Concepts in Quantum Mechanics

Cover......Page 1
Title Page......Page 4
Copyright......Page 5
Contents......Page 8
Preface......Page 14
Acknowledgments......Page 16
1.1.1 Planck's Formula for Energy Distribution in Black-body Radiation......Page 18
1.1.2 de Broglie Relation and Wave Nature of Material Particles......Page 19
1.1.3 The Photo-electric Effect......Page 20
1.1.4 The Compton Effect......Page 21
1.1.5 Ritz Combination Principle......Page 23
1.2.1 Principle of Superposition of States......Page 26
1.2.2 Heisenberg Uncertainty Relations......Page 29
1.3 Representation of States......Page 31
1.5 Linear Operators......Page 32
1.6 Adjoint of a Linear Operator......Page 33
1.7 Eigenvalues and Eigenvectors of a Linear Operator......Page 35
1.8.1 Physical Interpretation of Eigenstates and Eigenvalues......Page 37
1.9 Observables and Completeness Criterion......Page 38
1.10 Commutativity and Compatibility of Observables......Page 40
1.11 Position and Momentum Commutation Relations......Page 41
1.12 Commutation Relation and the Uncertainty Product......Page 43
1A1.1 Lagrange Equations of Motion......Page 48
1A1.2 Classical Dynamical Variables......Page 49
References......Page 50
2.2 How to Set up a Representation......Page 52
2.3 Representatives of a Linear Operator......Page 54
2.4 Change of Representation......Page 57
2.5 Coordinate Representation......Page 60
2.5.1 Physical Interpretation of the Wave Function......Page 61
2.6 Replacement of Momentum Observable p by (omitted)......Page 62
2.7 Integral Representation of Dirac Bracket (A2| F |A1)......Page 67
2.8.1 Physical Interpretation of Φ(P1,P2,...Pf)......Page 69
2.9 Dirac Delta Function......Page 70
2.9.1 Three-dimensional Delta Function......Page 72
2.10 Relation between the Coordinate and Momentum Representations......Page 73
3.1 Schrödinger Equation of Motion......Page 84
3.2 Schrödinger Equation in the Coordinate Representation......Page 86
3.3 Equation of Continuity......Page 87
3.4 Stationary States......Page 88
3.5 Time-independent Schrödinger Equation in the Coordinate Representation......Page 89
3.6 Time-independent Schrödinger Equation in the Momentum Representation......Page 91
3.6.1 Two-body Bound State Problem (in Momentum Representation) for Non-local Separable Potential......Page 93
3.7 Time-independent Schrödinger Equation in Matrix Form......Page 94
3.8 The Heisenberg Picture......Page 96
3.9 The Interaction Picture......Page 98
3A1.1 Characteristic Equation of a Matrix......Page 103
3A1.3 Diagonalization of a Matrix......Page 104
4 PROBLEMS OF ONE-DIMENSIONAL POTENTIAL BARRIERS......Page 106
4.1 Motion of a Particle across a Potential Step......Page 107
4.2 Passage of a Particle through a Potential Barrier of Finite Extent......Page 111
4.3 Tunneling of a Particle through a Potential Barrier......Page 116
4.4 Bound States in a One-dimensional Square Potential Well......Page 120
4.5 Motion of a Particle in a Periodic Potential......Page 124
5.2 Motion of a Particle in a Box......Page 132
5.2.1 Density of States......Page 134
5.3 Simple Harmonic Oscillator......Page 135
5.4 Operator Formulation of the Simple Harmonic Oscillator Problem......Page 139
5.4.1 Physical Meaning of the Operators â and â†......Page 140
5.4.2 Occupation Number Representation (ONR)......Page 142
5.5 Bound State of a Two-particle System with Central Interaction......Page 143
5.6 Bound States of Hydrogen (or Hydrogen-like) Atoms......Page 148
5.7 The Deuteron Problem......Page 154
5.8 Energy Levels in a Three-dimensional Square Well: General Case......Page 161
5.9 Energy Levels in an Isotropic Harmonic Potential Well......Page 164
5A1.1 Legendre and Associated Legendre Equations......Page 173
5A1.2 Spherical Harmonics......Page 176
