Table Of ContentQUANTUM THEORY
Quantum Theory
A Wide Spectrum
by
E.B. MANOUKIAN
Suranaree University of Technology,
Nakhon Ratchasima,
Thailand
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN-10 1-4020-4189-6 (HB)
ISBN-13 978-1-4020-4189-1 (HB)
ISBN-10 1-4020-4190-X (e-book)
ISBN-13 978-1-4020-4190-7 (e-book)
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Contents
Acknowledgments............................................ XV
Preface ..................................................... XVII
1 Fundamentals............................................. 1
1.1 Selective Measurements ................................. 2
1.2 A, B, C to Probabilities ................................. 8
1.3 Expectation Values and Matrix Representations ............ 10
1.3.1 Probabilities and Expectation Values ............... 10
1.3.2 Representations of Simple Machines ................ 13
1.4 Generation of States, Inner-Product Spaces, Hermitian
Operators and the Eigenvalue Problem.................... 15
1.4.1 Generation of States and Vector Spaces ............. 16
1.4.2 Transformation Functions and Wavefunctions in
Different Descriptions............................. 18
1.4.3 An Illustration ................................... 19
1.4.4 Generation of Inner Product Spaces ................ 23
1.4.5 Hermitian Operators and the Eigenvalue Problem .... 24
1.5 Pure Ensembles and Mixtures............................ 25
1.6 Polarization of Light: An Interlude ....................... 29
1.7 The Hilbert Space; Rigged Hilbert Space .................. 33
1.8 Self-Adjoint Operators and Their Spectra.................. 39
1.9 Wigner’s Theorem on Symmetry Transformations .......... 55
1.10 Probability, Conditional Probability and Measurement ...... 65
1.10.1 Correlation of a Physical System and an Apparatus... 66
1.10.2 Probability and Conditional Probability............. 68
1.10.3 An Exactly Solvable Model ........................ 70
Problems .................................................. 79
VI Contents
2 Symmetries and Transformations ......................... 81
2.1 Galilean Space-Time Coordinate Transformations........... 81
2.2 Successive Galilean Transformations and the Closed Path.... 86
2.3 Quantum Galilean Transformations and Their Generators ... 89
2.4 The Transformation Function (cid:1)x|p(cid:2)....................... 98
2.5 Quantum Dynamics and Construction of Hamiltonians ...... 100
..
2.5.1 The Time Evolution: Schrodinger Equation .......... 100
2.5.2 Time as an Operator?............................. 101
2.5.3 Construction of Hamiltonians ...................... 102
2.5.4 Multi-Particle Hamiltonians ....................... 104
2.5.5 Two-Particle Systems and Relative Motion .......... 104
2.5.6 Multi-Electron Atoms with Positions of the Electrons
Defined Relative to the Nucleus .................... 105
2.5.7 Decompositions into Clusters of Particles............ 106
Appendix to §2.5: Time-Evolution for Time-Dependent
Hamiltonians .......................................... 109
2.6 Discrete Transformations: Parity and Time Reversal ........ 112
2.7 Orbital Angular Momentum and Spin..................... 116
2.8 Spinors and Arbitrary Spins ............................. 121
2.8.1 Spinors and Generation of Arbitrary Spins........... 121
2.8.2 Rotation of a Spinor by 2π Radians................. 129
2.8.3 Time Reversal and Parity Transformation ........... 130
2.8.4 Kramers Degeneracy.............................. 132
Appendix to §2.8: Transformation Rule of a Spinor of Rank One
Under a Coordinate Rotation ............................ 133
2.9 Supersymmetry ........................................ 136
Problems .................................................. 139
3 Uncertainties, Localization, Stability and Decay of
Quantum Systems ........................................ 143
3.1 Uncertainties, Localization and Stability................... 143
3.1.1 A Basic Inequality................................ 143
3.1.2 Uncertainties .................................... 144
3.1.3 Localization and Stability ......................... 145
3.1.4 Localization, Stability and Multi-Particle Systems .... 148
3.2 Boundedness of the Spectra of Hamiltonians From Below .... 151
3.3 Boundedness of Hamiltonians From Below: General Classes
of Interactions ......................................... 152
3.4 Boundedness of Hamiltonian From Below: Multi-Particle
Systems ............................................... 163
3.4.1 Multi-Particle Systems with Two-Body Potentials .... 164
