The Wigner distribution function for the one-dimensional parabose oscillator

TL;DR

This paper explores the Wigner distribution function for the one-dimensional parabose oscillator, deriving explicit formulas and analyzing its properties, which extend quantum phase space concepts to non-canonical systems.

Contribution

It introduces two explicit expressions for the Wigner distribution of the parabose oscillator, involving Laguerre polynomials, and compares their behavior to canonical cases.

Findings

Wigner distribution functions are expressed as multiple sums with Laguerre polynomials.

The ground state distribution resembles the first excited state of the canonical oscillator.

Plots illustrate the distribution's behavior for the parabose oscillator.

Abstract

In the beginning of the 1950's, Wigner introduced a fundamental deformation from the canonical quantum mechanical harmonic oscillator, which is nowadays sometimes called a Wigner quantum oscillator or a parabose oscillator. Also, in quantum mechanics the so-called Wigner distribution is considered to be the closest quantum analogue of the classical probability distribution over the phase space. In this article, we consider which definition for such distribution function could be used in the case of non-canonical quantum mechanics. We then explicitly compute two different expressions for this distribution function for the case of the parabose oscillator. Both expressions turn out to be multiple sums involving (generalized) Laguerre polynomials. Plots then show that the Wigner distribution function for the ground state of the parabose oscillator is similar in behaviour to the Wigner distribution function of the first excited state of the canonical quantum oscillator.