Quantum Mechanics Mathematical Structure and Physical Structure Problems and Solutions
Quantum Mechanics
Mathematical Structure
and
Physical Structure
Problems and Solutions
John R. Boccio
Professor of Physics
Swarthmore College
April 9, 2012
Contents
33 4.22.8 Functions of Operators . . . . . . . . . . . . . . . . . . . .
27 4.22.2 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . .
28 4.22.3 Orthogonal Basis Vectors . . . . . . . . . . . . . . . . . .
29 4.22.4 Operator Matrix Representation . . . . . . . . . . . . . .
30 4.22.5 Matrix Representation and Expectation Value . . . . . .
31 4.22.6 Projection Operator Representation . . . . . . . . . . . .
32 4.22.7 Operator Algebra . . . . . . . . . . . . . . . . . . . . . . .
34 4.22.9 A Symmetric Matrix . . . . . . . . . . . . . . . . . . . . .
27 4.22 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34 4.22.10 Determinants and Traces . . . . . . . . . . . . . . . . . .
35 4.22.11 Function of a Matrix . . . . . . . . . . . . . . . . . . . . .
36 4.22.12 More Gram-Schmidt . . . . . . . . . . . . . . . . . . . . .
37 4.22.13 Infinite Dimensions . . . . . . . . . . . . . . . . . . . . . .
38 4.22.14 Spectral Decomposition . . . . . . . . . . . . . . . . . . .
39 4.22.15 Measurement Results . . . . . . . . . . . . . . . . . . . .
27 4.22.1 Simple Basis Vectors . . . . . . . . . . . . . . . . . . . . .
Dirac Language
3 Formulation of Wave Mechanics - Part 2
7 3.11.6 Uncertain Dart . . . . . . . . . . . . . . . . . . . . . . . .
1 3.11 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 3.11.1 Free Particle in One-Dimension - Wave Functions . . . . .
1 3.11.2 Free Particle in One-Dimension - Expectation Values . . .
3 3.11.3 Time Dependence . . . . . . . . . . . . . . . . . . . . . .
5 3.11.4 Continuous Probability . . . . . . . . . . . . . . . . . . .
6 3.11.5 Square Wave Packet . . . . . . . . . . . . . . . . . . . . .
10 3.11.7 Find the Potential and the Energy . . . . . . . . . . . . .
4 The Mathematics of Quantum Physics:
11 3.11.8 Harmonic Oscillator wave Function . . . . . . . . . . . . .
12 3.11.9 Spreading of a Wave Packet . . . . . . . . . . . . . . . . .
13 3.11.10 The Uncertainty Principle says ... . . . . . . . . . . . . . .
18 3.11.11 Free Particle Schrodinger Equation . . . . . . . . . . . . .
19 3.11.12 Double Pinhole Experiment . . . . . . . . . . . . . . . . .
19 3.11.13 A Falling Pencil . . . . . . . . . . . . . . . . . . . . . . .
24
40 ii CONTENTS 4.22.16 Expectation Values . . . . . . . . . . . . . . . . . . . . . .
40 4.22.17 Eigenket Properties . . . . . . . . . . . . . . . . . . . . .
96 6.19.5 Is it a Density Matrix? . . . . . . . . . . . . . . . . . . . .
87 5.6.14 Matrix Observables for Classical Probability . . . . . . . .
88
6 The Formulation of Quantum Mechanics
91 6.19 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91 6.19.1 Can It Be Written? . . . . . . . . . . . . . . . . . . . . . .
91 6.19.2 Pure and Nonpure States . . . . . . . . . . . . . . . . . .
92 6.19.3 Probabilities . . . . . . . . . . . . . . . . . . . . . . . . .
94 6.19.4 Acceptable Density Operators . . . . . . . . . . . . . . . .
97 6.19.6 Unitary Operators . . . . . . . . . . . . . . . . . . . . . .
79 5.6.12 Modeling Dice: Observables and Expectation Values . . .
97 6.19.7 More Density Matrices . . . . . . . . . . . . . . . . . . . .
99
6.19.8 Scale Transformation . . . . . . . . . . . . . . . . . . . . . 101
6.19.9 Operator Properties . . . . . . . . . . . . . . . . . . . . . 102
6.19.10 An Instantaneous Boost . . . . . . . . . . . . . . . . . . . 103
6.19.11 A Very Useful Identity . . . . . . . . . . . . . . . . . . . . 105
6.19.12 A Very Useful Identity with some help.... . . . . . . . . . 106
6.19.13 Another Very Useful Identity . . . . . . . . . . . . . . . . 108
85 5.6.13 Conditional Probabilities for Dice . . . . . . . . . . . . . .
77 5.6.11 The Poisson Probability Distribution . . . . . . . . . . . .
41 4.22.18 The World of Hard/Soft Particles . . . . . . . . . . . . . .
57
43 4.22.19 Things in Hilbert Space . . . . . . . . . . . . . . . . . . .
45 4.22.20 A 2-Dimensional Hilbert Space . . . . . . . . . . . . . . .
47 4.22.21 Find the Eigenvalues . . . . . . . . . . . . . . . . . . . . .
49 4.22.22 Operator Properties . . . . . . . . . . . . . . . . . . . . .
50 4.22.23 Ehrenfest’s Relations . . . . . . . . . . . . . . . . . . . . .
