Concepts in quantum mechanics:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton, Fla. [u.a.]
CRC Press, Chapman & Hall
2009
|
| Series: | CRC series in pure and applied physics
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 598 S. graph. Darst. |
| ISBN: | 9781420078725 |
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| 245 | 1 | 0 | |a Concepts in quantum mechanics |c Vishnu Swarup Mathur ; Surendra Singh |
| 264 | 1 | |a Boca Raton, Fla. [u.a.] |b CRC Press, Chapman & Hall |c 2009 | |
| 300 | |a XV, 598 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a CRC series in pure and applied physics | |
| 650 | 4 | |a Quantum theory | |
| 650 | 4 | |a Quantentheorie | |
| 650 | 4 | |a Quantum theory | |
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| adam_text | Contents
Preface
......................................... xiii
Acknowledgments
................................... xv
1
NEED FOR QUANTUM MECHANICS AND ITS PHYSICAL BASIS
1
1.1
Inadequacy of Classical Description for Small Systems
........... 1
1.1.1
Planck s Formula for Energy Distribution in Black-body Radiation
1
1.1.2 de Broglie
Relation and Wave Nature of Material Particles
..... 2
1.1.3
The Photo-electric Effect
....................... 3
1.1.4
The Compton Effect
.......................... 4
1.1.5 Ritz
Combination Principle
...................... 6
1.2
Basis of Quantum Mechanics
......................... 9
1.2.1
Principle of Superposition of States
.................. 9
1.2.2 Heisenberg
Uncertainty Relations
................... 12
1.3
Representation of States
............................ 14
1.4
Dual Vectors: Bra and
Ket
Vectors
...................... 15
1.5
Linear Operators
................................ 15
1.5.1
Properties of a Linear Operator
.................... 16
1.6
Adjoint of a Linear Operator
......................... 16
1.7
Eigenvalues and Eigenvectors of a Linear Operator
............. 18
1.8
Physical Interpretation
............................. 20
1.8.1
Physical Interpretation of Eigenstates and Eigenvalues
....... 20
1.8.2
Physical Meaning of the Orthogonality of States
.......... 21
1.9
Observables
and Completeness Criterion
................... 21
1.10
Commutativity and Compatibility of
Observables
.............. 23
1.11
Position and Momentum Commutation Relations
.............. 24
1.12
Commutation Relation and the Uncertainty Product
............ 26
Appendix 1A1: Basic Concepts in Classical Mechanics
.............. 31
1A1.1
Lagrange
Equations of Motion
.................... 31
1A1.2 Classical Dynamical Variables
..................... 32
2
REPRESENTATION THEORY
35
2.1
Meaning of Representation
........................... 35
2.2
How to Set up a Representation
........................ 35
2.3
Representatives of a Linear Operator
..................... 37
2.4
Change of Representation
........................... 40
2.5
Coordinate Representation
........................... 43
2.5.1
Physical Interpretation of the Wave Function
............ 44
2.6
Replacement of Momentum Observable
ρ
by —ifijp
............. 45
2.7
Integral Representation of Dirac Bracket
(A2¡
F
A )
........... 50
2.8
The Momentum Representation
........................ 52
2.8.1
Physical Interpretation of
Φ(ρι
.p2
, ··
-p¡)
.............. 52
2.9
Dirac Delta Function
.............................. 53
2.9.1
Three-dimensional Delta Function
.................. 55
2.9.2
Normalization of a Plane Wave
.................... 56
2.10
Relation between the Coordinate and Momentum Representations
.... 56
EQUATIONS OF MOTION
67
3.1 Schrödinger
Equation of Motion
........................ 67
3.2 Schrödinger
Equation in the Coordinate Representation
.......... 69
3.3
Equation of Continuity
............................. 70
3.4
Stationary States
................................ 71
3.5
Time-independent
Schrödinger
Equation in the Coordinate Representation
72
3.6
Time-independent
Schrödinger
Equation in the Momentum Representation
74
3.6.1
Two-body Bound State Problem (in Momentum Representation) for
Non-local Separable Potential
..................... 76
3.7
Time-independent
Schrödinger
Equation in Matrix Form
.......... 77
3.8
The
Heisenberg
Picture
............................ 79
3.9
The Interaction Picture
............................ 81
Appendix 3A1: Matrices
............................... 86
