Visual Quantum Mechanics





Table of Contents (Book Two): Sections and Subsections

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  • Cover Text
  • Preface
  • Contents
  • Chapter 1. Spherical Symmetry 1
    • 1.1. A Note on Symmetry Transformations 2
      • 1.1.1. Rotations as symmetry transformations 2
      • 1.1.2. Symmetry transformations in quantum mechanics 4
      • 1.1.3. Realizations of symmetry transformations 5
      • 1.1.4. Invariance of a physical system 6
    • 1.2. Rotations in Quantum Mechanics 7
      • 1.2.1. Rotation of vectors in R3 7
      • 1.2.2. Rotation of wave functions 10
    • 1.3. Angular Momentum 12
      • 1.3.1. Angular momentum in classical mechanics 12
      • 1.3.2. Angular momentum in quantum mechanics 12
      • 1.3.3. Commutation relations of the angular-momentum operators 14
      • 1.3.4. The meaning of the angular-momentum commutation relations 15
    • 1.4. Spherical Symmetry of a Quantum System 17
      • 1.4.1. Conservation of angular momentum 17
      • 1.4.2. Spherically symmetric potentials 18
      • 1.4.3. Symmetry and degeneracy 19
    • 1.5. The Possible Eigenvalues of Angular-Momentum Operators 21
      • 1.6. Spherical Harmonics 26
      • 1.6.1. Spherical coordinates 26
      • 1.6.2. Angular momentum in spherical coordinates 28
      • 1.6.3. Special topic: Properties of the spherical harmonics 32
    • 1.7. Particle on a Sphere 34
      • 1.7.1. Classical particle on a sphere 34
      • 1.7.2. The rigid rotator 35
      • 1.7.3. Transition to quantum mechanics 35
      • 1.7.4. Dynamics of the rigid rotator 36
    • 1.8. Quantization on a Sphere 38
      • 1.8.1. Comparison of classical and quantum probability densities 38
      • 1.8.2. Special topic: Curvilinear coordinates 41
    • 1.9. Free Schrödinger Equation in Spherical Coordinates 44
      • 1.9.1. Solutions of the radial equation 44
      • 1.9.2. Special Topic: Properties of the Riccati-Bessel functions 45
      • 1.9.3. Special Topic: Expanding the plane wave 46
      • 1.9.4. Special topic: Spherical harmonics and the Fourier transformation 48
    • 1.10. Spherically Symmetric Potentials 50
      • 1.10.1. The structure of the eigenvalue spectrum 50
      • 1.10.2. The vibrating rotator: A model of a diatomic molecule 53
  • Chapter 2. Coulomb Problem 57
    • 2.1. Introduction 58
      • 2.1.1. Classical models of atoms 58
      • 2.1.2. Transitions between eigenstates 59
    • 2.2. The Classical Coulomb Problem 61
      • 2.2.1. The Coulomb force 61
      • 2.2.2. Classical motion 63
    • 2.3. Algebraic Solution Using the Runge-Lenz Vector 66
      • 2.3.1. The Coulomb problem in quantum mechanics 66
      • 2.3.2. The Runge-Lenz vector in quantum mechanics 67
      • 2.3.3. The eigenvalues of $H$ 68
    • 2.4. Algebraic Solution of the Radial Schrödinger Equation 70
      • 2.4.1. Factorization of the radial Coulomb problem 70
      • 2.4.2. The ground state of the radial Schrödinger operator 72
      • 2.4.3. Excited states of the radial Schrödinger operator 73
      • 2.4.4. Quantum numbers of the Coulomb problem 76
      • 2.4.5. Time evolution of simple superpositions 82
    • 2.5. Direct Solution of the Radial Schrödinger Equation 84
      • 2.5.1. Solution of the Coulomb problem in two dimensions 84
      • 2.5.2. Solution of the Coulomb problem in three dimensions 88
    • 2.6. Special Topic: Parabolic Coordinates 91