51 4.22.24 Solution of Coupled Linear ODEs . . . . . . . . . . . . . .
53 4.22.25 Spectral Decomposition Practice . . . . . . . . . . . . . .
55 4.22.26 More on Projection Operators . . . . . . . . . . . . . . .
5 Probability 61 5.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74 5.6.10 Extended Menu at Berger’s Burgers . . . . . . . . . . . .
61 5.6.1 Simple Probability Concepts . . . . . . . . . . . . . . . .
61 5.6.2 Playing Cards . . . . . . . . . . . . . . . . . . . . . . . . .
66 5.6.3 Birthdays . . . . . . . . . . . . . . . . . . . . . . . . . . .
67 5.6.4 Is there life? . . . . . . . . . . . . . . . . . . . . . . . . . .
68 5.6.5 Law of large Numbers . . . . . . . . . . . . . . . . . . . .
68 5.6.6 Bayes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69 5.6.7 Psychological Tests . . . . . . . . . . . . . . . . . . . . . .
70 5.6.8 Bayes Rules, Gaussians and Learning . . . . . . . . . . .
71 5.6.9 Berger’s Burgers-Maximum Entropy Ideas . . . . . . . . .
6.19.14 Pure to Nonpure? . . . . . . . . . . . . . . . . . . . . . . 109 CONTENTS iii
6.19.15 Schur’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . 110
6.19.16 More About the Density Operator . . . . . . . . . . . . . 112
6.19.17 Entanglement and the Purity of a Reduced Density Op- erator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.19.18 The Controlled-Not Operator . . . . . . . . . . . . . . . . 115
6.19.19 Creating Entanglement via Unitary Evolution . . . . . . . 116
6.19.20 Tensor-Product Bases . . . . . . . . . . . . . . . . . . . . 117
6.19.21 Matrix Representations . . . . . . . . . . . . . . . . . . . 118
6.19.22 Practice with Dirac Language for Joint Systems . . . . . 121
6.19.23 More Mixed States . . . . . . . . . . . . . . . . . . . . . . 123
6.19.24 Complete Sets of Commuting Observables . . . . . . . . . 125
6.19.25 Conserved Quantum Numbers . . . . . . . . . . . . . . . 126
7 How Does It really Work:
Photons, K-Mesons and Stern-Gerlach 127
7.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.5.1 Change the Basis . . . . . . . . . . . . . . . . . . . . . . . 127
7.5.2 Polaroids . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
7.5.3 Calcite Crystal . . . . . . . . . . . . . . . . . . . . . . . . 129
7.5.4 Turpentine . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.5.5 What QM is all about - Two Views . . . . . . . . . . . . 130
7.5.6 Photons and Polarizers . . . . . . . . . . . . . . . . . . . 134
7.5.7 Time Evolution . . . . . . . . . . . . . . . . . . . . . . . . 135
7.5.8 K-Meson oscillations . . . . . . . . . . . . . . . . . . . . . 136
7.5.9 What comes out? . . . . . . . . . . . . . . . . . . . . . . . 138
7.5.10 Orientations . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7.5.11 Find the phase angle . . . . . . . . . . . . . . . . . . . . . 140
7.5.12 Quarter-wave plate . . . . . . . . . . . . . . . . . . . . . . 143
7.5.13 What is happening? . . . . . . . . . . . . . . . . . . . . . 144
7.5.14 Interference . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.5.15 More Interference . . . . . . . . . . . . . . . . . . . . . . . 146
7.5.16 The Mach-Zender Interferometer and Quantum Interference147
7.5.17 More Mach-Zender . . . . . . . . . . . . . . . . . . . . . . 153
8 Schrodinger Wave equation
1-Dimensional Quantum Systems 155
8.15 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
8.15.1 Delta function in a well . . . . . . . . . . . . . . . . . . . 155
8.15.2 Properties of the wave function . . . . . . . . . . . . . . . 156
8.15.3 Repulsive Potential . . . . . . . . . . . . . . . . . . . . . . 157
8.15.4 Step and Delta Functions . . . . . . . . . . . . . . . . . . 159
8.15.5 Atomic Model . . . . . . . . . . . . . . . . . . . . . . . . 160
8.15.6 A confined particle . . . . . . . . . . . . . . . . . . . . . . 164 8.15.7 1/x potential . . . . . . . . . . . . . . . . . . . . . . . . . 165
CONTENTS iv
8.15.8 Using the commutator . . . . . . . . . . . . . . . . . . . . 166
8.15.9 Matrix Elements for Harmonic Oscillator . . . . . . . . . 168
8.15.10 A matrix element . . . . . . . . . . . . . . . . . . . . . . . 169