3A1.1 Characteristic Equation of a Matrix
................. 86
3A1.2 Similarity (and Unitary) Transformation of Matrices
........ 87
3A1.3 Diagonalization of a Matrix
...................... 87
PROBLEMS OF ONE-DIMENSIONAL POTENTIAL BARRIERS
89
4.1
Motion of a Particle across a Potential Step
................. 90
4.2
Passage of a Particle through a Potential Barrier of Finite Extent
..... 94
4.3
Tunneling of a Particle through a Potential Barrier
............. 99
4.4
Bound States in a One-dimensional Square Potential Well
......... 103
4.5
Motion of a Particle in a Periodic Potential
................. 107
BOUND STATES OF SIMPLE SYSTEMS
115
5.1
Introduction
................................... 115
5.2
Motion of a Particle in a Box
......................... 115
5.2.1
Density of States
............................ 117
5.3
Simple Harmonic Oscillator
.......................... 118
5.4
Operator Formulation of the Simple Harmonic Oscillator Problem
.... 122
5.4.1
Physical Meaning of the Operators
ã
and
â^
............. 123
5.4.2
Occupation Number Representation (ONR)
............. 125
5.5
Bound State of a Two-particle System with Central Interaction
...... 126
5.6
Bound States of Hydrogen (or Hydrogen-like) Atoms
............ 131
5.7
The
Deuteron
Problem
............................. 137
5.8
Energy Levels in a Three-dimensional Square Well: General Case
..... 144
5.9
Energy Levels in an
Isotropie
Harmonic Potential Well
........... 147
Appendix 5A1: Special Functions
.......................... 156
5A1.1 Legendre and Associated Legendre Equations
............ 156
5A1.2 Spherical Harmonics
.......................... 159
5A1.3 Laguerre and Associated Laguerre Equations
............ 162
5A1.4 Hermite Equation
............................ 166
5A1.5 Bessel Equation
............................. 169
Appendix 5A2: Orthogonal Curvilinear Coordinate Systems
........... 174
5A2.1 Spherical Polar Coordinates
...................... 174
5A2.2 Cylindrical Coordinates
........................ 175
5A2.3 Parabolic Coordinates
......................... 177
5A2.4 General Features of Orthogonal Curvilinear System of Coordinates
178
SYMMETRIES AND CONSERVATION LAWS
181
6.1
Symmetries and Their Group Properties
................... 181
6.2
Symmetries in a Quantum Mechanical System
................ 182
6.3
Basic Symmetry Groups of the Hamiltonian and Conservation Laws
. . . 183
6.3.1
Space Translation Symmetry
..................... 184
6.3.2
Time Translation Symmetry
...................... 185
6.3.3
Spatial Rotation Symmetry
...................... 185
6.4
Lie Groups and Their Generators
....................... 188
6.5
Examples of Lie Group
............................. 191
6.5.1
Proper Rotation Group
Д(3)
(or Special Orthogonal Group
50(3)) 191
6.5.2
The SU(2) Group
............................ 193
6.5.3
Isospin and SU(2) Symmetry
..................... 194
Appendix 6A1: Groups and Representations
.................... 199
ANGULAR MOMENTUM IN QUANTUM MECHANICS
203
7.1
Introduction
................................... 203
7.2
Raising and Lowering Operators
....................... 206
7.3
Matrix Representation of Angular Momentum Operators
.......... 208
7.4
Matrix Representation of Eigenstates of Angular Momentum
....... 209
7.5
Coordinate Representation of Angular Momentum Operators and States
. 212
7.6
General Rotation Group and Rotation Matrices
............... 214
7.6.1
Rotation Matrices
........................... 217
7.7
Coupling of Two Angular Momenta
...................... 218
7.8
Properties of Clebsch-Gordan Coefficients
.................. 219
7.8.1
The Vector Model of the Atom
.................... 221
7.8.2
Projection Theorem for Vector Operators
.............. 221
7.9
Coupling of Three Angular Momenta
..................... 227
7.10
Coupling of Four Angular Momenta (L
-
S
and
j— j
Coupling)
..... 228
APPROXIMATION METHODS
235
8.1
Introduction
................................... 235
8.2
Non-degenerate Time-independent Perturbation Theory
.......... 236
8.3
Time-independent Degenerate Perturbation Theory
............. 242
8.4
The
Zeeman
Effect
............................... 249
8.5
WKBJ Approximation
............................. 254
8.6
Particle in a Potential Well
.......................... 262
8.7
Application of WKBJ Approximation to
α
-decay ..............
