Notes on Quantum Mechanics - K. Schulten






Contents
1 Lagrangian Mechanics 1
1.1 Basics of Variational Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Lagrangian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Symmetry Properties in Lagrangian Mechanics . . . . . . . . . . . . . . . . . . . . . 7
2 Quantum Mechanical Path Integral 11
2.1 The Double Slit Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Axioms for Quantum Mechanical Description of Single Particle . . . . . . . . . . . . 11
2.3 How to Evaluate the Path Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Propagator for a Free Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 Propagator for a Quadratic Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.6 Wave Packet Moving in Homogeneous Force Field . . . . . . . . . . . . . . . . . . . 25
2.7 Stationary States of the Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . 34
3 The Schrodinger Equation 51
3.1 Derivation of the Schrodinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3 Particle Flux and Schrodinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.4 Solution of the Free Particle Schrodinger Equation . . . . . . . . . . . . . . . . . . . 57
3.5 Particle in One-Dimensional Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.6 Particle in Three-Dimensional Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4 Linear Harmonic Oscillator 73
4.1 Creation and Annihilation Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.2 Ground State of the Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.3 Excited States of the Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . 78
4.4 Propagator for the Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.5 Working with Ladder Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.6 Momentum Representation for the Harmonic Oscillator . . . . . . . . . . . . . . . . 88
4.7 Quasi-Classical States of the Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . 90
5 Theory of Angular Momentum and Spin 97
5.1 Matrix Representation of the group SO(3) . . . . . . . . . . . . . . . . . . . . . . . . 97
5.2 Function space representation of the group SO(3) . . . . . . . . . . . . . . . . . . . . 104
5.3 Angular Momentum Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
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5.4 Angular Momentum Eigenstates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.5 Irreducible Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.6 Wigner Rotation Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.7 Spin 1
2 and the group SU(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.8 Generators and Rotation Matrices of SU(2) . . . . . . . . . . . . . . . . . . . . . . . 128
5.9 Constructing Spin States with Larger Quantum Numbers Through Spinor Operators 129
5.10 Algebraic Properties of Spinor Operators . . . . . . . . . . . . . . . . . . . . . . . . 131
5.11 Evaluation of the Elements dj
mm0( ) of the Wigner Rotation Matrix . . . . . . . . . 138
5.12 Mapping of SU(2) onto SO(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6 Quantum Mechanical Addition of Angular Momenta and Spin 141
6.1 Clebsch-Gordan Coe cients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.2 Construction of Clebsch-Gordan Coe cients . . . . . . . . . . . . . . . . . . . . . . . 147
6.3 Explicit Expression for the Clebsch{Gordan Coe cients . . . . . . . . . . . . . . . . 151
6.4 Symmetries of the Clebsch-Gordan Coe cients . . . . . . . . . . . . . . . . . . . . . 160
6.5 Example: Spin{Orbital Angular Momentum States . . . . . . . . . . . . . . . . . . 163
6.6 The 3j{Coe cients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
6.7 Tensor Operators and Wigner-Eckart Theorem . . . . . . . . . . . . . . . . . . . . . 176
6.8 Wigner-Eckart Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
7 Motion in Spherically Symmetric Potentials 183
7.1 Radial Schrodinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
7.2 Free Particle Described in Spherical Coordinates . . . . . . . . . . . . . . . . . . . . 188
8 Interaction of Charged Particles with Electromagnetic Radiation 203
8.1 Description of the Classical Electromagnetic Field / Separation of Longitudinal and
Transverse Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
8.2 Planar Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
8.3 Hamilton Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
8.4 Electron in a Stationary Homogeneous Magnetic Field . . . . . . . . . . . . . . . . . 210
8.5 Time-Dependent Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 215
8.6 Perturbations due to Electromagnetic Radiation . . . . . . . . . . . . . . . . . . . . 220
8.7 One-Photon Absorption and Emission in Atoms . . . . . . . . . . . . . . . . . . . . . 225
8.8 Two-Photon Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
9 Many{Particle Systems 239
9.1 Permutation Symmetry of Bosons and Fermions . . . . . . . . . . . . . . . . . . . . . 239
9.2 Operators of 2nd Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
9.3 One{ and Two{Particle Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
9.4 Independent-Particle Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
9.5 Self-Consistent Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
9.6 Self-Consistent Field Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
9.7 Properties of the SCF Ground State . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
9.8 Mean Field Theory for Macroscopic Systems . . . . . . . . . . . . . . . . . . . . . . 272
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10 Relativistic Quantum Mechanics 285
10.1 Natural Representation of the Lorentz Group . . . . . . . . . . . . . . . . . . . . . . 286
10.2 Scalars, 4{Vectors and Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
10.3 Relativistic Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
10.4 Function Space Representation of Lorentz Group . . . . . . . . . . . . . . . . . . . . 300
10.5 Klein{Gordon Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
10.6 Klein{Gordon Equation for Particles in an Electromagnetic Field . . . . . . . . . . . 307
10.7 The Dirac Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
10.8 Lorentz Invariance of the Dirac Equation . . . . . . . . . . . . . . . . . . . . . . . . 317
10.9 Solutions of the Free Particle Dirac Equation . . . . . . . . . . . . . . . . . . . . . . 322
10.10Dirac Particles in Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . 333
11 Spinor Formulation of Relativistic Quantum Mechanics 351
11.1 The Lorentz Transformation of the Dirac Bispinor . . . . . . . . . . . . . . . . . . . 351
11.2 Relationship Between the Lie Groups SL(2,C) and SO(3,1) . . . . . . . . . . . . . . 354
11.3 Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
11.4 Spinor Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
11.5 Lorentz Invariant Field Equations in Spinor Form . . . . . . . . . . . . . . . . . . . . 369
12 Symmetries in Physics: Isospin and the Eightfold Way 371
12.1 Symmetry and Degeneracies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
12.2 Isospin and the SU(2)
avor symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 375
12.3 The Eightfold Way and the
avor SU(3) symmetry . . . . . . . . . . . . . . . . . . 380

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