3.4.2 Multi-Particle Systems and Other Potentials......... 166
3.4.3 Multi-Particle Systems with Coulomb Interactions.... 167
3.5 Decay of Quantum Systems.............................. 168
Appendix to §3.5: The Paley-Wiener Theorem .................. 174
Contents VII
Problems .................................................. 178
4 Spectra of Hamiltonians .................................. 181
4.1 Hamiltonians with Potentials Vanishing at Infinity.......... 182
4.2 On Bound-States ....................................... 187
4.2.1 A Potential Well ................................. 187
4.2.2 Limit of the Potential Well ........................ 190
4.2.3 The Dirac Delta Potential ......................... 190
4.2.4 Sufficiency Conditions for the Existence of a
Bound-State for ν =1 ............................ 192
4.2.5 Sufficiency Conditions for the Existence of a
Bound-State for ν =2 ............................ 194
4.2.6 Sufficiency Conditions for the Existence of a
Bound-State for ν =3 ............................ 195
4.2.7 No-Binding Theorems............................. 197
4.3 Hamiltonians with Potentials Approaching Finite Constants
at Infinity ............................................. 199
4.4 Hamiltonians with Potentials Increasing with No Bound at
Infinity................................................ 200
4.5 Counting the Number of Eigenvalues...................... 203
4.5.1 General Treatment of the Problem.................. 203
4.5.2 Counting the Number of Eigenvalues................ 206
4.5.3 The Sum of the Negative Eigenvalues ............... 216
Appendix to §4.5: Evaluation of Certain Integrals ............... 219
4.6 Lower Bounds to the Expectation Value of the Kinetic
Energy: An Application of Counting Eigenvalues ........... 220
4.6.1 One-Particle Systems ............................. 220
4.6.2 Multi-Particle States: Fermions .................... 222
4.6.3 Multi-Particle States: Bosons ...................... 224
4.7 The Eigenvalue Problem and Supersymmetry .............. 224
4.7.1 General Aspects.................................. 224
4.7.2 Construction of Supersymmetric Hamiltonians ....... 226
4.7.3 The Eigenvalue Problem .......................... 230
Problems .................................................. 244
5 Angular Momentum Gymnastics.......................... 249
5.1 The Eigenvalue Problem ................................ 251
5.2 Matrix Elements of Finite Rotations ...................... 254
5.3 Orbital Angular Momentum ............................. 258
5.3.1 Transformation Theory ........................... 258
5.3.2 Half-Odd Integral Values? ......................... 259
5.3.3 The Spherical Harmonics.......................... 262
5.3.4 Addition Theorem of Spherical Harmonics........... 267
5.4 Spin .................................................. 269
5.4.1 General Structure ................................ 269
VIII Contents
5.4.2 Spin 1/2 ........................................ 270
5.4.3 Spin 1 .......................................... 272
5.4.4 Arbitrary Spins .................................. 274
5.5 Addition of Angular Momenta ........................... 275
5.6 Explicit Expression for the Clebsch-Gordan Coefficients ..... 284
5.7 Vector Operators ....................................... 290
5.8 Tensor Operators....................................... 296
5.9 Combining Several Angular Momenta: 6-j and 9-j Symbols .. 304
5.10 Particle States and Angular Momentum; Helicity States ..... 307
5.10.1 Single Particle States ............................. 307
5.10.2 Two Particle States............................... 317
Problems .................................................. 324
6 Intricacies of Harmonic Oscillators........................ 329
6.1 The Harmonic Oscillator ................................ 329
6.2 Transition to and Between Excited States in the Presence of
a Time-Dependent Disturbance .......................... 335
6.3 The Harmonic Oscillator in the Presence of a Disturbance
at Finite Temperature .................................. 340
6.4 The Fermi Oscillator.................................... 343
6.5 Bose-Fermi Oscillators and Supersymmetric Bose-Fermi
Transformations........................................ 346
6.6 Coherent State of the Harmonic Oscillator................. 349
Problems .................................................. 356
7 Intricacies of the Hydrogen Atom......................... 359
7.1 Stability of the Hydrogen Atom .......................... 360