8.15.11 Correlation function . . . . . . . . . . . . . . . . . . . . . 170
8.15.12 Instantaneous Force . . . . . . . . . . . . . . . . . . . . . 171
8.15.13 Coherent States . . . . . . . . . . . . . . . . . . . . . . . . 172
8.15.14 Oscillator with Delta Function . . . . . . . . . . . . . . . 174
8.15.15 Measurement on a Particle in a Box . . . . . . . . . . . . 177
8.15.16 Aharonov-Bohm experiment . . . . . . . . . . . . . . . . . 183
8.15.17 A Josephson Junction . . . . . . . . . . . . . . . . . . . . 186
8.15.18 Eigenstates using Coherent States . . . . . . . . . . . . . 189
8.15.19 Bogliubov Transformation . . . . . . . . . . . . . . . . . . 190
8.15.20 Harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . 192
8.15.21 Another oscillator . . . . . . . . . . . . . . . . . . . . . . 193
8.15.22 The coherent state . . . . . . . . . . . . . . . . . . . . . . 194
8.15.23 Neutrino Oscillations . . . . . . . . . . . . . . . . . . . . . 200
8.15.24 Generating Function . . . . . . . . . . . . . . . . . . . . . 205
8.15.25 Given the wave function ...... . . . . . . . . . . . . . . . . 207
8.15.26 What is the oscillator doing? . . . . . . . . . . . . . . . . 208
8.15.27 Coupled oscillators . . . . . . . . . . . . . . . . . . . . . . 210
8.15.28 Interesting operators .... . . . . . . . . . . . . . . . . . . . 210
8.15.29 What is the state? . . . . . . . . . . . . . . . . . . . . . . 212
8.15.30 Things about particles in box . . . . . . . . . . . . . . . . 213
8.15.31 Handling arbitrary barriers..... . . . . . . . . . . . . . . . 214
8.15.32 Deuteron model . . . . . . . . . . . . . . . . . . . . . . . 217
8.15.33 Use Matrix Methods . . . . . . . . . . . . . . . . . . . . . 219
8.15.34 Finite Square Well Encore . . . . . . . . . . . . . . . . . . 220
8.15.35 Half-Infinite Half-Finite Square Well Encore . . . . . . . . 224
8.15.36 Nuclear α Decay . . . . . . . . . . . . . . . . . . . . . . . 229
8.15.37 One Particle, Two Boxes . . . . . . . . . . . . . . . . . . 231
8.15.38 A half-infinite/half-leaky box . . . . . . . . . . . . . . . . 236
8.15.39 Neutrino Oscillations Redux . . . . . . . . . . . . . . . . . 239
8.15.40 Is it in the ground state? . . . . . . . . . . . . . . . . . . 242
8.15.41 Some Thoughts on T-Violation . . . . . . . . . . . . . . . 243
8.15.42 Kronig-Penney Model . . . . . . . . . . . . . . . . . . . . 246
8.15.43 Operator Moments and Uncertainty . . . . . . . . . . . . 254
8.15.44 Uncertainty and Dynamics . . . . . . . . . . . . . . . . . 255
9 Angular Momentum; 2- and 3-Dimensions 259
9.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
9.7.1 Position representation wave function . . . . . . . . . . . 259
9.7.2 Operator identities . . . . . . . . . . . . . . . . . . . . . . 260
9.7.3 More operator identities . . . . . . . . . . . . . . . . . . . 261
9.7.4 On a circle . . . . . . . . . . . . . . . . . . . . . . . . . . 263
9.7.5 Rigid rotator . . . . . . . . . . . . . . . . . . . . . . . . . 263 CONTENTS v
9.7.6 A Wave Function . . . . . . . . . . . . . . . . . . . . . . . 265
9.7.7 L = 1 System . . . . . . . . . . . . . . . . . . . . . . . . . 265
9.7.8 A Spin-3/2 Particle . . . . . . . . . . . . . . . . . . . . . 268
9.7.9 Arbitrary directions . . . . . . . . . . . . . . . . . . . . . 272
9.7.10 Spin state probabilities . . . . . . . . . . . . . . . . . . . 276
9.7.11 A spin operator . . . . . . . . . . . . . . . . . . . . . . . . 277
9.7.12 Simultaneous Measurement . . . . . . . . . . . . . . . . . 278
9.7.13 Vector Operator . . . . . . . . . . . . . . . . . . . . . . . 280
9.7.14 Addition of Angular Momentum . . . . . . . . . . . . . . 281
9.7.15 Spin = 1 system . . . . . . . . . . . . . . . . . . . . . . . 282
9.7.16 Deuterium Atom . . . . . . . . . . . . . . . . . . . . . . . 287
9.7.17 Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . 288
9.7.18 Spin in Magnetic Field . . . . . . . . . . . . . . . . . . . . 289
9.7.19 What happens in the Stern-Gerlach box? . . . . . . . . . 296
9.7.20 Spin = 1 particle in a magnetic field . . . . . . . . . . . . 297
9.7.21 Multiple magnetic fields . . . . . . . . . . . . . . . . . . . 298
9.7.22 Neutron interferometer . . . . . . . . . . . . . . . . . . . . 299