    • 2.7. Physical Units and Dilations 96
      • 2.7.1. The Coulomb problem in physical units 96
      • 2.7.2. Scaling transformations 99
      • 2.7.3. Energies and eigenfunctions in physical units 101
      • 2.7.4. The generator of dilations and the virial theorem 103
    • 2.8. Special Topic: Dynamics of Rydberg States 105
  • Chapter 3. Particles with Spin 113
    • 3.1. Introduction 113
    • 3.2. Classical Theory of the Magnetic Moment 115
      • 3.2.1. Magnetic moment of an extended particle 115
      • 3.2.2. The influence of an external magnetic field on a magnetic moment 116
    • 3.3. The Stern-Gerlach Experiment 118
    • 3.4. The Spin Operators 123
      • 3.4.1. Magnetic moment and spin 123
      • 3.4.2. The $g$-factor 123
      • 3.4.3. Electron in a constant magnetic field 125
      • 3.4.4. Properties of the spin operator 126
    • 3.5. Spinor-Wave Functions 127
      • 3.5.1. A Hilbert space for a spin 1/2 particle 127
      • 3.5.2. Spin operators in the standard representation 129
      • 3.5.3. Changing representations 130
      • 3.5.4. Interpretation of spinor-wave functions 131
      • 3.5.5. Visualization of spinor-wave functions 132
        • 3.5.5.1. Visualization by vectors 132
        • 3.5.5.2. Vizualization by colors 134
    • 3.6. The Pauli Equation 134
      • 3.6.1. The Pauli operator 134
      • 3.6.2. Magnetic fields with constant direction 136
      • 3.6.3. Supersymmetric structure 137
    • 3.7. Solution in a Homogeneous Magnetic Field 138
      • 3.7.1. The $g$-factor of orbital motion 138
      • 3.7.2. Solutions with zero energy 138
      • 3.7.3. The spectrum of eigenvalues 141
    • 3.8. Special Topic: Magnetic Ground States 142
      • 3.8.1. Two dimensions 143
      • 3.8.2. Three dimensions 145
    • 3.9. The Coulomb Problem with Spin 146
      • 3.9.1. Coulomb-Hamiltonian for a particle with spin 146
      • 3.9.2. Complete set of observables 147
      • 3.9.3. The spin-orbit interaction 148
      • 3.9.4. Eigenfunctions of the spin-orbit operator 150
      • 3.9.5. The radial equation 155
  • Chapter 4. Qubits 157
    • 4.1. States and Observables 158
      • 4.1.1. The Hilbert space of a qubit 158
      • 4.1.2. States of a qubit 159
      • 4.1.3. Qubit observables 160
    • 4.2. Measurement and Preparation 162
      • 4.2.1. Stern-Gerlach experiment 162
      • 4.2.2. Projection postulate 164
      • 4.2.3. Stern-Gerlach filter and state preparation 165
    • 4.3. Ensemble Measurements 167
      • 4.3.1. State verification 167
      • 4.3.2. Determining an unknown state 168
      • 4.3.3. Classical teleportation is impossible 170
    • 4.4. Qubit Manipulations 171
      • 4.4.1. All states are "spin-up" in some direction 171
      • 4.4.2. Rotations of a qubit 174
      • 4.4.3. Time evolution of the spin in a magnetic field 176
      • 4.4.4. Special topic: Spinor rotations 177
      • 4.4.5. Transition probabilities between qubit states 179
    • 4.5. Other Qubit Systems 181
      • 4.5.1. Photon polarizations 181
      • 4.5.2. Spatial states of photons 184
      • 4.5.3. Two states of a harmonic oscillator 187
    • 4.6. Single-Particle Interference 189
      • 4.6.1. Interferometer 189
      • 4.6.2. A double-slit experiment 192
      • 4.6.3. A rotation through $2\pi $ 193
      • 4.6.4. Interaction-free measurement 194
    • 4.7. Quantum Cryptography 197
      • 4.7.1. One-time pad 197
      • 4.7.2. Quantum key distribution 198