264
8.8
The Variational Method
............................ 267
8.9
The Problem of the Hydrogen Molecule
................... 270
8.10
System of
n
Identical Particles: Symmetric and Anti-symmetric States
. . 274
8.11
Excited States of the Helium Atom
...................... 278
8.12
Statistical (Thomas-Fermi) Model of the Atom
............... 280
8.13
Hartree s Self-consistent Field Method for Multi-electron Atoms
...... 281
8.14
Hartree-Fock Equations
............................ 285
8.15
Occupation Number Representation
..................... 290
QUANTUM THEORY OF SCATTERING
299
9.1
Introduction
................................... 299
9.2
Laboratory and Center-of-mass (CM) Reference Frames
.......... 300
9.2.1
Cross-sections in the CM and Laboratory Frames
.......... 302
9.3
Scattering Equation and the Scattering Amplitude
............. 303
9.4
Partial Waves and Phase
Shifts
........................ 306
9.5
Calculation of
Phase
Shift
........................... 311
9.6
Phase
Shifts for Some Simple Potential Forms
................ 313
9.7
Scattering due to Coulomb Potential
..................... 320
9.8
The Integral Form of Scattering Equation
.................. 324
9.8.1
Scattering Amplitude
......................... 327
9.9
Lippmann-Schwinger Equation and the Transition Operator
........ 329
9.10
Born Expansion
................................. 332
9.10.1
Born Approximation
.......................... 332
9.10.2
Validity of Born Approximation
.................... 334
9.10.3
Born Approximation and the Method of Partial Waves
....... 337
Appendix 9A1: The Calculus of Residues
...................... 342
10
TIME-DEPENDENT PERTURBATION METHODS
351
10.1
Introduction
................................... 351
10.2
Perturbation Constant over an Interval of Time
............... 353
10.3
Harmonic Perturbation: Semi-classical Theory of Radiation
........ 358
10.4
Einstein Coefficients
.............................. 363
10.5
Multipole Transitions
.............................. 365
10.6
Electric
Dipole
Transitions in Atoms and Selection Rules
.......... 366
10.7
Photo-electric Effect
.............................. 368
10.8
Sudden and Adiabatic Approximations
.................... 369
10.9
Second Order Effects
.............................. 373
11
THE THREE-BODY PROBLEM
377
11.1
Introduction
................................... 377
11.2
Eyges Approach
................................. 377
11.3
Mitra s Approach
................................ 381
11.4
Faddeev s Approach
.............................. 385
11.5
Faddeev Equations in Momentum Representation
.............. 391
11.6
Faddeev Equations for a Three-body Bound System
............ 393
11.7
Alt,
Grassberger
and Sandhas (AGS) Equations
............... 396
12
RELATIVISTIC QUANTUM MECHANICS
403
12.1
Introduction
................................... 403
12.2
Dirac Equation
................................. 405
12.3
Spin of the Electron
.............................. 408
12.4
Free Particle (Plane Wave) Solutions of Dirac Equation
.......... 409
12.5
Dirac Equation for a Zero Mass Particle
................... 413
12.6 Zitterbewegung
and Negative Energy Solutions
............... 415
12.7
Dirac Equation for an Electron in an Electromagnetic Field
........ 417
12.8
Invariance
of Dirac Equation
......................... 422
12.9
Dirac Bilinear Covariants
........................... 427
12.10
Dirac Electron in a Spherically Symmetric Potential
............ 428
12.11
Charge Conjugation. Parity and Time Reversal
Invariance
......... 436
Appendix 12A1: Theory of Special Relativity
.................... 445
12A1.1
Lorentz
Transformation
........................ 445
12
Al.