7.2 The Eigenvalue Problem ................................ 363
7.3 The Eigenstates ........................................ 366
7.4 The Hydrogen Atom Including Spin and Relativistic
Corrections ............................................ 370
Appendix to §7.4: Normalization of the Wavefunction Including
Spin and Relativistic Corrections ......................... 378
7.5 The Fine-Structure of the Hydrogen Atom................. 379
Appendix to §7.5: Combining Spin and Angular Momentum in
the Atom.............................................. 383
7.6 The Hyperfine-Structure of the Hydrogen Atom ............ 384
7.7 The Non-Relativistic Lamb Shift ......................... 391
7.7.1 The Radiation Field .............................. 391
7.7.2 Expression for the Energy Shifts ................... 394
7.7.3 The Lamb Shift and Renormalization ............... 398
Appendix to §7.7: Counter-Terms and Mass Renormalization ..... 401
7.8 Decay of Excited States ................................. 403
7.9 The Hydrogen Atom in External Electromagnetic Fields..... 406
7.9.1 The Atom in an External Magnetic Field............ 406
Contents IX
7.9.2 The Atom in an External Electric Field ............. 412
Problems .................................................. 414
8 Quantum Physics of Spin 1/2 and Two-Level Systems;
Quantum Predictions Using Such Systems ................ 419
8.1 General Properties of Spin 1/2 and Two-Level Systems...... 420
8.1.1 General Aspects of Spin 1/2 ....................... 420
8.1.2 Spin 1/2 in External Magnetic Fields ............... 423
8.1.3 Two-Level Systems; Exponential Decay.............. 427
8.2 The Pauli Hamiltonian; Supersymmetry ................... 432
8.2.1 The Pauli Hamiltonian............................ 432
8.2.2 Supersymmetry .................................. 434
8.3 Landau Levels; Expression for the g-Factor ................ 436
8.3.1 Landau Levels ................................... 436
8.3.2 Expression for the g-Factor ........................ 440
8.4 Spin Precession and Radiation Losses ..................... 441
8.5 Anomalous Magnetic Moment of the Electron .............. 444
8.5.1 Observational Aspect of the Anomalous Magnetic
Moment......................................... 445
8.5.2 Computation of the Anomalous Magnetic Moment.... 446
8.6 Density Operators and Spin ............................. 453
8.6.1 Spin in a General Time-Dependent Magnetic Field ... 453
8.6.2 Scattering of Spin 1/2 Particle off a Spin 0 Target .... 454
8.6.3 Scattering of Spin 1/2 Particles off a Spin 1/2 Target . 459
8.7 Quantum Interference and Measurement; The Role of the
Environment........................................... 462
8.7.1 Interaction with an Apparatus and Unitary Evolution
Operator ........................................ 463
8.7.2 Interaction with a Harmonic Oscillator in a Coherent
State ........................................... 467
8.7.3 The Role of the Environment ...................... 469
8.8 RamseyOscillatoryFieldsMethodandSpinFlip;Monitoring
the Spin............................................... 473
8.8.1 Ramsey Apparatus and Interference; Spin Flip ....... 473
8.8.2 Monitoring the Spin .............................. 478
8.9 Schrödinger’s Cat and Quantum Decoherence .............. 482
8.10 Bell’s Test............................................. 486
8.10.1 Bell’s Test....................................... 486
8.10.2 Basic Processes .................................. 490
Appendix to §8.10. Entangled States; The C-H Inequality ........ 499
8.11 Quantum Teleportation and Quantum Cryptography........ 501
8.11.1 Quantum Teleportation ........................... 501
8.11.2 Quantum Cryptography........................... 503
X Contents
8.12 Rotation of a Spinor .................................... 508
8.13 Geometric Phases ...................................... 513
8.13.1 The Berry Phase and the Adiabatic Regime ......... 513
8.13.2 Degeneracy...................................... 518
8.13.3 Aharonov-Anandan (AA) Phase.................... 520
8.13.4 Samuel-Bhandari (SB) Phase ...................... 529
8.14 Quantum Dynamics of the Stern-Gerlach Effect ............ 531
8.14.1 The Quantum Dynamics .......................... 531
8.14.2 The Intensity Distribution......................... 535
Appendix to §8.14: Time Evolution and Intensity Distribution .... 540
Problems .................................................. 544
9 Green Functions .......................................... 547
