Wigner functions in quantum mechanics with a minimum length scale arising from generalized uncertainty principle

TL;DR

This paper extends the Wigner function concept to quantum mechanics with a minimum length scale from a generalized uncertainty principle, analyzing its properties and implications for quantum harmonic oscillators.

Contribution

It introduces a phase space formulation of GUP-based quantum mechanics, generalizing Wigner functions and exploring their properties and effects on operator averages.

Findings

Wigner function properties are preserved under GUP

Phase space averages of Hamiltonian are calculated for deformed algebra

Deformation parameter influences quantum operator averages

Abstract

In this paper we generalize the concept of Wigner function in the case of quantum mechanics with a minimum length scale arising due to the application of a generalized uncertainty principle (GUP). We present the phase space formulation of such theories following GUP and show that the Weyl transform and the Wigner function does satisfy some of their known properties in standard quantum mechanics. We utilise the generalized Wigner function to calculate the phase space average of the Hamiltonian of a quantum harmonic oscillator satisfying deformed Heisenberg algebra. It is also shown that averages of certain quantum mechanical operators in such theories may restrict the value of the deformation parameter specifying the degree of deformation of Heisenberg algebra. All the results presented are for pure states. The results can be generalized for mixed states.