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Communications in Theoretical Physics

A Note on Wigner Functions and *-Genvalue Equation

Sicong Jing, Bing-Sheng Lin

Published Apr 1, 2009• 0 citations
Publisher

Abstract

In deformation quantization, static Wigner functions obey functional *-genvalue equation, which is equivalent to time-independent Schrodinger equation in Hilbert space operator formalism of quantum mechanics. This equivalence is proved mostly for Hamiltonian with form Ĥ = (1/2)2 + V() [D. Fairlie, Proc. Camb. Phil. Soc. 60 (1964) 581]. In this note we generalize this proof to a very general Hamiltonian Ĥ(, ) and give examples to support it.

DOI
10.1088/0253-6102/51/4/06

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In deformation quantization, static Wigner functions obey functional *-genvalue equation, which is equivalent to time-independent Schrodinger equation in Hilbert space operator formalism of quantum mechanics. This equivalence is proved mostly for Hamiltonian with form Ĥ = (1/2)2 + V() [D. Fairlie, Proc. Camb. Phil. Soc. 60 (1964) 581]. In this note we generalize this proof to a very general Hamiltonian Ĥ(, ) and give examples to support it.
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.
The Wigner phase-space distribution function provides the basis for Moyal's deformation quantization alternative to the more conventional Hilbert space and path integral quantizations. The general features of time-independent Wigner functions are explored here, including the functional (``star'') eigenvalue equations they satisfy; their projective orthogonality spectral properties; their Darboux (``supersymmetric'') isospectral potential recursions; and their canonical transformations. These features are illustrated explicitly through simple solvable potentials: the harmonic oscillator, the linear potential, the P\"oschl-Teller potential, and the Liouville potential.
We consider the time-independent Wigner functions of phase-space quantum mechanics (also known as deformation quantization) for a Morse potential. First, we find them by solving the *-eigenvalue equations, using a method that can be applied to potentials that are polynomial in an exponential. A Mellin transform converts the *-eigenvalue equations to difference equations, and factorized solutions are found directly for all values of the parameters. The symbols found this way are of both diagonal and off-diagonal density operator elements in the energy basis. The Wigner transforms of the density matrices built from the known wavefunctions are then shown to confirm the solutions.