Features of Time-independent Wigner Functions

TL;DR

This paper explores the fundamental properties of time-independent Wigner functions, including their eigenvalue equations, spectral orthogonality, and transformations, with explicit examples from solvable quantum potentials.

Contribution

It provides a comprehensive analysis of the mathematical features of time-independent Wigner functions, highlighting their spectral and transformation properties with explicit solvable examples.

Findings

Wigner functions satisfy specific eigenvalue equations.

They exhibit projective orthogonality and spectral properties.

Explicit examples demonstrate their behavior in solvable potentials.

Abstract

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. 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 Poeschl-Teller potential, and the Liouville potential.