5A1.3 Laguerre and Associated Laguerre Equations......Page 179
5A1.4 Hermite Equation......Page 183
5A1.5 Bessel Equation......Page 186
5A2.1 Spherical Polar Coordinates......Page 191
5A2.2 Cylindrical Coordinates......Page 192
5A2.3 Parabolic Coordinates......Page 194
5A2.4 General Features of Orthogonal Curvilinear System of Coordinates......Page 195
6.1 Symmetries and Their Group Properties......Page 198
6.2 Symmetries in a Quantum Mechanical System......Page 199
6.3 Basic Symmetry Groups of the Hamiltonian and Conservation Laws......Page 200
6.3.1 Space Translation Symmetry......Page 201
6.3.3 Spatial Rotation Symmetry......Page 202
6.4 Lie Groups and Their Generators......Page 205
6.5.1 Proper Rotation Group R(3) (or Special Orthogonal Group SO(3))......Page 208
6.5.2 The SU(2) Group......Page 210
6.5.3 Isospin and SU(2) Symmetry......Page 211
Appendix 6A1: Groups and Representations......Page 216
7.1 Introduction......Page 220
7.2 Raising and Lowering Operators......Page 223
7.3 Matrix Representation of Angular Momentum Operators......Page 225
7.4 Matrix Representation of Eigenstates of Angular Momentum......Page 226
7.5 Coordinate Representation of Angular Momentum Operators and States......Page 229
7.6 General Rotation Group and Rotation Matrices......Page 231
7.6.1 Rotation Matrices......Page 234
7.7 Coupling of Two Angular Momenta......Page 235
7.8 Properties of Clebsch-Gordan Coefficients......Page 236
7.8.2 Projection Theorem for Vector Operators......Page 238
7.9 Coupling of Three Angular Momenta......Page 244
7.10 Coupling of Four Angular Momenta (L – S and j – j Coupling)......Page 245
8.1 Introduction......Page 252
8.2 Non-degenerate Time-independent Perturbation Theory......Page 253
8.3 Time-independent Degenerate Perturbation Theory......Page 259
8.4 The Zeeman Effect......Page 266
8.5 WKBJ Approximation......Page 271
8.6 Particle in a Potential Well......Page 279
8.7 Application of WKBJ Approximation to α decay......Page 281
8.8 The Variational Method......Page 284
8.9 The Problem of the Hydrogen Molecule......Page 287
8.10 System of n Identical Particles: Symmetric and Anti-symmetric States......Page 291
8.11 Excited States of the Helium Atom......Page 295
8.12 Statistical (Thomas-Fermi) Model of the Atom......Page 297
8.13 Hartree's Self-consistent Field Method for Multi-electron Atoms......Page 298
8.14 Hartree-Fock Equations......Page 302
8.15 Occupation Number Representation......Page 307
9.1 Introduction......Page 316
9.2 Laboratory and Center-of-mass (CM) Reference Frames......Page 317
9.2.1 Cross-sections in the CM and Laboratory Frames......Page 319
9.3 Scattering Equation and the Scattering Amplitude......Page 320
9.4 Partial Waves and Phase Shifts......Page 323
9.5 Calculation of Phase Shift......Page 328
9.6 Phase Shifts for Some Simple Potential Forms......Page 330
9.7 Scattering due to Coulomb Potential......Page 337
9.8 The Integral Form of Scattering Equation......Page 341
9.8.1 Scattering Amplitude......Page 344
9.9 Lippmann-Schwinger Equation and the Transition Operator......Page 346
9.10.1 Born Approximation......Page 349
9.10.2 Validity of Born Approximation......Page 351
9.10.3 Born Approximation and the Method of Partial Waves......Page 354
Appendix 9A1: Calculus of Residues......Page 359
10.1 Introduction......Page 368
10.2 Perturbation Constant over an Interval of Time......Page 370
10.3 Harmonic Perturbation: Semi-classical Theory of Radiation......Page 375
10.4 Einstein Coeffcients......Page 380
10.5 Multipole Transitions......Page 382
10.6 Electric Dipole Transitions in Atoms and Selection Rules......Page 383
10.7 Photo-electric Effect......Page 385
10.8 Sudden and Adiabatic Approximations......Page 386
10.9 Second Order Effects......Page 390
