Phase Space Picture of Quantum Mechanics: Group Theoretical Approach

✍ Scribed by Y. S. Kim, Marilyn E. Noz

• Publisher: World Scientific
• Year: 1991
• Tongue: English
• Leaves: 348
• Series: Lecture Notes in Physics Vol 40
• Category: Library

✦ Synopsis

Discusses the physical consequences of the symmetries of the Wigner function in phase space, for scientists and students who wish to study the basic principles of the phase-space picture of quantum mechanics and physical applications of the Wigner distribution functions, and for those who simply wish to study the physical applications of the Lorentz group.

✦ Table of Contents

Preface
Introduction
1 PHASE SPACE IN CLASSICAL MECHANICS
1.1 Hamiltonian Form of Classical Mechanics
1.2 Trajectories in Phase Space ..

1.3 Canonical Transformations . . . . . . . . 6
1.4 Coupled Harmonic Oscillators . . . . . . . 9
1.5 Group of Linear Canonical Transformations in Four~ Dimensional Phase
Space . . . . . . . . . . . . . 12
1.6 Poisson Brackets . . . . . . . 14
1.7 Distributions in Phase Space 15
2 FORMS OF QUANTUM MECHANICS
2.1 Schrodinger and Heisenberg Pictures . . .
2.2 Interaction Representation . . . . . . . . .
2.3 Density~Matrix Formulation of Quantum
2.4 Mixed States . . . . . . . . . . . . . . .
2.5 Density Matrix and Ensemble Average .
2.6 Time Dependence of the Density Matrix
Mechanics
3 WIGNER PHASE-SPACE DISTRIBUTION FUNCTIONS 37
3.1 Basic Properties of the Wigner Phase~Space Distribution Function 38
3.2 Time Dependence of the Wigner Function 40
3.3 Wave Packet Spreads . 42
3.4 Harmonic Oscillators . 45
3.5 Density Matrix . . . . 47
3.6 Measurable Quantities
3. 7 Early and Recent Applications
4 LINEAR CANONICAL TRANSFORMATIONS IN QUANTUM MECHANICS 57
4.1 Canonical Transformations in Two-Dimensional Phase Space 57
4.2 Linear Canonical Transformations in Quantum Mechanics . . 60
4.3 Wave Packet Spreads in Terms of Canonical Transformations 63
4.4 Harmonic Oscillators . . . . . . . . . . . . . . . . . . . . . . . 64
4.5 (2 + I)-Dimensional Lorentz Group . . . . . . . . . . . . . . 66
4.6 Canonical Transformations in Four-Dimensional Phase Space 69
4.7 The Schrodinger Picture of Two-Mode Canonical Transformations 72
4.8 (3 + 2)-Dimensional de Sitter Group . . . . . . . . . . . . . . . . . 73
5 COHERENT AND SQUEEZED STATES
5.1 Phase-Number Uncertainty Relation
5.2 Baker-Campbell-Hausdorff Relation
5.3 Coherent States of Light . . . . . . .
5.4 Symmetry Groups of Coherent States
5.5 Squeezed States . . . . . . . . . . . . .
5.6 Two-Mode Squeezed States ..... .
5.7 Density Matrix through Two-Mode Squeezed States

6 PHASE-SPACE PICTURE OF COHERENT AND SQUEEZED
STATES 99
6.1 Invariant Subgroups . . . . . 100
6.2 Coherent States . . . . . . . . 102
6.3 Single-Mode Squeezed States 105
6.4 Squeezed Vacuum ...... .
6.5 Expectation Values in terms of Vacuum Expectation Values
6.6 Overlapping Distribution Functions .
6. 7 Thomas Effect . . . . . . .
6.8 Two-Mode Squeezed States
6.9 Contraction of Phase Space
7 LORENTZ TRANSFORMATIONS
7.1 Group of Lorentz Transformations .
7.2 Little Groups of the Lorentz Group .
7.3 Massless Particles .......... .
7.4 Decomposition of Lorentz Transformations. 129
7.5 Analytic Continuation to the Little Groups for Massless and Imaginary-
Mass particles . . . . . . . . . . 131
7.6 Light-Cone Coordinate System . . . . 133
7.7 Localized Light Waves ........ .
7.8 Covariant Localization of Light Waves
7.9 Covariant Phase-Space Picture of Localized Light Waves .
7.10 Uncertainty Relations for Light Waves and for Photons
8 COVARIANT HARMONIC OSCILLATORS
8.1 Theory of the Poincare Group . . . . . . . . . .
8.2 Covariant Harmonic Oscillators . . . . . . . . . . . . . . . .
8.3 Irreducible Unitary Representations of the Poincare Group
8.4 C-number Time-Energy Uncertainty Relation ..... .
8.5 Dirac's Form of Relativistic Theory of "Atom" . . . . 157
8.6 Lorentz Transformations of Harmonic Oscillator Wave functions. 160
8.7 Covariant Phase-Space Picture of Harmonic Oscillators. 162
9 LORENTZ-SQUEEZED HADRONS
9.1 Quark Model ............ .
9.2 Hadronic Mass Spectra . . . . . . . .
9.3 Hadrons in the Relativistic Quark Model.
9.4 Form Factors of Nucleons ........ .
9.5 Phase-Space Picture of Overlapping Wave Functions
9.6 Feynman's Parton Picture . . . . . . . . . . . . . . . .
9.7 Experimental Observation of the Parton Distribution .
10 SPACE-TIME GEOMETRY OF EXTENDED PARTICLES 193
10.1 Two-Dimensional Euclidean Group and Cylindrical Group 195
10.2 Contractions of the Three-Dimensional Rotation Group 197
10.3 Three-Dimensional Geometry of the Little Groups . 200
10.4 Little Groups in the Light-Cone Coordinate System 202
10.5 Cylindrical Group and Gauge Transformations . . . 204
10.6 Little Groups for Relativistic Extended Particles . . 207
10.7 Lorentz Transformations and Hadronic Temperature 210
10.8 Decoherence and Entropy . . . . . . . . . . . . . . . . 213
A Reprinted Articles 217
A.1 E.P. Wigner, On the Quantum Correction for Thermodynamic Equilibrium
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
A.2 E.P. Wigner, On Unitary Representations of the Inhomogeneous Lorentz
Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
A.3 P.A.M. Dirac, Unitary Representations of the Lorentz Group . . 293
A.4 P.A.M. Dirac, A Remarkable Representation of the 3 + 2 de Sitter
Group ................................... 307
References
Index