2
Minkowski Space-Time Continuum
.................. 448
12A1.3 Four-vectors in Relativistic Mechanics
................ 450
12A1.4 Covariant Form of Maxwell s Equations
............... 452
13
QUANTIZATION OF RADIATION FIELD
455
13.1
Introduction
................................... 455
13.2
Radiation Field as a Swarm of Oscillators
.................. 455
13.3
Quantization of Radiation Field
........................ 459
13.4
Interaction of Matter with Quantized Radiation Field
........... 462
13.5
Applications
................................... 466
13.6
Atomic Level Shift: Lamb-Retherford Shift
................. 476
13.7
Compton Scattering
.............................. 482
Appendix 13A1: Electromagnetic Field in Coulomb Gauge
............ 497
14
SECOND QUANTIZATION
501
14.1
Introduction
................................... 501
14.2
Classical Concept of Field
........................... 502
14.3
Analogy of Field and Particle Mechanics
................... 504
14.4
Field Equations from Lagrangian Density
.................. 507
14.4.1
Electromagnetic Field
......................... 507
14.4.2
Klein-Gordon Field (Real and Complex)
............... 508
14.4.3
Dirac Field
............................... 510
14.5
Quantization of a Real Scalar (KG) Field
.................. 511
14.6
Quantization of Complex Scalar (KG) Field
................. 514
14.7
Dirac Field and Its Quantization
....................... 519
14.8
Positron Operators and Spinors
........................ 522
14.8.1
Equations Satisfied by Electron and Positron Spinors
........ 524
14.8.2
Projection Operators
.......................... 525
14.8.3
Electron Vacuum
............................ 527
14.9
Interacting Fields and the Covariant Perturbation Theory
......... 527
14.9.1
U
Matrix
................................ 529
14.9.2
S
Matrix and Iterative Expansion of
S
Operator
.......... 531
14.9.3
Time-ordered Operator Product in Terms of Normal Constituents
532
14.10
Second Order Processes in Electrodynamics
................. 534
14.10.1
Feynman Diagrams
........................... 536
14.11
Amplitude for Compton Scattering
...................... 540
14.12
Feynman Graphs
................................ 545
14.12.1
Compton Scattering Amplitude Using Feynman Rules
....... 546
14.12.2
Electron-positron (e~e+) Pair Annihilation
............. 547
14.12.3
Two-photon Annihilation Leading to (e~e+) Pair Creation
.... 549
14.12.4 Möller
(e
є )
Scattering
....................... 550
14.12.5
Bhabha
(e~e+)
Scattering
....................... 550
14.13
Calculation of the Cross-section of Compton Scattering
.......... 551
14.14
Cross-sections for Other Electromagnetic Processes
............. 557
14.14.1
Electron-Positron Pair Annihilation (Electron at Rest)
....... 557
14.14.2 Möller
(Ce-)
and Bhabha
(e~c+)
Scattering
............ 558
Appendix
14
Al:
Calculus of Variation and
Euler-
Lagrange
Equations
...... 564
Appendix 14A2: Functional and Functional Derivatives
............. 567
Appendix 14A3: Interaction of the Electron and Radiation Fields
........ 569
Appendix 14A4: On the Convergence of Iterative Expansion of the
5
Operator
. 570
15
EPILOGUE
