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Chinese Physics Letters

Modified Form of Wigner Functions for Non-Hamiltonian Systems

Tai-Hua Heng, Ping Li, Sicong Jing

Published Feb 19, 2007• 4 citations
Publisher

Abstract

Quantization of non-Hamiltonian systems (such as damped systems) often gives rise to complex spectra and corresponding resonant states, therefore a standard form calculating Wigner functions cannot lead to static quasi-probability distribution functions. We show that a modified form of the Wigner functions satisfies a ∗-genvalue equation and can be derived from deformation quantization for such systems.

DOI
10.1088/0256-307x/24/3/002

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Quantization of non-Hamiltonian systems (such as damped systems) often gives rise to complex spectra and corresponding resonant states, therefore a standard form calculating Wigner functions cannot lead to static quasi-probability distribution functions. We show that a modified form of the Wigner functions satisfies a ∗-genvalue equation and can be derived from deformation quantization for such systems.
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.