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Journal of Mathematical Physics

Relationships among Generalized Phase-Space Distributions

George C. Summerfield, Paul F. Zweifel

Published Feb 1, 1969• 18 citations
Publisher

Abstract

The generalized phase-space distributions, including the Wigner distribution, are presented in terms of expected values of generating operators. A generalization of the Weyl correspondence is obtained to provide expressions for generalized Wigner equivalents. Finally, rather simple relationships are obtained connecting the generalized phase-space distributions to the Wigner distribution; and similar relationships are obtained for the generalized Wigner equivalents. In particular, it appears that, among the class considered, there is no reason to use any distribution other than the Wigner for performing any calculations.

DOI
10.1063/1.1664837

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•Jan 1, 2014•2 citations
The generalized phase-space distributions, including the Wigner distribution, are presented in terms of expected values of generating operators. A generalization of the Weyl correspondence is obtained to provide expressions for generalized Wigner equivalents. Finally, rather simple relationships are obtained connecting the generalized phase-space distributions to the Wigner distribution; and similar relationships are obtained for the generalized Wigner equivalents. In particular, it appears that, among the class considered, there is no reason to use any distribution other than the Wigner for performing any calculations.
A set of quasi-probability distribution functions which give the correct quantum mechanical marginal distributions of position and momentum is studied. The phase-space distribution does not have to be bilinear in the state function. The Wigner distribution is a special case. A general relationship between the phase-space distribution functions and the rule of associating classical quantities to quantum mechanical operators is derived. This allows the writing of correspondence rules at will,of which the ones presently known are particular cases. The dynamics and other properties of the generalized phase-space distribution are considered.
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In contrast to classical physics, the language of quantum mechanics involves operators and wave functions (or, more generally, density operators). However, in 1932, Wigner formulated quantum mechanics in terms of a distribution function W(q,p)W(q,p)W(q,p), the marginals of which yield the correct quantum probabilities for qqq and ppp separately \cite{wigner}. Its usefulness stems from the fact that it provides a re-expression of quantum mechanics in terms of classical concepts so that quantum mechanical expectation values are now expressed as averages over phase-space distribution functions. In other words, statistical information is transferred from the density operator to a quasi-classical (distribution) function.
Abstract Coherent states, in Wigner representation of quantum mechanics which is classical in structure with Wigner function playing the role of classical phase space probability distribution, and standard quantum mechanical expression for average values of corresponding Weyl quantum operators, general expression for matrix elements of the operator corresponding to phase of the oscillator by Weyl procedure, are obtained in the |n〉 basis. The general expression for matrix elements in the same basis of any other operator is also obtained. As the only mathematical technique necessary in the proposed procedure is a simple change of variables to polar coordinates in the corresponding integrals, this way to introduce Weyl phase operator, and some other operators in Weyl quantization, greatly simplifies necessary derivations and calculations.
This thesis describes quantum mechanics in the phase space formulation. We introduce quantization and in particular the Weyl quantization. We study a general class of phase space distribution functions on phase space. The Wigner distribution function is one such distribution function. The Wigner distribution function in general attains negative values and thus can not be interpreted as a real probability density, as opposed to for example the Husimi distribution function. The Husimi distribution however does not yield the correct marginal distribution functions known from quantum mechanics. Properties of the Wigner and Husimi distribution function are studied to more extent. We calculate the Wigner and Husimi distribution function for the energy eigenstates of a particle trapped in a box. We then look at the semi classical limit for this example. The time evolution of Wigner functions are studied by making use of the Moyal bracket. The Moyal bracket and the Poisson bracket are compared in the classical limit. The phase space formulation of quantum mechanics has as advantage that classical concepts can be studied and compared to quantum mechanics. For certain quantum mechanical systems the time evolution of Wigner distribution functions becomes equivalent to the classical time evolution stated in the exact Egerov theorem. Another advantage of using Wigner functions is when one is interested in systems involving mixed states. A disadvantage of the phase space formulation is that for most problems it quickly loses its simplicity and becomes hard to calculate.