9.1 The Free Green Functions ............................... 548
9.2 Linear and Quadratic Potentials.......................... 555
9.3 The Dirac Delta Potential ............................... 558
9.4 Time-Dependent Forced Dynamics........................ 561
9.5 The Law of Reflection and Reconciliation with the Classical
Law .................................................. 565
9.6 Two-Dimensional Green Function in Polar Coordinates:
Application to the Aharonov-Bohm Effect ................. 570
9.7 General Properties of the Full Green Functions and
Applications ........................................... 580
9.7.1 A Matrix Notation ............................... 580
9.7.2 Applications ..................................... 582
9.7.3 An Integral Expression for the (Homogeneous) Green
Function ........................................ 586
9.8 The Thomas-Fermi Approximation and Deviations Thereof .. 587
9.9 The Coulomb Green Function: The Full Spectrum .......... 590
9.9.1 An Integral Equation ............................. 590
9.9.2 The Negative Spectrum p0 <0,λ<0 ............... 594
9.9.3 The Positive Spectrum p0 >0...................... 596
Problems .................................................. 598
10 Path Integrals ............................................ 601
10.1 The Free Particle....................................... 602
10.2 Particle in a Given Potential............................. 604
10.3 Charged Particle in External Electromagnetic Fields:
Velocity Dependent Potentials ........................... 608
10.4 Constrained Dynamics .................................. 614
10.4.1 Classical Notions ................................. 614
10.4.2 Constrained Path Integrals ........................ 623
10.4.3 Second Class Constraints and the Dirac Bracket...... 627
10.5 Bose Excitations ....................................... 628
Contents XI
10.6 Grassmann Variables: Fermi Excitations................... 633
10.6.1 Real Grassmann Variables......................... 633
10.6.2 Complex Grassmann Variables ..................... 637
10.6.3 Fermi Excitations ................................ 640
Problems .................................................. 645
11 The Quantum Dynamical Principle ....................... 649
11.1 The Quantum Dynamical Principle ....................... 650
11.2 Expressions for Transformations Functions................. 656
11.3 Trace Functionals ...................................... 665
11.4 From the Quantum Dynamical Principle to Path Integrals ... 669
11.5 Bose/Fermi Excitations ................................. 672
11.6 Closed-Time Path and Expectation-Value Formalism........ 675
Problems .................................................. 681
12 Approximating Quantum Systems ........................ 683
12.1 Non-Degenerate Perturbation Theory ..................... 684
12.2 Degenerate Perturbation Theory ......................... 688
12.3 Variational Methods .................................... 690
12.4 High-Order Perturbations, Divergent Series; Padé
Approximants.......................................... 695
12.5 WKB Approximation ................................... 703
12.5.1 General Theory .................................. 703
12.5.2 Barrier Penetration............................... 709
12.5.3 WKB Quantization Rules ......................... 712
12.5.4 The Radial Equation ............................. 715
12.6 Time-Dependence;SuddenApproximationandtheAdiabatic
Theorem .............................................. 716
12.6.1 Weak Perturbations .............................. 717
12.6.2 Sudden Approximation............................ 720
12.6.3 The Adiabatic Theorem ........................... 724
12.7 Master Equation; Exponential Law, Coupling to the
Environment........................................... 727
12.7.1 Master Equation ................................. 728
12.7.2 Exponential Law ................................. 733
12.7.3 Coupling to the Environment ...................... 734
Problems .................................................. 736
13 Multi-Electron Atoms: Beyond the Thomas-Fermi Atom.. 739
13.1 The Thomas-Fermi Atom................................ 740
Appendix A To §13.1: The TF Energy Gives the Leading
Contribution to E(Z) for Large Z ........................ 746
AppendixBto§13.1:TheTFDensityActuallyGivestheSmallest
Value for the Energy Density Functional in (13.1.6)......... 752
13.2 Correction due to Electrons Bound Near the Nucleus........ 753
