Quantum Theory: A Wide Spectrum

Contents......Page 5
Acknowledgments......Page 15
Preface......Page 16
1 Fundamentals......Page 19
1.1 Selective Measurements......Page 20
1.2 A, B, C to Probabilities......Page 26
1.3.1 Probabilities and Expectation Values......Page 28
1.3.2 Representations of Simple Machines......Page 31
1.4 Generation of States, Inner-Product Spaces, Hermitian Operators and the Eigenvalue Problem......Page 33
1.4.1 Generation of States and Vector Spaces......Page 34
1.4.2 Transformation Functions and Wavefunctions in Different Descriptions......Page 36
1.4.3 An Illustration......Page 37
1.4.4 Generation of Inner Product Spaces......Page 41
1.4.5 Hermitian Operators and the Eigenvalue Problem......Page 42
1.5 Pure Ensembles and Mixtures......Page 43
1.6 Polarization of Light: An Interlude......Page 47
1.7 The Hilbert Space; Rigged Hilbert Space......Page 51
1.8 Self-Adjoint Operators and Their Spectra......Page 57
1.9 Wigner’s Theorem on Symmetry Transformations......Page 73
1.10 Probability, Conditional Probability and Measurement......Page 83
1.10.1 Correlation of a Physical System and an Apparatus......Page 84
1.10.2 Probability and Conditional Probability......Page 86
1.10.3 An Exactly Solvable Model......Page 88
Problems......Page 97
2.1 Galilean Space-Time Coordinate Transformations......Page 99
2.2 Successive Galilean Transformations and the Closed Path......Page 104
2.3 Quantum Galilean Transformations and Their Generators......Page 107
2.4 The Transformation Function 〈x|p〉......Page 116
2.5.1 The Time Evolution: Schrödinger Equation......Page 118
2.5.2 Time as an Operator?......Page 119
2.5.3 Construction of Hamiltonians......Page 120
2.5.5 Two-Particle Systems and Relative Motion......Page 122
2.5.6 Multi-Electron Atoms with Positions of the Electrons Defined Relative to the Nucleus......Page 123
2.5.7 Decompositions into Clusters of Particles......Page 124
Appendix to §2.5: Time-Evolution for Time-Dependent Hamiltonians......Page 127
2.6 Discrete Transformations: Parity and Time Reversal......Page 130
2.7 Orbital Angular Momentum and Spin......Page 134
2.8.1 Spinors and Generation of Arbitrary Spins......Page 139
2.8.2 Rotation of a Spinor by 2π Radians......Page 147
2.8.3 Time Reversal and Parity Transformation......Page 148
2.8.4 Kramers Degeneracy......Page 150
Appendix to §2.8: Transformation Rule of a Spinor of Rank One Under a Coordinate Rotation......Page 151
2.9 Supersymmetry......Page 154
Problems......Page 157
3.1.1 A Basic Inequality......Page 160
3.1.2 Uncertainties......Page 161
3.1.3 Localization and Stability......Page 162
3.1.4 Localization, Stability and Multi-Particle Systems......Page 165
3.2 Boundedness of the Spectra of Hamiltonians From Below......Page 168
3.3 Boundedness of Hamiltonians From Below: General Classes of Interactions......Page 169
3.4 Boundedness of Hamiltonian From Below: Multi-Particle Systems......Page 180
3.4.1 Multi-Particle Systems with Two-Body Potentials......Page 181
3.4.2 Multi-Particle Systems and Other Potentials......Page 183
3.4.3 Multi-Particle Systems with Coulomb Interactions......Page 184
3.5 Decay of Quantum Systems......Page 185
Appendix to §3.5: The Paley-Wiener Theorem......Page 191
Problems......Page 195
4 Spectra of Hamiltonians......Page 197
4.1 Hamiltonians with Potentials Vanishing at Infinity......Page 198
4.2.1 A Potential Well......Page 203
4.2.3 The Dirac Delta Potential......Page 206
4.2.4 Sufficiency Conditions for the Existence of a Bound-State for v = 1......Page 208
4.2.5 Sufficiency Conditions for the Existence of a Bound-State for v = 2......Page 210
4.2.6 Sufficiency Conditions for the Existence of a Bound-State for v = 3......Page 211
4.2.7 No-Binding Theorems......Page 213
4.3 Hamiltonians with Potentials Approaching Finite Constants at Infinity......Page 215
4.4 Hamiltonians with Potentials Increasing with No Bound at Infinity......Page 216
4.5.1 General Treatment of the Problem......Page 219
4.5.2 Counting the Number of Eigenvalues......Page 222
4.5.3 The Sum of the Negative Eigenvalues......Page 232
Appendix to §4.5: Evaluation of Certain Integrals......Page 235
4.6.1 One-Particle Systems......Page 236