573
15.1
Introduction
................................... 573
15.2
EPR
Gedanken Experiment .......................... 574
15.3 Einstein-Podolsky-Rosen-Bohm Gedanken Experiment ........... 577
15.4
Theory of Hidden Variables and Bell s Inequality
.............. 579
15.5
Clauser-Horne Form of Bell s Inequality and Its Violation in Two-photon
Correlation Experiments
............................ 584
General References
.................................. 591
Index
593
|
| adam_txt |
Contents
Preface
. xiii
Acknowledgments
. xv
1
NEED FOR QUANTUM MECHANICS AND ITS PHYSICAL BASIS
1
1.1
Inadequacy of Classical Description for Small Systems
. 1
1.1.1
Planck's Formula for Energy Distribution in Black-body Radiation
1
1.1.2 de Broglie
Relation and Wave Nature of Material Particles
. 2
1.1.3
The Photo-electric Effect
. 3
1.1.4
The Compton Effect
. 4
1.1.5 Ritz
Combination Principle
. 6
1.2
Basis of Quantum Mechanics
. 9
1.2.1
Principle of Superposition of States
. 9
1.2.2 Heisenberg
Uncertainty Relations
. 12
1.3
Representation of States
. 14
1.4
Dual Vectors: Bra and
Ket
Vectors
. 15
1.5
Linear Operators
. 15
1.5.1
Properties of a Linear Operator
. 16
1.6
Adjoint of a Linear Operator
. 16
1.7
Eigenvalues and Eigenvectors of a Linear Operator
. 18
1.8
Physical Interpretation
. 20
1.8.1
Physical Interpretation of Eigenstates and Eigenvalues
. 20
1.8.2
Physical Meaning of the Orthogonality of States
. 21
1.9
Observables
and Completeness Criterion
. 21
1.10
Commutativity and Compatibility of
Observables
. 23
1.11
Position and Momentum Commutation Relations
. 24
1.12
Commutation Relation and the Uncertainty Product
. 26
Appendix 1A1: Basic Concepts in Classical Mechanics
. 31
1A1.1
Lagrange
Equations of Motion
. 31
1A1.2 Classical Dynamical Variables
. 32
2
REPRESENTATION THEORY
35
2.1
Meaning of Representation
. 35
2.2
How to Set up a Representation
. 35
2.3
Representatives of a Linear Operator
. 37
2.4
Change of Representation
. 40
2.5
Coordinate Representation
. 43
2.5.1
Physical Interpretation of the Wave Function
. 44
2.6
Replacement of Momentum Observable
ρ
by —ifijp
. 45
2.7
Integral Representation of Dirac Bracket
(A2¡
F
\A\)
. 50
2.8
The Momentum Representation
. 52
2.8.1
Physical Interpretation of
Φ(ρι
.p2
, ··
-p¡)
. 52
2.9
Dirac Delta Function
. 53
2.9.1
Three-dimensional Delta Function
. 55
2.9.2
Normalization of a Plane Wave
. 56
2.10
Relation between the Coordinate and Momentum Representations
. 56
EQUATIONS OF MOTION
67
3.1 Schrödinger
Equation of Motion
. 67
3.2 Schrödinger
Equation in the Coordinate Representation
. 69
3.3
Equation of Continuity
. 70
3.4
Stationary States
. 71
3.5
Time-independent
Schrödinger
Equation in the Coordinate Representation
72
3.6
Time-independent
Schrödinger
Equation in the Momentum Representation
74
3.6.1
Two-body Bound State Problem (in Momentum Representation) for
Non-local Separable Potential
. 76
3.7
Time-independent
Schrödinger
Equation in Matrix Form
. 77
3.8
The
Heisenberg
Picture
. 79
3.9
The Interaction Picture
. 81
Appendix 3A1: Matrices
. 86
3A1.1 Characteristic Equation of a Matrix
. 86
3A1.2 Similarity (and Unitary) Transformation of Matrices