9.7.23 Magnetic Resonance . . . . . . . . . . . . . . . . . . . . . 302
9.7.24 More addition of angular momentum . . . . . . . . . . . . 306
9.7.25 Clebsch-Gordan Coefficients . . . . . . . . . . . . . . . . . 308
9.7.26 Spin −1/2 and Density Matrices . . . . . . . . . . . . . . . 309
9.7.27 System of N Spin −1/2 Particle . . . . . . . . . . . . . . . 311
9.7.28 In a coulomb field . . . . . . . . . . . . . . . . . . . . . . 312
9.7.29 Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . 312
9.7.30 What happens? . . . . . . . . . . . . . . . . . . . . . . . . 314
9.7.31 Anisotropic Harmonic Oscillator . . . . . . . . . . . . . . 315
9.7.32 Exponential potential . . . . . . . . . . . . . . . . . . . . 317
9.7.33 Bouncing electrons . . . . . . . . . . . . . . . . . . . . . . 320
9.7.34 Alkali Atoms . . . . . . . . . . . . . . . . . . . . . . . . . 322
9.7.35 Trapped between . . . . . . . . . . . . . . . . . . . . . . . 323
9.7.36 Logarithmic potential . . . . . . . . . . . . . . . . . . . . 324
9.7.37 Spherical well . . . . . . . . . . . . . . . . . . . . . . . . . 325
9.7.38 In magnetic and electric fields . . . . . . . . . . . . . . . . 328
9.7.39 Extra(Hidden) Dimensions . . . . . . . . . . . . . . . . . 329
9.7.40 Spin −1/2 Particle in a D-State . . . . . . . . . . . . . . . 339
9.7.41 Two Stern-Gerlach Boxes . . . . . . . . . . . . . . . . . . 340
9.7.42 A Triple-Slit experiment with Electrons . . . . . . . . . . 341
9.7.43 Cylindrical potential . . . . . . . . . . . . . . . . . . . . . 342
9.7.44 Crazy potentials..... . . . . . . . . . . . . . . . . . . . . . 345
9.7.45 Stern-Gerlach Experiment for a Spin-1 Particle . . . . . . 347
9.7.46 Three Spherical Harmonics . . . . . . . . . . . . . . . . . 348
9.7.47 Spin operators ala Dirac . . . . . . . . . . . . . . . . . . . 350
9.7.48 Another spin = 1 system . . . . . . . . . . . . . . . . . . 351
9.7.49 Properties of an operator . . . . . . . . . . . . . . . . . . 352
9.7.50 Simple Tensor Operators/Operations . . . . . . . . . . . . 354
9.7.51 Rotations and Tensor Operators . . . . . . . . . . . . . . 355
CONTENTS vi
9.7.52 Spin Projection Operators . . . . . . . . . . . . . . . . . . 356
9.7.53 Two Spins in a magnetic Field . . . . . . . . . . . . . . . 357
9.7.54 Hydrogen d States . . . . . . . . . . . . . . . . . . . . . . 359
9.7.55 The Rotation Operator for Spin −1/2 . . . . . . . . . . . . 360
9.7.56 The Spin Singlet . . . . . . . . . . . . . . . . . . . . . . . 362
9.7.57 A One-Dimensional Hydrogen Atom . . . . . . . . . . . . 364
9.7.58 Electron in Hydrogen p −orbital . . . . . . . . . . . . . . . 365
9.7.59 Quadrupole Moment Operators . . . . . . . . . . . . . . . 372
9.7.60 More Clebsch-Gordon Practice . . . . . . . . . . . . . . . 375
9.7.61 Spherical Harmonics Properties . . . . . . . . . . . . . . . 383
9.7.62 Starting Point for Shell Model of Nuclei . . . . . . . . . . 387
9.7.63 The Axial-Symmetric Rotor . . . . . . . . . . . . . . . . . 395
9.7.64 Charged Particle in 2-Dimensions . . . . . . . . . . . . . . 398
9.7.65 Particle on a Circle Again . . . . . . . . . . . . . . . . . . 408
9.7.66 Density Operators Redux . . . . . . . . . . . . . . . . . . 411
9.7.67 Angular Momentum Redux . . . . . . . . . . . . . . . . . 412
9.7.68 Wave Function Normalizability . . . . . . . . . . . . . . . 415
9.7.69 Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
9.7.70 Pauli Matrices and the Bloch Vector . . . . . . . . . . . . 417
10 Time-Independent Perturbation Theory 419
10.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
10.9.1 Box with a Sagging Bottom . . . . . . . . . . . . . . . . . 419
10.9.2 Perturbing the Infinite Square Well . . . . . . . . . . . . . 420
10.9.3 Weird Perturbation of an Oscillator . . . . . . . . . . . . 421
10.9.4 Perturbing the Infinite Square Well Again . . . . . . . . . 423
10.9.5 Perturbing the 2-dimensional Infinite Square Well . . . . 424
10.9.6 Not So Simple Pendulum . . . . . . . . . . . . . . . . . . 426 10.9.7 1-Dimensional Anharmonic Oscillator . . . . . . . . . . . 427
10.9.8 A Relativistic Correction for Harmonic Oscillator . . . . . 429