4.6.2 Multi-Particle States: Fermions......Page 238
4.7.1 General Aspects......Page 240
4.7.2 Construction of Supersymmetric Hamiltonians......Page 242
4.7.3 The Eigenvalue Problem......Page 246
Problems......Page 260
5 Angular Momentum Gymnastics......Page 264
5.1 The Eigenvalue Problem......Page 266
5.2 Matrix Elements of Finite Rotations......Page 269
5.3.1 Transformation Theory......Page 273
5.3.2 Half-Odd Integral Values?......Page 274
5.3.3 The Spherical Harmonics......Page 277
5.3.4 Addition Theorem of Spherical Harmonics......Page 282
5.4.1 General Structure......Page 284
5.4.2 Spin 1/2......Page 285
5.4.3 Spin 1......Page 287
5.4.4 Arbitrary Spins......Page 289
5.5 Addition of Angular Momenta......Page 290
5.6 Explicit Expression for the Clebsch-Gordan Coefficients......Page 299
5.7 Vector Operators......Page 305
5.8 Tensor Operators......Page 311
5.9 Combining Several Angular Momenta: 6-j and 9-j Symbols......Page 319
5.10.1 Single Particle States......Page 322
5.10.2 Two Particle States......Page 332
Problems......Page 339
6.1 The Harmonic Oscillator......Page 343
6.2 Transition to and Between Excited States in the Presence of a Time-Dependent Disturbance......Page 349
6.3 The Harmonic Oscillator in the Presence of a Disturbance at Finite Temperature......Page 354
6.4 The Fermi Oscillator......Page 357
6.5 Bose-Fermi Oscillators and Supersymmetric Bose-Fermi Transformations......Page 360
6.6 Coherent State of the Harmonic Oscillator......Page 363
Problems......Page 370
7 Intricacies of the Hydrogen Atom......Page 372
7.1 Stability of the Hydrogen Atom......Page 373
7.2 The Eigenvalue Problem......Page 376
7.3 The Eigenstates......Page 379
7.4 The Hydrogen Atom Including Spin and Relativistic Corrections......Page 383
Appendix to §7.4: Normalization of the Wavefunction Including Spin and Relativistic Corrections......Page 391
7.5 The Fine-Structure of the Hydrogen Atom......Page 392
Appendix to §7.5: Combining Spin and Angular Momentum in the Atom......Page 396
7.6 The Hyperfine-Structure of the Hydrogen Atom......Page 397
7.7.1 The Radiation Field......Page 404
7.7.2 Expression for the Energy Shifts......Page 407
7.7.3 The Lamb Shift and Renormalization......Page 411
Appendix to §7.7: Counter-Terms and Mass Renormalization......Page 414
7.8 Decay of Excited States......Page 416
7.9.1 The Atom in an External Magnetic Field......Page 419
7.9.2 The Atom in an External Electric Field......Page 425
Problems......Page 427
8 Quantum Physics of Spin 1/2 and Two-Level Systems; Quantum Predictions Using Such Systems......Page 432
8.1.1 General Aspects of Spin 1/2......Page 433
8.1.2 Spin 1/2 in External Magnetic Fields......Page 436
8.1.3 Two-Level Systems; Exponential Decay......Page 440
8.2.1 The Pauli Hamiltonian......Page 445
8.2.2 Supersymmetry......Page 447
8.3.1 Landau Levels......Page 449
8.3.2 Expression for the g-Factor......Page 453
8.4 Spin Precession and Radiation Losses......Page 454
8.5 Anomalous Magnetic Moment of the Electron......Page 457
8.5.1 Observational Aspect of the Anomalous Magnetic Moment......Page 458
8.5.2 Computation of the Anomalous Magnetic Moment......Page 459
8.6.1 Spin in a General Time-Dependent Magnetic Field......Page 466
8.6.2 Scattering of Spin 1/2 Particle off a Spin 0 Target......Page 467
8.6.3 Scattering of Spin 1/2 Particles off a Spin 1/2 Target......Page 472
8.7 Quantum Interference and Measurement; The Role of the Environment......Page 475
8.7.1 Interaction with an Apparatus and Unitary Evolution Operator......Page 476
8.7.2 Interaction with a Harmonic Oscillator in a Coherent State......Page 480
8.7.3 The Role of the Environment......Page 482
8.8.1 Ramsey Apparatus and Interference; Spin Flip......Page 486
8.8.2 Monitoring the Spin......Page 491
8.9 Schrödinger’s Cat and Quantum Decoherence......Page 495
8.10.1 Bell\'s Test......Page 499
8.10.2 Basic Processes......Page 503
Appendix to §8.10. Entangled States; The C-H Inequality......Page 512
8.11.1 Quantum Teleportation......Page 514
8.11.2 Quantum Cryptography......Page 516
8.12 Rotation of a Spinor......Page 521
8.13.1 The Berry Phase and the Adiabatic Regime......Page 526
8.13.2 Degeneracy......Page 531