    • 4.8. Hidden Variables 200
      • 4.8.1. Failure of classical picture 200
      • 4.8.2. Hidden-variable interpretation 201
    • 4.9. Special Topic: Qubit Dynamics 204
      • 4.9.1. Time-dependent Hamiltonian 204
      • 4.9.2. Time dependence generated by unitary operators 205
      • 4.9.3. Magnetic resonance 206
  • Chapter 5. Composite Systems 211
    • 5.1. States of Two-Particle Systems 212
      • 5.1.1. The Hamiltonian for two free particles 212
      • 5.1.2. The Schrödinger equation of a two-particle system 213
      • 5.1.3. Two-particle Hilbert space 214
      • 5.1.4. The interpretation of two-particle wave functions 215
    • 5.2. Hilbert Space of a Bipartite System 216
      • 5.2.1. Construction of the tensor product 216
      • 5.2.2. Orthonormal basis of the tensor product space 217
      • 5.2.3. Entangled states 218
      • 5.2.4. Example: Two-qubit system 219
      • 5.2.5. Example: Particle with spin 220
    • 5.3. Interacting Particles 221
      • 5.3.1. Two-particle interactions 221
      • 5.3.2. Separation of the center-of-mass motion 222
    • 5.4. Observables of a Bipartite System 223
      • 5.4.1. Tensor product of operators 223
      • 5.4.2. Local manipulations 225
    • 5.5. The Density Operator 227
      • 5.5.1. What does entanglement mean for the subsystems? 227
      • 5.5.2. Expectation values of subsystem observables 229
      • 5.5.3. Trace-class operators 230
      • 5.5.4. Density operator of a subsystem 232
    • 5.6. Pure and Mixed States 233
      • 5.6.1. State of a subsystem 233
      • 5.6.2. Canonical form of the density operator 234
      • 5.6.3. Ensemble interpretation of mixed states 236
    • 5.7. Preparation of Mixed States 238
      • 5.7.1. Preparing an ensemble in a mixed state 238
      • 5.7.2. Ambiguity of realizing mixed states 240
      • 5.7.3. Example: Mixed qubit states 241
    • 5.8. More About Bipartite Systems 244
      • 5.8.1. Normal form of the state vector 244
      • 5.8.2. Maximally entangled states 247
      • 5.8.3. Purification 248
      • 5.8.4. Projection postulate for mixed states 249
    • 5.9. Indistinguishable Particles 250
      • 5.9.1. (Anti-)symmetric states of bipartite systems 250
      • 5.9.2. Example: Bosons and fermions 254
      • 5.9.3. Example: Two-qubit system 255
    • 5.10. Special Topic: Multiparticle Systems with Spin 256
    • 5.11. Special Topic: Addition of Angular Momenta 259
      • 5.11.1. Total angular momentum 259
      • 5.11.2. Eigenvalues of J3 260
      • 5.11.3. The quantum numbers of the total angular momentum 261
      • 5.11.4. Clebsch-Gordan coefficients 264
      • 5.11.5. Angular momentum plus spin 1/2 266
  • Chapter 6. Quantum Information Theory 271
    • 6.1. Entangled States of Two-Qubit Systems 272
      • 6.1.1. Bases of entangled states 272
      • 6.1.2. Global aspects of local measurements 273
      • 6.1.3. Determining a Bell state 275
      • 6.1.4. Preparing a Bell state 276
    • 6.2. Local and Nonlocal 278
      • 6.2.1. Nonlocal interaction? 278
      • 6.2.2. Local manipulations 280
    • 6.3. The Einstein-Podolsky-Rosen Paradox 281
    • 6.4. Correlations Arising from Entangled States 285
      • 6.4.1. Joint probabilities in two-qubit systems 285
      • 6.4.2. A protocol for verifying joint probabilities 288
    • 6.5. Bell Inequalities and Local Hidden Variables 290
      • 6.5.1. General setup for measuring correlations 291