. 87
3A1.3 Diagonalization of a Matrix
. 87
PROBLEMS OF ONE-DIMENSIONAL POTENTIAL BARRIERS
89
4.1
Motion of a Particle across a Potential Step
. 90
4.2
Passage of a Particle through a Potential Barrier of Finite Extent
. 94
4.3
Tunneling of a Particle through a Potential Barrier
. 99
4.4
Bound States in a One-dimensional Square Potential Well
. 103
4.5
Motion of a Particle in a Periodic Potential
. 107
BOUND STATES OF SIMPLE SYSTEMS
115
5.1
Introduction
. 115
5.2
Motion of a Particle in a Box
. 115
5.2.1
Density of States
. 117
5.3
Simple Harmonic Oscillator
. 118
5.4
Operator Formulation of the Simple Harmonic Oscillator Problem
. 122
5.4.1
Physical Meaning of the Operators
ã
and
â^
. 123
5.4.2
Occupation Number Representation (ONR)
. 125
5.5
Bound State of a Two-particle System with Central Interaction
. 126
5.6
Bound States of Hydrogen (or Hydrogen-like) Atoms
. 131
5.7
The
Deuteron
Problem
. 137
5.8
Energy Levels in a Three-dimensional Square Well: General Case
. 144
5.9
Energy Levels in an
Isotropie
Harmonic Potential Well
. 147
Appendix 5A1: Special Functions
. 156
5A1.1 Legendre and Associated Legendre Equations
. 156
5A1.2 Spherical Harmonics
. 159
5A1.3 Laguerre and Associated Laguerre Equations
. 162
5A1.4 Hermite Equation
. 166
5A1.5 Bessel Equation
. 169
Appendix 5A2: Orthogonal Curvilinear Coordinate Systems
. 174
5A2.1 Spherical Polar Coordinates
. 174
5A2.2 Cylindrical Coordinates
. 175
5A2.3 Parabolic Coordinates
. 177
5A2.4 General Features of Orthogonal Curvilinear System of Coordinates
178
SYMMETRIES AND CONSERVATION LAWS
181
6.1
Symmetries and Their Group Properties
. 181
6.2
Symmetries in a Quantum Mechanical System
. 182
6.3
Basic Symmetry Groups of the Hamiltonian and Conservation Laws
. . . 183
6.3.1
Space Translation Symmetry
. 184
6.3.2
Time Translation Symmetry
. 185
6.3.3
Spatial Rotation Symmetry
. 185
6.4
Lie Groups and Their Generators
. 188
6.5
Examples of Lie Group
. 191
6.5.1
Proper Rotation Group
Д(3)
(or Special Orthogonal Group
50(3)) 191
6.5.2
The SU(2) Group
. 193
6.5.3
Isospin and SU(2) Symmetry
. 194
Appendix 6A1: Groups and Representations
. 199
ANGULAR MOMENTUM IN QUANTUM MECHANICS
203
7.1
Introduction
. 203
7.2
Raising and Lowering Operators
. 206
7.3
Matrix Representation of Angular Momentum Operators
. 208
7.4
Matrix Representation of Eigenstates of Angular Momentum
. 209
7.5
Coordinate Representation of Angular Momentum Operators and States
. 212
7.6
General Rotation Group and Rotation Matrices
. 214
7.6.1
Rotation Matrices
. 217
7.7
Coupling of Two Angular Momenta
. 218
7.8
Properties of Clebsch-Gordan Coefficients
. 219
7.8.1
The Vector Model of the Atom
. 221
7.8.2
Projection Theorem for Vector Operators
. 221
7.9
Coupling of Three Angular Momenta
. 227
7.10
Coupling of Four Angular Momenta (L
-
S
and
j— j
Coupling)
. 228
APPROXIMATION METHODS
235
8.1
Introduction
. 235
8.2
Non-degenerate Time-independent Perturbation Theory
. 236
8.3
Time-independent Degenerate Perturbation Theory
. 242
8.4
The
Zeeman
Effect
. 249
8.5
WKBJ Approximation
. 254
8.6
Particle in a Potential Well
. 262
8.7
Application of WKBJ Approximation to
α
-decay .
