Shape invariance in phase space

TL;DR

This paper explores how shape invariance, a key concept in quantum mechanics, can be applied within phase space formalism to simplify the calculation of Wigner functions, demonstrated through harmonic oscillator and Morse potential examples.

Contribution

It introduces a novel set of relations between Wigner functions derived from shape invariance in the phase space framework, enabling direct computation from a single known function.

Findings

Derived new relations between Wigner functions using shape invariance.

Demonstrated the method with harmonic oscillator and Morse potential.

Showed that Wigner functions can be calculated directly once one is known.

Abstract

Shape invariance is a powerful solvability condition, that allows for complete knowledge of the energy spectrum, and eigenfunctions of a system. After a short introduction into the deformation quantization formalism, this paper explores the implications of the supersymmetric quantum mechanics and shape invariance techniques to the phase space formalism. We show that shape invariance induces a new set of relations between the Wigner functions of the system, that allows for their direct calculation, once we know one of them. The simple harmonic oscillator and the Morse potential are solved as examples.