 223
15 WKB APPROXIMATION, NUCLEAR ALPHA DECAY 225
15.1 Generalized Wave Functions................... 225
15.2 Finite Potential Barrier ..................... 230
15.3 Gamow's Theory of Alpha Decay ................ 231
16 TIME-DEPENDENT PERTURBATION THEORY, LASERS235
16.1 Equivalent Schrodinger Equation................ 236
16.2 Dyson Equation .......................... 240
16.3 Constant Perturbation ...................... 241
16.4 Harmonic Perturbation ...................... 244
16.5 Photon Absorption ........................ 247
16.5.1 Radiation Bath ...................... 247
16.6 Photon Emission ......................... 249
16.7 Selection Rules .......................... 249
16.8 Lasers ............................... 250
17 SCATTERING, NUCLEAR REACTIONS 251
17.1 Cross Section ........................... 251
17.2 Scattering Amplitude ....................... 252
17.2.1 Calculation of ci ..................... 255
17.3 Phase Shift ............................ 257
17.4 Integral Scattering Theory .................... 259
17.4.1 Lippman-Schwinger Equation .............. 259
17.4.2 Scattering Amplitude .................. 261
17.4.3 Born Approximation ................... 262
17.5 Nuclear Reactions ........................ 264
18 SOLIDS AND QUANTUM STATISTICS 265
18.1 Solids ............................... 265
18.2 Quantum Statistics ........................ 265
19 SUPERCONDUCTIVITY 267
20 ELEMENTARY PARTICLES 269
21 chapter 1 problems 271
21.1 Problems ............................. 271
21.2 Answers .............................. 272
21.3 Solutions ............................. 273
22 chapter 2 problems 281
22.1 Problems ............................. 281
22.2 Answers .............................. 282
22.3 Solutions ............................. 283
23 chapter 3 problems 287
23.1 Problems ............................. 287
23.2 Answers .............................. 288
23.3 Solutions ............................. 289
24 chapter 4 problems 291
24.1 Problems ............................. 291
24.2 Answers .............................. 292
24.3 Solutions ............................. 293
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