10.9.9 Degenerate perturbation theory on a spin = 1 system . . 430
10.9.10 Perturbation Theory in Two-Dimensional Hilbert Space . 431
10.9.11 Finite Spatial Extent of the Nucleus . . . . . . . . . . . . 435
10.9.12 Spin-Oscillator Coupling . . . . . . . . . . . . . . . . . . . 438
10.9.13 Motion in spin-dependent traps . . . . . . . . . . . . . . . 440
10.9.14 Perturbed Oscillator . . . . . . . . . . . . . . . . . . . . . 443
10.9.15 Another Perturbed Oscillator . . . . . . . . . . . . . . . . 444
10.9.16 Helium from Hydrogen - 2 Methods . . . . . . . . . . . . 446
10.9.17 Hydrogen atom + xy perturbation . . . . . . . . . . . . . 449
10.9.18 Rigid rotator in a magnetic field . . . . . . . . . . . . . . 451
10.9.19 Another rigid rotator in an electric field . . . . . . . . . . 453
10.9.20 A Perturbation with 2 Spins . . . . . . . . . . . . . . . . 454
10.9.21 Another Perturbation with 2 Spins . . . . . . . . . . . . . 456
10.9.22 Spherical cavity with electric and magnetic fields . . . . . 458
10.9.23 Hydrogen in electric and magnetic fields . . . . . . . . . . 461 CONTENTS vii
10.9.24 n = 3 Stark effect in Hydrogen . . . . . . . . . . . . . . . 463
10.9.25 Perturbation of the n = 3 level in Hydrogen - Spin-Orbit and Magnetic Field corrections . . . . . . . . . . . . . . . 466
10.9.26 Stark Shift in Hydrogen with Fine Structure . . . . . . . 477 10.9.27 2-Particle Ground State Energy . . . . . . . . . . . . . . . 482 10.9.28 1s2s Helium Energies . . . . . . . . . . . . . . . . . . . . . 484
10.9.29 Hyperfine Interaction in the Hydrogen Atom . . . . . . . 485
10.9.30 Dipole Matrix Elements . . . . . . . . . . . . . . . . . . . 487
10.9.31 Variational Method 1 . . . . . . . . . . . . . . . . . . . . 489
10.9.32 Variational Method 2 . . . . . . . . . . . . . . . . . . . . 494
10.9.33 Variational Method 3 . . . . . . . . . . . . . . . . . . . . 495
10.9.34 Variational Method 4 . . . . . . . . . . . . . . . . . . . . 496
10.9.35 Variation on a linear potential . . . . . . . . . . . . . . . 497
10.9.36 Average Perturbation is Zero . . . . . . . . . . . . . . . . 499 10.9.37 3-dimensional oscillator and spin interaction . . . . . . . . 500
10.9.38 Interacting with the Surface of Liquid Helium . . . . . . . 501
10.9.39 Positronium + Hyperfine Interaction . . . . . . . . . . . . 502
10.9.40 Two coupled spins . . . . . . . . . . . . . . . . . . . . . . 504
10.9.41 Perturbed Linear Potential . . . . . . . . . . . . . . . . . 508
10.9.42 The ac-Stark Effect . . . . . . . . . . . . . . . . . . . . . 509
10.9.43 Light-shift for multilevel atoms . . . . . . . . . . . . . . . 516
10.9.44 A Variational Calculation . . . . . . . . . . . . . . . . . . 525
10.9.45 Hyperfine Interaction Redux . . . . . . . . . . . . . . . . 526
10.9.46 Find a Variational Trial Function . . . . . . . . . . . . . . 528
10.9.47 Hydrogen Corrections on 2s and 2p Levels . . . . . . . . . 535
10.9.48 Hyperfine Interaction Again . . . . . . . . . . . . . . . . . 539
10.9.49 A Perturbation Example . . . . . . . . . . . . . . . . . . . 542
10.9.50 More Perturbation Practice . . . . . . . . . . . . . . . . . 544
11 Time-Dependent Perturbation Theory 547
11.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547
11.5.1 Square Well Perturbed by an Electric Field . . . . . . . . 547 11.5.2 3-Dimensional Oscillator in an electric field . . . . . . . . 549
11.5.3 Hydrogen in decaying potential . . . . . . . . . . . . . . . 550 11.5.4 2 spins in a time-dependent potential . . . . . . . . . . . 551
11.5.5 A Variational Calculation of the Deuteron Ground State Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553
11.5.6 Sudden Change - Don’t Sneeze . . . . . . . . . . . . . . . 556
11.5.7 Another Sudden Change - Cutting the spring . . . . . . . 557
11.5.8 Another perturbed oscillator . . . . . . . . . . . . . . . . 558
11.5.9 Nuclear Decay . . . . . . . . . . . . . . . . . . . . . . . . 559
11.5.10 Time Evolution Operator . . . . . . . . . . . . . . . . . . 562
11.5.11 Two-Level System . . . . . . . . . . . . . . . . . . . . . . 562
11.5.12 Instantaneous Force . . . . . . . . . . . . . . . . . . . . . 563
11.5.13 Hydrogen beam between parallel plates . . . . . . . . . . 564
