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Communications in Theoretical Physics

Generalized Wigner Functions for Damped Systems in Deformation Quantization

Tai-Hua Heng, Sicong Jing

Published Aug 1, 2007• 0 citations
Publisher

Abstract

Quantization of damped systems usually gives rise to complex spectra and corresponding resonant states, which do not belong to the Hilbert space. Therefore, the standard form of calculating Wigner function (WF) does not work for these systems. In this paper we show that in order to let WF satisfy a ∗-genvalue equation for the damped systems, one must modify its standard form slightly, and this modification exactly coincides with the results derived from a ∗-Exponential expansion in deformation quantization.

DOI
10.1088/0253-6102/48/2/011

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Quantization of damped systems usually gives rise to complex spectra and corresponding resonant states, which do not belong to the Hilbert space. Therefore, the standard form of calculating Wigner function (WF) does not work for these systems. In this paper we show that in order to let WF satisfy a ∗-genvalue equation for the damped systems, one must modify its standard form slightly, and this modification exactly coincides with the results derived from a ∗-Exponential expansion in deformation quantization.
An adaptation of the Wentzel–Kramers–Brilluoin method in the deformation quantization formalism is presented with the aim to obtain an approximate technique of solving the eigenvalue problem for energy in the phase space quantum approach. A relationship between the phase σ(r→) of a wave function expiħσ(r→) and its respective Wigner function is derived. Formulas to calculate the Wigner function of a product and of a superposition of wave functions are proposed. Properties of a Wigner function of interfering states are also investigated. Examples of this quasi–classical approximation in deformation quantization are analysed. A strict form of the Wigner function for states represented by tempered generalised functions has been derived. Wigner functions of unbound states in the Poeschl–Teller potential have been found.
We examine the deformation quantization of a single particle moving in one dimension (i) in the presence of an infinite potential wall, (ii) confined by an infinite square well, and (iii) bound by a delta function potential energy. In deformation quantization, considered as an autonomous formulation of quantum mechanics, the Wigner function of stationary states must be found by solving the so-called ∗\\*∗-genvalue (``stargenvalue'') equation for the Hamiltonian. For the cases considered here, this pseudo-differential equation is difficult to solve directly, without an ad hoc modification of the potential. Here we treat the infinite wall as the limit of a solvable exponential potential. Before the limit is taken, the corresponding ∗\\*∗-genvalue equation involves the Wigner function at momenta translated by imaginary amounts. We show that it can be converted to a partial differential equation, however, with a well-defined limit. We demonstrate that the Wigner functions calculated from the standard Schr\\"odinger wave functions satisfy the resulting new equation. Finally, we show how our results may be adapted to allow for the presence of another, non-singular part in the potential.