A Concise Treatise on Quantum Mechanics in Phase Space

This is a text on quantum mechanics formulated simultaneously in terms of position and momentum, i.e. in phase space. It is written at an introductory level, drawing on the remarkable history of the subject for inspiration and motivation. Wigner functions — density matrices in a special Weyl representation — and star products are the cornerstones of the formalism.

The resulting framework is a rich source of physical intuition. It has been used to describe transport in quantum optics, structure and dynamics in nuclear physics, chaos, and decoherence in quantum computing. It is also of importance 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 way to formulate and understand quantum mechanics, independent of the conventional Hilbert space or path integral approaches to the subject.

In this logically complete and self-standing formulation, one need not choose sides between coordinate or momentum space variables. It works in full phase space, accommodating the uncertainty principle; and it offers unique insights into the classical limit of quantum theory. The observables in this formulation are c-number functions in phase space instead of operators, with the same interpretation as their classical counterparts, only composed together in novel algebraic ways using star products.

This treatise provides an introductory overview and supplementary material suitable for an advanced undergraduate or a beginning graduate course in quantum mechanics.

Sample Chapter(s)
Historical Survey (846 KB)

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Contents:Historical SurveyIntroductionThe Wigner FunctionSolving for the Wigner FunctionThe Uncertainty PrincipleEhrenfest's TheoremIllustration: The Harmonic OscillatorTime EvolutionNon-Diagonal Wigner FunctionsStationary Perturbation TheoryPropagators and Canonical TransformationsThe Weyl CorrespondenceAlternate Rules of AssociationThe Groenewold–van Hove Theorem; The Uniqueness of MBs and *-ProductsAdvanced Topic: Quasi-Hermitian Quantum SystemsOmitted MiscellanySynopses of Selected Papers
Readership: Advanced undergraduates and graduates studying quantum mechanics.