      • 6.5.2. Realism and hidden variables 292
      • 6.5.3. Statistical independence 293
      • 6.5.4. Bell's theorem 295
      • 6.5.5. Violation of the Bell inequality 297
    • 6.6. Entanglement-Assisted Communication 300
      • 6.6.1. Dense coding 301
      • 6.6.2. Quantum state teleportation 302
    • 6.7. Quantum Computers 305
    • 6.8. Logic Gates 307
      • 6.8.1. Single-qubit gates: The Hadamard transformation 308
      • 6.8.2. The controlled-not gate 310
      • 6.8.3. Sequences of quantum logic gates 313
      • 6.8.4. An impossible gate 315
    • 6.9. Quantum Algorithms 316
      • 6.9.1. Function evaluation 316
      • 6.9.2. Simple quantum algorithms 317
      • 6.9.3. Further applications of quantum computers 320
        • 6.9.3.1. Grover's algorithm 320
        • 6.9.3.2. Simon's algorithm 321
        • 6.9.3.3. Period finding and the quantum Fourier transform 321
        • 6.9.3.4. Shor's algorithm 322
        • 6.9.3.5. Error correction 322
  • Chapter 7. Relativistic Systems in One Dimension 323
    • 7.1. Introduction 324
    • 7.2. The Free Dirac Equation 325
      • 7.2.1. The relativistic energy-momentum relation 325
      • 7.2.2. The Dirac operator 326
    • 7.3. Dirac Spinors and State Space 327
      • 7.3.1. A Hilbert space for the Dirac equation 327
      • 7.3.2. The standard interpretation 328
      • 7.3.3. The relativistic momentum space 330
      • 7.3.4. A solution of the Dirac equation 331
    • 7.4. Plane Waves and Wave Packets 332
      • 7.4.1. Diagonalization in momentum space 332
      • 7.4.2. Eigenvectors and plane-wave solutions 334
      • 7.4.3. Building wave packets 337
    • 7.5. Subspaces with Positive and Negative Energies 339
    • 7.6. Kinematics of Wave Packets 343
      • 7.6.1. The limiting velocity 343
      • 7.6.2. Negative-energy wave packets 346
    • 7.7. Zitterbewegung 347
      • 7.7.1. The standard position operator 347
      • 7.7.2. Velocity of the standard position 349
      • 7.7.3. The classical velocity operator 350
      • 7.7.4. Time evolution of the standard velocity 351
      • 7.7.5. Time evolution of the standard position 352
      • 7.7.6. Superpositions and Zitterbewegung 353
      • 7.7.7. Superluminal motion? 357
    • 7.8. Special Topic: Energy Representation and Velocity Space 359
      • 7.8.1. Variable substitutions 360
      • 7.8.2. Energy representation 362
      • 7.8.3. Velocity representation 364
    • 7.9. Relativistic Invariance 365
      • 7.9.1. Lorentz transformations 365
      • 7.9.2. Transformation of the Dirac equation 367
      • 7.9.3. Representation of Lorentz boosts 369
      • 7.9.4. Invariance of the free-particle Dirac equation 370
      • 7.9.5. Unitary implementation of Lorentz boosts 371
  • Chapter 8. The Dirac Equation 377
    • 8.1. The Dirac Equation 378
      • 8.1.1. Free Dirac operator in two and three dimensions 378
      • 8.1.2. Properties of the free-particle Dirac operator 380
      • 8.1.3. Electromagnetic fields 381
    • 8.2. Relativistic Covariance 382
      • 8.2.1. Poincaré transformations 382
      • 8.2.2. Poincaré covariance of the Dirac equation 384
      • 8.2.3. Velocity transformations 386
      • 8.2.4. Rotations 387
      • 8.2.5. Unitary representation 388
    • 8.3. Classification of External Fields 389
      • 8.3.1. Poincaré transformations of external fields 389
      • 8.3.2. Electromagnetic vector potential 390
