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Physical Review A

Time dependence of a general class of quantum distribution functions

Robert F. O’Connell, Lipo Wang, H. A. Williams

Published Nov 1, 1984• 30 citations
Publisher

Abstract

We derive an explicit analytic expression for the time development of a generalized class of quantum distribution functions, which include as special cases the Wigner, the normal, and the antinormal distributions. We find that the simplest result is obtained in the case of the Wigner distribution.
DOI
10.1103/physreva.30.2187

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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.
A class of positive quantum phase-space distributions based on antinormal ordering of the squeezed-photon annihilation and creation operators is introduced. This class of distributions, referred to as the generalized antinormally ordered distributions, is shown to be generated when the Wigner distribution is smoothed with a squeezed Gaussian wave packet. The distribution function associated with generalized antinormal ordering can thus be identified as the Husimi distribution function. The usefulness of the distributions in studies of the quantum particle dynamics is illustrated with simple examples.
Abstract In this work, we develop a phase space formulation of quantum mechanics to investigate the inversely quadratic Hellmann-Kratzer (IQHK) potential, with the aim of simultaneously constructing the Wigner distribution and the corresponding characteristic functions. By employing Weyl transformations, we establish two independent computational approaches, each yielding explicit and generalized analytical expressions for higher-order moments and momentum. In addition, we derive a generalized analytical formulation of the momentum operator and establish a corresponding generalized expression of Heisenberg’s uncertainty principle, explicitly dependent on the quantum numbers n and L. This constitutes a novel and complementary contribution to the study of the IQHK potential. Moreover, our approach enables the identification of a dimensionless principal parameter β in phase-space transformations, which further refines the generalized analytical expressions for the energy levels En,L. These energy levels are numerically evaluated and exhibit excellent agreement with previously published results for all special cases of the IQHK potential across various diatomic molecular systems. Furthermore, we verify that our results are consistent with the canonical Heisenberg-Born-Jordan-Dirac commutation relations and Heisenberg’s uncertainty principle, thereby confirming the robustness and reliability of the proposed methodology within this framework.