CONTENTS viii
11.5.14 Particle in a Delta Function and an Electric Field . . . . 565
11.5.15 Nasty time-dependent potential [complex integration needed]569
11.5.16 Natural Lifetime of Hydrogen . . . . . . . . . . . . . . . . 570
11.5.17 Oscillator in electric field . . . . . . . . . . . . . . . . . . 573
11.5.18 Spin Dependent Transitions . . . . . . . . . . . . . . . . . 574
11.5.19 The Driven Harmonic Oscillator . . . . . . . . . . . . . . 579
11.5.20 A Novel One-Dimensional Well . . . . . . . . . . . . . . . 581
11.5.21 The Sudden Approximation . . . . . . . . . . . . . . . . . 582
11.5.22 The Rabi Formula . . . . . . . . . . . . . . . . . . . . . . 584
11.5.23 Rabi Frequencies in Cavity QED . . . . . . . . . . . . . . 585
12 Identical Particles 589
12.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589
12.9.1 Two Bosons in a Well . . . . . . . . . . . . . . . . . . . . 589
12.9.2 Two Fermions in a Well . . . . . . . . . . . . . . . . . . . 590
12.9.3 Two spin −1/2 particles . . . . . . . . . . . . . . . . . . . 592
12.9.4 Hydrogen Atom Calculations . . . . . . . . . . . . . . . . 595
12.9.5 Hund’s rule . . . . . . . . . . . . . . . . . . . . . . . . . . 599
12.9.6 Russell-Saunders Coupling in Multielectron Atoms . . . . 600
12.9.7 Magnetic moments of proton and neutron . . . . . . . . . 603
12.9.8 Particles in a 3-D harmonic potential . . . . . . . . . . . . 605 12.9.9 2 interacting particles . . . . . . . . . . . . . . . . . . . . 608
12.9.10 LS versus JJ coupling . . . . . . . . . . . . . . . . . . . . 610
12.9.11 In a harmonic potential . . . . . . . . . . . . . . . . . . . 612 12.9.12 2 particles interacting via delta function . . . . . . . . . . 614 12.9.13 2 particles in a square well . . . . . . . . . . . . . . . . . 616 12.9.14 2 particles interacting via a harmonic potential . . . . . . 617
12.9.15 The Structure of helium . . . . . . . . . . . . . . . . . . . 619
13 Scattering Theory and Molecular Physics 623
13.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623
13.3.1 S-Wave Phase Shift . . . . . . . . . . . . . . . . . . . . . 623
13.3.2 Scattering Slow Particles . . . . . . . . . . . . . . . . . . 625
13.3.3 Inverse square scattering . . . . . . . . . . . . . . . . . . . 626
13.3.4 Ramsauer-Townsend Effect . . . . . . . . . . . . . . . . . 629
13.3.5 Scattering from a dipole . . . . . . . . . . . . . . . . . . . 630
13.3.6 Born Approximation Again . . . . . . . . . . . . . . . . . 631
13.3.7 Translation invariant potential scattering . . . . . . . . . 632 13.3.8 ℓ = 1 hard sphere scattering . . . . . . . . . . . . . . . . . 632
13.3.9 Vibrational Energies in a Diatomic Molecule . . . . . . . 634
13.3.10 Ammonia Molecule . . . . . . . . . . . . . . . . . . . . . . 635
13.3.11 Ammonia molecule Redux . . . . . . . . . . . . . . . . . . 637
13.3.12 Molecular Hamiltonian . . . . . . . . . . . . . . . . . . . . 638
13.3.13 Potential Scattering from a 3D Potential Well . . . . . . . 640
13.3.14 Scattering Electrons on Hydrogen . . . . . . . . . . . . . 645 CONTENTS ix
13.3.15 Green’s Function . . . . . . . . . . . . . . . . . . . . . . . 646
13.3.16 Scattering from a Hard Sphere . . . . . . . . . . . . . . . 649
13.3.17 Scattering from a Potential Well . . . . . . . . . . . . . . 650
13.3.18 Scattering from a Yukawa Potential . . . . . . . . . . . . 653
13.3.19 Born approximation - Spin-Dependent Potential . . . . . 654
13.3.20 Born approximation - Atomic Potential . . . . . . . . . . 656
13.3.21 Lennard-Jones Potential . . . . . . . . . . . . . . . . . . . 657
13.3.22 Covalent Bonds - Diatomic Hydrogen . . . . . . . . . . . 661
13.3.23 Nucleus as sphere of charge - Scattering . . . . . . . . . . 663
15 States and Measurement 667
15.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667
15.6.1 Measurements in a Stern-Gerlach Apparatus . . . . . . . 667
15.6.2 Measurement in 2-Particle State . . . . . . . . . . . . . . 670
15.6.3 Measurements on a 2 Spin-1/2 System . . . . . . . . . . . 671
15.6.4 Measurement of a Spin-1/2 Particle . . . . . . . . . . . . 673
15.6.5 Mixed States vs. Pure States and Interference . . . . . . . 677