264
8.8
The Variational Method
. 267
8.9
The Problem of the Hydrogen Molecule
. 270
8.10
System of
n
Identical Particles: Symmetric and Anti-symmetric States
. . 274
8.11
Excited States of the Helium Atom
. 278
8.12
Statistical (Thomas-Fermi) Model of the Atom
. 280
8.13
Hartree's Self-consistent Field Method for Multi-electron Atoms
. 281
8.14
Hartree-Fock Equations
. 285
8.15
Occupation Number Representation
. 290
QUANTUM THEORY OF SCATTERING
299
9.1
Introduction
. 299
9.2
Laboratory and Center-of-mass (CM) Reference Frames
. 300
9.2.1
Cross-sections in the CM and Laboratory Frames
. 302
9.3
Scattering Equation and the Scattering Amplitude
. 303
9.4
Partial Waves and Phase
Shifts
. 306
9.5
Calculation of
Phase
Shift
. 311
9.6
Phase
Shifts for Some Simple Potential Forms
. 313
9.7
Scattering due to Coulomb Potential
. 320
9.8
The Integral Form of Scattering Equation
. 324
9.8.1
Scattering Amplitude
. 327
9.9
Lippmann-Schwinger Equation and the Transition Operator
. 329
9.10
Born Expansion
. 332
9.10.1
Born Approximation
. 332
9.10.2
Validity of Born Approximation
. 334
9.10.3
Born Approximation and the Method of Partial Waves
. 337
Appendix 9A1: The Calculus of Residues
. 342
10
TIME-DEPENDENT PERTURBATION METHODS
351
10.1
Introduction
. 351
10.2
Perturbation Constant over an Interval of Time
. 353
10.3
Harmonic Perturbation: Semi-classical Theory of Radiation
. 358
10.4
Einstein Coefficients
. 363
10.5
Multipole Transitions
. 365
10.6
Electric
Dipole
Transitions in Atoms and Selection Rules
. 366
10.7
Photo-electric Effect
. 368
10.8
Sudden and Adiabatic Approximations
. 369
10.9
Second Order Effects
. 373
11
THE THREE-BODY PROBLEM
377
11.1
Introduction
. 377
11.2
Eyges Approach
. 377
11.3
Mitra's Approach
. 381
11.4
Faddeev's Approach
. 385
11.5
Faddeev Equations in Momentum Representation
. 391
11.6
Faddeev Equations for a Three-body Bound System
. 393
11.7
Alt,
Grassberger
and Sandhas (AGS) Equations
. 396
12
RELATIVISTIC QUANTUM MECHANICS
403
12.1
Introduction
. 403
12.2
Dirac Equation
. 405
12.3
Spin of the Electron
. 408
12.4
Free Particle (Plane Wave) Solutions of Dirac Equation
. 409
12.5
Dirac Equation for a Zero Mass Particle
. 413
12.6 Zitterbewegung
and Negative Energy Solutions
. 415
12.7
Dirac Equation for an Electron in an Electromagnetic Field
. 417
12.8
Invariance
of Dirac Equation
. 422
12.9
Dirac Bilinear Covariants
. 427
12.10
Dirac Electron in a Spherically Symmetric Potential
. 428
12.11
Charge Conjugation. Parity and Time Reversal
Invariance
. 436
Appendix 12A1: Theory of Special Relativity
. 445
12A1.1
Lorentz
Transformation
. 445
12
Al.
