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Journal of Physics Conference Series

Wigner distributions in quantum mechanics

Elisa Ercolessi, Giuseppe Marmo, G. Morandi, Narasimhaiengar Mukunda

Published Nov 1, 2007• 5 citations
Publisher

Abstract

The Weyl-Wigner description of quantum mechanical operators and states in classical phase-space language is well known for Cartesian systems. We describe a new approach based on ideas of Dirac which leads to the same results but with interesting additional insights. A way to set up Wigner distributions in an interesting non-Cartesian case, when the configuration space is a compact connected Lie group, is outlined. Both these methods are adapted to quantum systems with finite-dimensional Hilbert spaces, and the results are contrasted.

DOI
10.1088/1742-6596/87/1/012010

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The Weyl-Wigner description of quantum mechanical operators and states in classical phase-space language is well known for Cartesian systems. We describe a new approach based on ideas of Dirac which leads to the same results but with interesting additional insights. A way to set up Wigner distributions in an interesting non-Cartesian case, when the configuration space is a compact connected Lie group, is outlined. Both these methods are adapted to quantum systems with finite-dimensional Hilbert spaces, and the results are contrasted.
We consider the problem of setting up the Wigner distribution for states of a quantum system whose configuration space is a Lie group. The basic properties of Wigner distributions in the familiar Cartesian case are systematically generalized to accommodate new features which arise when the configuration space changes from n-dimensional Euclidean space Rn to a Lie group G. The notion of canonical momentum is carefully analyzed, and the meanings of marginal probability distributions and their recovery from the Wigner distribution are clarified. For the case of compact G an explicit definition of the Wigner distribution is proposed, possessing all the required properties. Geodesic curves in G which help introduce a notion of the midpoint of two group elements play a central role in the construction.
The Wigner-Weyl isomorphism for quantum mechanics on a compact simple Lie group GGG is developed in detail. Several New features are shown to arise which have no counterparts in the familiar Cartesian case. Notable among these is the notion of a `semiquantised phase space', a structure on which the Weyl symbols of operators turn out to be naturally defined and, figuratively speaking, located midway between the classical phase space T∗GT^*GT∗G and the Hilbert space of square integrable functions on GGG. General expressions for the star product for Weyl symbols are presented and explicitly worked out for the angle-angular momentum case.
Drawing inspiration from Dirac's work on functions of non-commuting observables, we develop an approach to phase-space descriptions of operators and the Wigner–Weyl correspondence in quantum mechanics, complementary to standard formulations. This involves a two-step process: introducing phase-space descriptions based on placing position dependences to the left of momentum dependences (or the other way around); then carrying out a natural transformation to eliminate a kernel which appears in the expression for the trace of the product of two operators. The method works uniformly for both continuous Cartesian degrees of freedom and for systems with finite-dimensional state spaces. It is interesting that the kernel encountered is naturally expressible in terms of geometric phases, and its removal involves extracting its square root in a suitable manner.
A complete solution to the problem of setting up Wigner distribution for N-level quantum systems is presented. The scheme makes use of some of the ideas introduced by Dirac in the course of defining functions of noncommuting observables and works uniformly for all N. Further, the construction developed here has the virtue of being essentially input-free in that it merely requires finding a square root of a certain N^2 x N^2 complex symmetric matrix, a task which, as is shown, can always be accomplished analytically. As an illustration, the case of a single qubit is considered in some detail and it is shown that one recovers the result of Feynman and Wootters for this case without recourse to any auxiliary constructs.