      • 8.3.3. Scalar potential 391
      • 8.3.4. Anomalous magnetic moment 391
    • 8.4. Positive and Negative Energies 392
      • 8.4.1. Negative-energy wave packets are positrons 392
      • 8.4.2. Relativistic scattering in one dimension: The Klein paradox 395
      • 8.4.3. Physical Hilbert space and relativistic observables 398
      • 8.4.4. Relativistic confinement 399
    • 8.5. Nonrelativistic Limit and Relativistic Corrections 402
      • 8.5.1. The nonrelativistic limit 402
      • 8.5.2. Relativistic corrections 404
      • 8.5.3. Spin-orbit interaction and the Darwin term 407
    • 8.6. Spherical Symmetry 410
      • 8.6.1. Matrix potentials with spherical symmetry 410
      • 8.6.2. Operators that commute with the Dirac operator 411
      • 8.6.3. Angular-momentum eigenfunctions 413
      • 8.6.4. The angular-momentum subspaces 414
    • 8.7. The Dirac-Coulomb Problem 417
      • 8.7.1. The radial Dirac-Coulomb equation 417
      • 8.7.2. A useful similarity transformation 418
      • 8.7.3. The second-order equations 419
      • 8.7.4. Supersymmetry 420
      • 8.7.5. The ground state 422
      • 8.7.6. The first excited state 424
      • 8.7.7. Further eigenfunctions 425
    • 8.8. Relativistic Hydrogen Atom 426
      • 8.8.1. Eigenvalues and eigenfunctions 426
      • 8.8.2. Degeneracy and higher symmetry 428
      • 8.8.3. Angular momentum and spectroscopical notation 428
      • 8.8.4. Fall to the center 430
  • Appendix A. Synopsis of Quantum Mechanics 433
    • A.1. The Hilbert Space of Quantum Mechanics 433
    • A.2. States of a Physical System 434
    • A.3. Observables 435
    • A.4. Interpretation Rule 436
    • A.5. Projections and Properties 436
    • A.6. Time Evolution 437
    • A.7. Measurements 439
    • A.8. Dirac's Formalism 439
  • Appendix B. Perturbation of Eigenvalues 443
    • B.1. Introduction 443
    • B.2. Rayleigh-Schrödinger series 446
    • B.3. Degenerate eigenvalues 449
    • B.4. Alkali atoms 451
    • B.5. Ground state of helium 452
  • Appendix C. Special Topic: Analytic Perturbation Theory 455
    • C.1. Relative boundedness 455
    • C.2. The resolvent 456
    • C.3. Analytic dependence on the perturbation parameter 458
  • Appendix D. Variational Method 461
    • D.1. Critical points of the energy functional 461
    • D.2. Semibounded Hamiltonians and the minimal energy 462
    • D.3. The ground state of helium (again) 464
    • D.4. Finding excited states by variational methods 465
    • D.5. The minimax principle and the Rayleigh-Ritz technique 466
  • Appendix E. Adiabatic and Geometric Phases 467
    • E.1. The adiabatic approximation 467
    • E.2. The Berry phase 469
    • E.3. Example: Spin in magnetic field 471
  • Appendix F. Formal Scattering Theory 475
    • F.1. Bound states and scattering states 475
    • F.2. Asymptotic completeness 476
    • F.3. Wave operators 477
    • F.4. The scattering operator 479
    • F.5. Properties of wave and scattering operators 480
    • F.6. Scattering operator in the energy representation 482
    • F.7. Stationary Scattering Theory 483
    • F.8. Scattering amplitude 485
    • F.9. Scattering cross section 486
  • Appendix G. Books 491
  • Appendix H. Movie Index 493
    • 0. Introduction 493
    • 1. Spherical Symmetry 494
    • 2. Coulomb Problem 495
    • 3. Spin 498
    • 4. Qubits 499
    • 5. Composite Systems 500
    • 6. Relativistic Systems 501
  • List of Symbols 505
  • Index 511