Quantum Mechanics in Phase Space

Cosmas K. Zachos, David B. Fairlie, Thomas L. Curtright9812383840, 9789812383846, 9789812703507

Wigner’s quasi-probability distribution function in phase space is a special (Weyl) representation of the density matrix. It has been useful in describing quantum transport in quantum optics; nuclear physics; decoherence, quantum computing, and quantum chaos. It is also important in signal processing and the mathematics of algebraic deformation. A remarkable aspect of its internal logic, pioneered by Groenewold and Moyal, has only emerged in the last quarter-century: it furnishes a third, alternative, formulation of quantum mechanics, independent of the conventional Hilbert space, or path integral formulations.
In this logically complete and self-standing formulation, one need not choose sides – coordinate or momentum space. It works in full phase space, accommodating the uncertainty principle, and it offers unique insights into the classical limit of quantum theory. This invaluable book is a collection of the seminal papers on the formulation, with an introductory overview which provides a trail map for those papers; an extensive bibliography; and simple illustrations, suitable for applications to a broad range of physics problems. It can provide supplementary material for a beginning graduate course in quantum mechanics.

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Table of contents :
CONTENTS ……Page 6
Preface ……Page 8
1 Introduction ……Page 10
2 The Wigner Function ……Page 12
3 Solving for the Wigner Function ……Page 14
4 The Uncertainty Principle ……Page 18
5 Ehrenfests Theorem ……Page 19
6 Illustration: The Harmonic Oscillator ……Page 20
7 Time Evolution ……Page 22
8 Nondiagonal Wigner Functions ……Page 25
9 Stationary Perturbation Theory ……Page 26
10 Propagators ……Page 27
11 Canonical Transformations ……Page 28
12 The Weyl Correspondence ……Page 30
13 Alternate Rules of Association ……Page 33
14 The Groenewold-van Hove Theorem and the Uniqueness of MBs and *-Products ……Page 34
15 Omitted Miscellany ……Page 35
16 Selected Papers: Brief Historical Outline ……Page 37
References ……Page 40
List of Selected Papers ……Page 48
Index ……Page 52
Quantenmechanik und Gruppentheorie ……Page 54
Die Eiudeutigkeit der Schrodingerschen Operatoren ……Page 100
On the Quantum Correction For Thermodynamic Equilibrium ……Page 109
ON THE PRINCIPLES OF ELEMENTARY QUANTUM MECHANICS ……Page 120
QUANTUM MECHANICS AS A STATISTICAL THEORY ……Page 176
THE EXACT TRANSITION PROBABILITIES O F QUANTUM- MECHANICAL OSCILLATORS CALCULATED BY THE PHASE-SPACE METHOD ……Page 202
The Formulation of Quantum Mechanics in terms of Ensemble in Phase Space ……Page 211
Formulation of Quantum Mechanics Based on the Quasi-Probability Distribution Induced on Phase Space ……Page 244
The formulation of quantum mechanics in terms of phase space functions ……Page 253
A NON-NEGATIVE WIGNER-TYPE DISTRIBUTION ……Page 259
Wigner function as the expectation value of a parity operator ……Page 262
Deformation Theory and Quantization ……Page 264
Deformation Theory and Quantization II. Physical Applications ……Page 314
Wigner distribution functions and the representation of canonical transformations in quantum mechanics ……Page 355
Wigners phase space function and atomic structure ……Page 359
DISTRIBUTION FUNCTIONS IN PHYSICS: FUNDAMENTALS ……Page 378
Canonical transformation in quantum mechanics ……Page 425
Negative probability ……Page 435
EXISTENCE OF STAR-PRODUCTS AND OF FORMAL DEFORb4ATIONS OF THE POISSON LIE ALGEBRA OF ARBITRARY SYMPLECTIC MANIFOLDS ……Page 449
A SIMPLE GEOMETRICAL CONSTRUCTION OF DEFORMATION QUANTIZATION ……Page 459
Features of time-independent Wigner functions ……Page 485
NEGATIVE PROBABILITY AND UNCERTAINTY RELATIONS ……Page 499
Generating all Wigner functions ……Page 504
Modified spectral method in phase space: Calculation of the Wigner function. I. Fundamentals ……Page 524
Modified spectral method in phase space: Calculation of the Wigner function. II. Generalizations ……Page 542