15.6.6 Which-path information, Entanglement, and Decoherence 680
15.6.7 Livio and Oivil . . . . . . . . . . . . . . . . . . . . . . . . 684
15.6.8 Measurements on Qubits . . . . . . . . . . . . . . . . . . 688
15.6.9 To be entangled.... . . . . . . . . . . . . . . . . . . . . . . 690
15.6.10 Alice, Bob and Charlie . . . . . . . . . . . . . . . . . . . . 691
16 The EPR Argument and Bell Inequality 695
16.10Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695
16.10.1 Bell Inequality with Stern-Gerlach . . . . . . . . . . . . . 695
16.10.2 Bell’s Theorem with Photons . . . . . . . . . . . . . . . . 699
16.10.3 Bell’s Theorem with Neutrons . . . . . . . . . . . . . . . . 704
16.10.4 Greenberger-Horne-Zeilinger State . . . . . . . . . . . . . 706
17 Path Integral Methods 711
17.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711
17.7.1 Path integral for a charged particle moving on a plane in the presence of a perpendicular magnetic field . . . . . . . 711
17.7.2 Path integral for the three-dimensional harmonic oscillator 712
17.7.3 Transitions in the forced one-dimensional oscillator . . . . 712
17.7.4 Green’s Function for a Free Particle . . . . . . . . . . . . 713
17.7.5 Propagator for a Free Particle . . . . . . . . . . . . . . . . 713
18 715
Solid State Physics
18.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715
18.7.1 Piecewise Constant Potential Energy One Atom per Primitive Cell . . . . . . . . . . . . . . . . 715
18.7.2 Piecewise Constant Potential Energy Two Atoms per Primitive Cell . . . . . . . . . . . . . . . 715
CONTENTS x
18.7.3 Free-Electron Energy Bands for a Crystal with a Primitive Rectangular Bravais Lattice . . . . . . . . . . . . . . . . . 716
18.7.4 Weak-Binding Energy Bands for a Crystal with a Hexag- onal Bravais Lattice . . . . . . . . . . . . . . . . . . . . . 717
18.7.5 A Weak-Binding Calculation #1 . . . . . . . . . . . . . . 718
18.7.6 Weak-Binding Calculations with Delta-Function Potential Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719
19 721
Second Quantization
19.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721
19.9.1 Bogoliubov Transformations . . . . . . . . . . . . . . . . . 721
19.9.2 Weakly Interacting Bose gas in the Bogoliubov Approxi- mation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721
19.9.3 Problem 19.9.2 Continued . . . . . . . . . . . . . . . . . . 722
19.9.4 Mean-Field Theory, Coherent States and the Grtoss-Pitaevkii Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 723
19.9.5 Weakly Interacting Bose Gas . . . . . . . . . . . . . . . . 724
19.9.6 Bose Coulomb Gas . . . . . . . . . . . . . . . . . . . . . . 724
19.9.7 Pairing Theory of Superconductivity . . . . . . . . . . . . 724
19.9.8 Second Quantization Stuff . . . . . . . . . . . . . . . . . . 725
19.9.9 Second Quantized Operators . . . . . . . . . . . . . . . . 727
19.9.10 Working out the details in Section 19.8 . . . . . . . . . . 727
20 Relativistic Wave Equations
729
Electromagnetic Radiation in Matter
20.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 729
20.8.1 Dirac Spinors . . . . . . . . . . . . . . . . . . . . . . . . . 729
20.8.2 Lorentz Transformations . . . . . . . . . . . . . . . . . . . 729
20.8.3 Dirac Equation in 1 + 1 Dimensions . . . . . . . . . . . . 730
20.8.4 Trace Identities . . . . . . . . . . . . . . . . . . . . . . . . 730
20.8.5 Right- and Left-Handed Dirac Particles . . . . . . . . . . 730
20.8.6 Gyromagnetic Ratio for the Electron . . . . . . . . . . . . 731
20.8.7 Dirac → Schrodinger . . . . . . . . . . . . . . . . . . . . . 731
20.8.8 Positive and Negative Energy Solutions . . . . . . . . . . 731
20.8.9 Helicity Operator . . . . . . . . . . . . . . . . . . . . . . . 732
20.8.10 Non-Relativisitic Limit . . . . . . . . . . . . . . . . . . . . 732
20.8.11 Gyromagnetic Ratio . . . . . . . . . . . . . . . . . . . . . 732
20.8.12 Properties of γ
5 . . . . . . . . . . . . . . . . . . . . . . . . 732
20.8.13 Lorentz and Parity Properties . . . . . . . . . . . . . . . . 732
20.8.14 A Commutator . . . . . . . . . . . . . . . . . . . . . . . . 733
20.8.15 Solutions of the Klein-Gordon equation . . . . . . . . . . 733