8.13.3 Aharonov-Anandan (AA) Phase......Page 533
8.13.4 Samuel-Bhandari (SB) Phase......Page 542
8.14.1 The Quantum Dynamics......Page 544
8.14.2 The Intensity Distribution......Page 548
Appendix to §8.14: Time Evolution and Intensity Distribution......Page 553
Problems......Page 557
9 Green Functions......Page 560
9.1 The Free Green Functions......Page 561
9.2 Linear and Quadratic Potentials......Page 568
9.3 The Dirac Delta Potential......Page 571
9.4 Time-Dependent Forced Dynamics......Page 574
9.5 The Law of Reflection and Reconciliation with the Classical Law......Page 578
9.6 Two-Dimensional Green Function in Polar Coordinates: Application to the Aharonov-Bohm Effect......Page 583
9.7.1 A Matrix Notation......Page 593
9.7.2 Applications......Page 595
9.7.3 An Integral Expression for the (Homogeneous) Green Function......Page 599
9.8 The Thomas-Fermi Approximation and Deviations Thereof......Page 600
9.9.1 An Integral Equation......Page 603
9.9.2 The Negative Spectrum p[sup(0)] < 0, λ < 0......Page 607
9.9.3 The Positive Spectrum p[sup(0)] > 0......Page 609
Problems......Page 611
10 Path Integrals......Page 613
10.1 The Free Particle......Page 614
10.2 Particle in a Given Potential......Page 616
10.3 Charged Particle in External Electromagnetic Fields: Velocity Dependent Potentials......Page 620
10.4.1 Classical Notions......Page 626
10.4.2 Constrained Path Integrals......Page 635
10.4.3 Second Class Constraints and the Dirac Bracket......Page 639
10.5 Bose Excitations......Page 640
10.6.1 Real Grassmann Variables......Page 645
10.6.2 Complex Grassmann Variables......Page 649
10.6.3 Fermi Excitations......Page 652
Problems......Page 657
11 The Quantum Dynamical Principle......Page 660
11.1 The Quantum Dynamical Principle......Page 661
11.2 Expressions for Transformations Functions......Page 667
11.3 Trace Functionals......Page 676
11.4 From the Quantum Dynamical Principle to Path Integrals......Page 680
11.5 Bose/Fermi Excitations......Page 683
11.6 Closed-Time Path and Expectation-Value Formalism......Page 686
Problems......Page 692
12 Approximating Quantum Systems......Page 694
12.1 Non-Degenerate Perturbation Theory......Page 695
12.2 Degenerate Perturbation Theory......Page 699
12.3 Variational Methods......Page 701
12.4 High-Order Perturbations, Divergent Series; Padé Approximants......Page 706
12.5.1 General Theory......Page 714
12.5.2 Barrier Penetration......Page 720
12.5.3 WKB Quantization Rules......Page 723
12.5.4 The Radial Equation......Page 726
12.6 Time-Dependence; Sudden Approximation and the Adiabatic Theorem......Page 727
12.6.1 Weak Perturbations......Page 728
12.6.2 Sudden Approximation......Page 731
12.6.3 The Adiabatic Theorem......Page 735
12.7 Master Equation; Exponential Law, Coupling to the Environment......Page 738
12.7.1 Master Equation......Page 739
12.7.2 Exponential Law......Page 744
12.7.3 Coupling to the Environment......Page 745
Problems......Page 747
13 Multi-Electron Atoms: Beyond the Thomas-Fermi Atom......Page 750
13.1 The Thomas-Fermi Atom......Page 751
Appendix A To §13.1: The TF Energy Gives the Leading Contribution to E(Z) for Large Z......Page 757
Appendix B to §13.1: The TF Density Actually Gives the Smallest Value for the Energy Density Functional in (13.1.6)......Page 763
13.2 Correction due to Electrons Bound Near the Nucleus......Page 764
13.3 The Exchange Term......Page 767
13.4 Quantum Correction......Page 770
Problems......Page 773
14 Quantum Physics and the Stability of Matter......Page 775
14.1 Lower Bound to the Multi-Particle Repulsive Coulomb Potential Energy......Page 777
Appendix to §14.1: A Thomas-Fermi-Like Energy Functional and No Binding......Page 779
14.2.1 A Lower Bound......Page 784
14.2.2 Upper Bounds......Page 787
14.3.1 Upper Bound of the Average Kinetic Energy of Electrons in Matter......Page 790
14.3.2 Inflation of Matter......Page 791
14.4 The Collapse of “Bosonic Matter”......Page 793
14.4.1 A Lower Bound......Page 794
14.4.2 An Upper Bound......Page 796
Appendix to §14.4: Upper Bounds for 〈H[sub(1)]〉 in (14.4.47)......Page 803
Problems......Page 806
15 Quantum Scattering......Page 808
15.1 Interacting States and Asymptotic Boundary Conditions......Page 809