11.2 Eyges Approach......Page 394
11.3 Mitra's Approach......Page 398
11.4 Faddeev's Approach......Page 402
11.5 Faddeev Equations in Momentum Representation......Page 408
11.6 Faddeev Equations for a Three-body Bound System......Page 410
11.7 Alt, Grassberger and Sandhas (AGS) Equations......Page 413
12.1 Introduction......Page 420
12.2 Dirac Equation......Page 422
12.3 Spin of the Electron......Page 425
12.4 Free Particle (Plane Wave) Solutions of Dirac Equation......Page 426
12.5 Dirac Equation for a Zero Mass Particle......Page 430
12.6 Zitterbewegung and Negative Energy Solutions......Page 432
12.7 Dirac Equation for an Electron in an Electromagnetic Field......Page 434
12.8 Invariance of Dirac Equation......Page 439
12.9 Dirac Bilinear Covariants......Page 444
12.10 Dirac Electron in a Spherically Symmetric Potential......Page 445
12.11 Charge Conjugation, Parity and Time Reversal Invariance......Page 453
12A1.1 Lorentz Transformation......Page 462
12A1.2 Minkowski Space-Time Continuum......Page 465
12A1.3 Four-vectors in Relativistic Mechanics......Page 467
12A1.4 Covariant Form of Maxwell's Equations......Page 469
13.2 Radiation Field as a Swarm of Oscillators......Page 472
13.3 Quantization of Radiation Field......Page 476
13.4 Interaction of Matter with Quantized Radiation Field......Page 479
13.5 Applications......Page 483
13.6 Atomic Level Shift: Lamb-Retherford Shift......Page 493
13.7 Compton Scattering......Page 499
Appendix 13A1: Electromagnetic Field in Coulomb Gauge......Page 514
14.1 Introduction......Page 518
14.2 Classical Concept of Field......Page 519
14.3 Analogy of Field and Particle Mechanics......Page 521
14.4.1 Electromagnetic Field......Page 524
14.4.2 Klein-Gordon Field (Real and Complex)......Page 525
14.4.3 Dirac Field......Page 527
14.5 Quantization of a Real Scalar (KG) Field......Page 528
14.6 Quantization of Complex Scalar (KG) Field......Page 531
14.7 Dirac Field and Its Quantization......Page 536
14.8 Positron Operators and Spinors......Page 539
14.8.1 Equations Satisfied by Electron and Positron Spinors......Page 541
14.8.2 Projection Operators......Page 542
14.9 Interacting Fields and the Covariant Perturbation Theory......Page 544
14.9.1 U Matrix......Page 546
14.9.2 S Matrix and Iterative Expansion of S Operator......Page 548
14.9.3 Time-ordered Operator Product in Terms of Normal Constituents......Page 549
14.10 Second Order Processes in Electrodynamics......Page 551
14.10.1 Feynman Diagrams......Page 553
14.11 Amplitude for Compton Scattering......Page 557
14.12 Feynman Graphs......Page 562
14.12.1 Compton Scattering Amplitude Using Feynman Rules......Page 563
14.12.2 Electron-positron (e-e+) Pair Annihilation......Page 564
14.12.3 Two-photon Annihilation Leading to (e-e+) Pair Creation......Page 566
14.12.5 Bhabha (e-e+) Scattering......Page 567
14.13 Calculation of the Cross-section of Compton Scattering......Page 568
14.14.1 Electron-Positron Pair Annihilation (Electron at Rest)......Page 574
14.14.2 Möller (e-e-) and Bhabha(e-e+) Scattering......Page 575
Appendix 14A1: Calculus of Variation and Euler-Lagrange Equations......Page 581
Appendix 14A2: Functionals and Functional Derivatives......Page 584
Appendix 14A3: Interaction of the Electron and Radiation Fields......Page 586
Appendix 14A4: On the Convergence of Iterative Expansion of the S Operator......Page 587
15.1 Introduction......Page 590
15.2 EPR Gedanken Experiment......Page 591
15.3 Einstein-Podolsky-Rosen-Bohm Gedanken Experiment......Page 594
15.4 Theory of Hidden Variables and Bell's Inequality......Page 596
15.5 Clauser-Horne Form of Bell's Inequality and Its Violation in Two-photon Correlation Experiments......Page 601
General References......Page 608
Index......Page 13