2
Minkowski Space-Time Continuum
. 448
12A1.3 Four-vectors in Relativistic Mechanics
. 450
12A1.4 Covariant Form of Maxwell's Equations
. 452
13
QUANTIZATION OF RADIATION FIELD
455
13.1
Introduction
. 455
13.2
Radiation Field as a Swarm of Oscillators
. 455
13.3
Quantization of Radiation Field
. 459
13.4
Interaction of Matter with Quantized Radiation Field
. 462
13.5
Applications
. 466
13.6
Atomic Level Shift: Lamb-Retherford Shift
. 476
13.7
Compton Scattering
. 482
Appendix 13A1: Electromagnetic Field in Coulomb Gauge
. 497
14
SECOND QUANTIZATION
501
14.1
Introduction
. 501
14.2
Classical Concept of Field
. 502
14.3
Analogy of Field and Particle Mechanics
. 504
14.4
Field Equations from Lagrangian Density
. 507
14.4.1
Electromagnetic Field
. 507
14.4.2
Klein-Gordon Field (Real and Complex)
. 508
14.4.3
Dirac Field
. 510
14.5
Quantization of a Real Scalar (KG) Field
. 511
14.6
Quantization of Complex Scalar (KG) Field
. 514
14.7
Dirac Field and Its Quantization
. 519
14.8
Positron Operators and Spinors
. 522
14.8.1
Equations Satisfied by Electron and Positron Spinors
. 524
14.8.2
Projection Operators
. 525
14.8.3
Electron Vacuum
. 527
14.9
Interacting Fields and the Covariant Perturbation Theory
. 527
14.9.1
U
Matrix
. 529
14.9.2
S
Matrix and Iterative Expansion of
S
Operator
. 531
14.9.3
Time-ordered Operator Product in Terms of Normal Constituents
532
14.10
Second Order Processes in Electrodynamics
. 534
14.10.1
Feynman Diagrams
. 536
14.11
Amplitude for Compton Scattering
. 540
14.12
Feynman Graphs
. 545
14.12.1
Compton Scattering Amplitude Using Feynman Rules
. 546
14.12.2
Electron-positron (e~e+) Pair Annihilation
. 547
14.12.3
Two-photon Annihilation Leading to (e~e+) Pair Creation
. 549
14.12.4 Möller
(e"
є")
Scattering
. 550
14.12.5
Bhabha
(e~e+)
Scattering
. 550
14.13
Calculation of the Cross-section of Compton Scattering
. 551
14.14
Cross-sections for Other Electromagnetic Processes
. 557
14.14.1
Electron-Positron Pair Annihilation (Electron at Rest)
. 557
14.14.2 Möller
(Ce-)
and Bhabha
(e~c+)
Scattering
. 558
Appendix
14
Al:
Calculus of Variation and
Euler-
Lagrange
Equations
. 564
Appendix 14A2: Functional and Functional Derivatives
. 567
Appendix 14A3: Interaction of the Electron and Radiation Fields
. 569
Appendix 14A4: On the Convergence of Iterative Expansion of the
5
Operator
. 570
15
EPILOGUE
573
15.1
Introduction
. 573
15.2
EPR
Gedanken Experiment . 574
15.3 Einstein-Podolsky-Rosen-Bohm Gedanken Experiment . 577
15.4
Theory of Hidden Variables and Bell's Inequality
. 579
15.5
Clauser-Horne Form of Bell's Inequality and Its Violation in Two-photon
Correlation Experiments
. 584
General References
. 591
Index
593 |
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| author | Mathur, Vishnu Swarup Singh, Surendra |
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| language | English |
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| spelling | Mathur, Vishnu Swarup Verfasser aut Concepts in quantum mechanics Vishnu Swarup Mathur ; Surendra Singh Boca Raton, Fla. [u.a.] CRC Press, Chapman & Hall 2009 XV, 598 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier CRC series in pure and applied physics Quantum theory Quantentheorie Quantenmechanik (DE-588)4047989-4 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Quantenmechanik (DE-588)4047989-4 s DE-604 Singh, Surendra Verfasser aut Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016761944&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Mathur, Vishnu Swarup Singh, Surendra Concepts in quantum mechanics Quantum theory Quantentheorie Quantenmechanik (DE-588)4047989-4 gnd |
| subject_GND | (DE-588)4047989-4 (DE-588)4123623-3 |
| title | Concepts in quantum mechanics |
| title_auth | Concepts in quantum mechanics |
| title_exact_search | Concepts in quantum mechanics |
| title_exact_search_txtP | Concepts in quantum mechanics |
| title_full | Concepts in quantum mechanics Vishnu Swarup Mathur ; Surendra Singh |
| title_fullStr | Concepts in quantum mechanics Vishnu Swarup Mathur ; Surendra Singh |
| title_full_unstemmed | Concepts in quantum mechanics Vishnu Swarup Mathur ; Surendra Singh |
| title_short | Concepts in quantum mechanics |
| title_sort | concepts in quantum mechanics |
| topic | Quantum theory Quantentheorie Quantenmechanik (DE-588)4047989-4 gnd |
| topic_facet | Quantum theory Quantentheorie Quantenmechanik Lehrbuch |
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