15.2 Particle Detection and Connection between Configuration and Momentum Spaces in Scattering......Page 816
Appendix to §15.2: Some Properties of F (u, v)......Page 821
15.3.1 Expression for the Differential Cross Section......Page 823
15.3.2 Sufficiency Conditions for the Validity of the Born Expansion......Page 825
15.3.3 Two-Particle Scattering......Page 827
15.4.1 The Optical Theorem......Page 830
15.4.2 Phase Shifts Analysis......Page 834
15.5.1 Asymptotically “Free” Coulomb Green Functions......Page 839
15.5.2 Asymptotic Time Development of a Charged Particle State......Page 841
15.5.3 The Full Green Function G[sub(+)] Near the Energy Shell......Page 842
15.5.4 The Scattering Amplitude via Evolution Operators......Page 843
15.6 Functional Treatment of Scattering Theory......Page 847
15.7.1 Eikonal Approximation......Page 851
15.7.2 Determination of Asymptotic “Free” Green Function of the Coulomb Interaction......Page 854
15.8 Multi-Channel Scatterings of Clusters and Bound Systems......Page 855
15.8.1 Channels and Channel Hamiltonians......Page 856
15.8.2 Interacting States Corresponding to Preparatory Channels......Page 860
15.8.3 Transition Probabilities and the Optical Theorem......Page 862
15.8.4 Basic Processes......Page 863
15.8.5 Born Approximation, Connectedness and Faddeev Equations......Page 867
15.8.6 Phase Shifts Analysis......Page 874
15.9.1 Passage of Charged Particles through Hydrogen......Page 876
15.9.2 Neutron Interferometer......Page 880
Problems......Page 886
16 Quantum Description of Relativistic Particles......Page 890
16.1 The Dirac Equation and Pauli’s Fundamental Theorem......Page 893
Appendix to §16.1: Pauli’s Fundamental Theorem......Page 898
16.2.1 Lorentz Transformations......Page 901
16.2.2 Lorentz Covariance, Boosts and Spatial Rotations......Page 903
16.2.3 Lorentz Invariant Scalar Products of Spinors, Lorentz Scalars and Lorentz Vectors......Page 907
16.3.1 Spin & Helicity......Page 909
16.3.2 P, C, T Transformations......Page 911
16.4 General Solution of the Dirac Equation......Page 912
16.5 Massless Dirac Particles......Page 921
16.6 Physical Interpretation, Localization and Particle Content......Page 925
16.6.1 Probability, Probability Current and the Initial Value Problem......Page 926
16.6.2 Diagonalization of the Hamiltonian and Definitions of Position Operators......Page 928
16.6.3 Origin of Relativistic Corrections in the Hydrogen Atom......Page 935
16.6.4 The Positron and Emergence of a Many-Particle Theory......Page 940
Appendix to §16.6: Exact Treatment of the Dirac Equation in the Bound Coulomb Problem......Page 942
16.7.1 Setting Up Spin 0 Equations......Page 946
16.7.2 A Continuity Equation......Page 949
16.7.3 General Solution of the Free Feshbach-Villars Equation......Page 950
16.7.4 Diagonalization of the Hamiltonian and Definition of Position Operators......Page 951
16.7.5 The External Field Problem......Page 953
16.8.1 M > 0:......Page 956
16.8.2 M = 0:......Page 959
16.9 Spin & Statistics......Page 962
16.9.1 Quantum Fields......Page 963
16.9.2 Lagrangian for Spin 0 Particles......Page 964
16.9.3 Lagrangian for Spin 1/2 Particles......Page 966
16.9.4 Schwinger’s Constructive Approach......Page 967
16.9.5 The Spin and Statistics Connection......Page 971
Appendix to §16.9: The Action Integral......Page 974
Problems......Page 977
Mathematical Appendices......Page 980
2. Derivative of the Exponential of Operator-Valued Functions......Page 981
4. A Modification of the Baker-Campbell-Hausdorff Formula......Page 983
1. General Convexity Theorem......Page 985
2. Minkowski’s Inequality for Integrals......Page 986
3. Hölder’s Inequality for Integrals......Page 987
4. Young’s Inequality for Integrals......Page 988
III: The Poisson Equation in 4D......Page 989
1. The Poisson Equation......Page 990
2. Generating Function......Page 991
3. Expansion Theorem......Page 992
4. Generalized Orthogonality Relation......Page 993
References......Page 995
A......Page 1007
C......Page 1008
D......Page 1009
F......Page 1010
H......Page 1011
L......Page 1012
N......Page 1013
P......Page 1014
R......Page 1015
S......Page 1016
V......Page 1018
Z......Page 1019