20.8.16 Matrix Representation of Dirac Matrices . . . . . . . . . . 733
20.8.17 Weyl Representation . . . . . . . . . . . . . . . . . . . . . 734
20.8.18 Total Angular Momentum . . . . . . . . . . . . . . . . . . 734
20.8.19 Dirac Free Particle . . . . . . . . . . . . . . . . . . . . . . 735
Chapter 3 Formulation of Wave Mechanics - Part 2
3.11 Solutions
3.11.1 Free Particle in One-Dimension - Wave Functions
(a) Z
∞ px
1
i ~
˜ ψ(p, 0) = e ψ(x, 0) dx
√ 2π~
−∞
Z 2
∞ (p (x −x0)
N −p0)x
i
~ −
4σ2= e e dx √
2π~
−∞ x2
Z (p x2 xx0
∞ −p0)x
4σ2 2σ2 4σ2
+i − − ~N
- = e
√ 2π~
−∞ x2
Z (p
∞ x2 x0 −p0)
x+− − −i ~
N 4σ2 2σ2 4σ2 = e
√ 2π~ −∞ x2 (p ∞ (p 2 x0 x x0 2 Z 2
−p0) −p0)
N σ 4σ2 2σ2 2σ2 −i ~ − −σ −i ~ 2σ = e e e
√ 2π~ x2 2 x0 (p ∞ 2 Z −∞
−p0) 4σ2 2 N σ 2σ2 −y −i ~
= e e 2σ e dy √
2π~ x2 2 x0
(p
2 −∞−p0)
N
σ √ 4σ2 2σ2
−i ~
= e e 2σ π √
2π~ √ (p 2 2 (p
−p0)
N σ 2 −p0)x0 ~2 σ = e e
√ ~
(b) Find ˜ ψ(p, t) We have
2
∂ ˜ ψ(p, t) p ˜ ˜ i~ = H ˜ ψ(p, t) = E ψ(p, t) = ψ(p, t)
p
∂t 2m since ˜ ψ(p, t) is an eigenfunction of H (only 1 p value). This implies that p t/~ t/2m~ 2
−iE −ip
˜ ψ(p, t) = ˜ ψ(p, 0)e = ˜ ψ(p, 0)e
(c) Find ψ(x, t) Z
∞
1
ipx/~
˜ ψ(x, t) = e ψ(p, t)dp
√ 2π~
−∞
√ Z
∞
(p
2 2 (p 2 −p0) −p0)x01 N σ
2
ipx/~ σ t/2m~ − −i −ip ~2 ~
= e e e e dp √ √
~ 2π~
−∞
Z
∞ 2 2 2 2 2 2 2 2 2 N σ ipx/~ /~ ip x /~ t/2m~ σ /~ 2pp σ /~ p σ /~ −ipx −ip −p
= e e e e e e e dp √
π~
−∞
Z 2 2 2 ∞
2
2 2 2 2 N σ ip x /~ p σ /~ p (σ /~ /~+2p σ /~ )- it/2m~)−p(ix/~−ix
= e e e dp √
π~
−∞
Z 2 2 2 ∞ 2 2 N σ ip x /~ p σ /~ β
−(αp−β)
= e e e e dp √
π~
−∞
where
2
σ it x x 2p
2
2
- α = and i σ
- 2
− 2αβ = i
2
~ ~ − ~ ~ 2m~
Therefore, letting y = αp − β and dy = αdp we get
Z 2 2
2
2 ∞ 2 N σ1
ip x /~ p σ /~ β −y
ψ(x, t) = e e e e dy √
π~ α
−∞ (x 2 /~)2 −x0−2ip0σ 2 2 2 − ~
1 4 σ2 + it
ip x /~ p σ /~
( 2m ) = N σe e e q
it~
2
σ +
2m
where we have used Z
∞ 2
√
−y
e dy = π
−∞
(d) Show that the spread in the spatial probability distribution
2
|ψ(x, t)| ℘(x, t) = hψ(t) | ψ(t)i increases with time.
The important terms are
1 q → amplitude decreases with time
it~
2
σ +
2m
and (x 2 /~)2
−x0−2ip0σ − 4 σ2 + it ~
( 2m ) e → width(spread) increases with time
A sequence of pictures below illustrates what is happening:
3.11.2 Free Particle in One-Dimension - Expectation Val-
uesFor a free particle in one-dimension
2
p H =
2m (a) Show x x hp i = hp i
t=0
d
1
x
hp i = x , H] x t = x h[p i = 0 → hp i hp i dt i~ h i
hp x i t=0
(b) Show t + hxi = hxi t=0
m
d
1
1
1
1
2
]
x x
hxi = h[x, H]i = h[x, p x i = hp i = hp i dt i~ 2im~ m m Therefore
1
t = x t +
hxi hp i hxi m
2
2
(c) Show (∆p ) = (∆p )
x x t=0
d
1
2
2
2
2
, H] =
t
hp x i = h[p x i = 0 → hp x i hp x i dt i~ Therefore
2
2
2
2
2
2
(∆p ) = = = (∆p )
x t x x x
t hp x i − hp i t hp x i − hp i
2
(d) Find (∆x) as a function of time and initial conditions. HINT: Find d
2
x dt To solve the resulting differential equation, one needs to know the time dependence of x + p x x hxp i. Find this by considering d
- p x
x x
hxp i dt We have
2
d d d
2
2
(∆x) = hx i + hxi dt dt dt
Now d
1
1
2
2
2
2
, H] , p ] hx i = h[x i = h[x x i dt i~ 2im~
1
1
2
2
2
2
2
2
= p x x + i~)p x
x x
hx x − p x i = hx(p − p x i 2im~ 2im~
1
2
2
= xp + i~xp x
x x x
hxp − p x i 2im~
1
2
2
= x + i~)(p x + i~) + i~xp x
x x x
h(p − p x i 2im~
1
2
2
2
= xp x + 2i~p x + i~xp x
x x x x
hp − ~ − p x i 2im~
1
2
2
2
= (p x + i~)x + 2i~p x + i~xp x
x x x x
hp − ~ − p x i 2im~
1
2
= x + i~xp
x x
h3i~p − ~ i 2im~
1
2
= x + i~(xp + i~xp
x x x
h2i~p − i~) − ~ i 2im~
1 = x + xp
x x
hp i m We then get d
1
1
2
2
x + xp x + xp ), H] x, p ] + [xp , p ]
x x x x x x
hp i = h[(p i = h[p x x i dt i~ 2im~
1
2
3
3
2
= xp x + xp xp
x x
hp x − p x x − p x i 2im~
1
3
3
= (p x + i~)p x + xp (xp
x x x x x x
hp − p x x − p − i~)p i 2im~
1
2
2
3
3
2
2
= xp + i~p x + xp xp + i~p
x x
hp x x − p x x − p x x i 2im~
1
2
2
3
3
2
= (p x + i~) + 2i~p x + xp
x x
hp x x − p x x − (xp − i~)p x i 2im~
1
2
2
2
2
2
= =
t
h4i~p x i = hp x i hp x i 2im~ m m where we have used the result from part (c). Therefore,
2
2
x x + xp x t = t + x x + xp x
hp i hp x i hp i m and we get d
2
1
2
2
t + x + xp
x x
hx i = hp x i hp i
2
dt m m
- 1 m hp
i − d dt
=
2 m
2