The Wigner function of a q-deformed harmonic oscillator model

TL;DR

This paper constructs the phase space representation of a q-deformed quantum harmonic oscillator, deriving explicit Wigner and Husimi functions, and explores their properties and implications for quantum state analysis.

Contribution

It provides explicit formulas for Wigner and Husimi functions of the q-oscillator, linking them to hypergeometric series and analyzing their behavior in various limits.

Findings

Wigner function expressed as a hypergeometric series related to Al-Salam-Chihara polynomials

In the limit q→1, distribution functions reduce to classical analogues

For q≪1, distributions resemble the ground state with negative momentum displacement

Abstract

The phase space representation for a q-deformed model of the quantum harmonic oscillator is constructed. We have found explicit expressions for both the Wigner and Husimi distribution functions for the stationary states of the $q$-oscillator model under consideration. The Wigner function is expressed as a basic hypergeometric series, related to the Al-Salam-Chihara polynomials. It is shown that, in the limit case $h \to 0$ ($q \to 1$), both the Wigner and Husimi distribution functions reduce correctly to their well-known non-relativistic analogues. Surprisingly, examination of both distribution functions in the q-deformed model shows that, when $q \ll 1$, their behaviour in the phase space is similar to the ground state of the ordinary quantum oscillator, but with a displacement towards negative values of the momentum. We have also computed the mean values of the position and momentum using the Wigner function. Unlike the ordinary case, the mean value of the momentum is not zero and it depends on $q$ and $n$. The ground-state like behaviour of the distribution functions for excited states in the q-deformed model opens quite new perspectives for further experimental measurements of quantum systems in the phase space.