Generating All Wigner Functions

TL;DR

This paper introduces simple generating functions for complete sets of Wigner functions in phase-space quantization, facilitating easier computation of matrix elements, spectra, and perturbation theory.

Contribution

It presents novel, compact generating functions for both discrete and continuous Wigner function sets, simplifying phase-space calculations.

Findings

Generated functions enable efficient matrix element evaluation

Applied to harmonic oscillator, linear, and Liouville potentials

Facilitates computation of star functions and spectra

Abstract

In the context of phase-space quantization, matrix elements and observables result from integration of c-number functions over phase space, with Wigner functions serving as the quasi-probability measure. The complete sets of Wigner functions necessary to expand all phase-space functions include off-diagonal Wigner functions, which may appear technically involved. Nevertheless, it is shown here that suitable generating functions of these complete sets can often be constructed, which are relatively simple, and lead to compact evaluations of matrix elements. New features of such generating functions are detailed and explored for integer-indexed sets, such as for the harmonic oscillator, as well as continuously indexed ones, such as for the linear potential and the Liouville potential. The utility of such generating functions is illustrated in the computation of star functions, spectra, and perturbation theory in phase space.