The Morse potential and phase-space quantum mechanics

TL;DR

This paper derives and verifies the Wigner functions for the Morse potential in phase-space quantum mechanics using a novel Mellin transform approach to solve the $ ext{*}$-eigenvalue equations.

Contribution

It introduces a method to solve $ ext{*}$-eigenvalue equations for exponential polynomial potentials via Mellin transforms, providing explicit solutions for Morse potential Wigner functions.

Findings

Derived explicit Wigner functions for Morse potential

Solved $ ext{*}$-eigenvalue equations using Mellin transforms

Confirmed solutions with density matrix wave functions

Abstract

We consider the time-independent Wigner functions of phase-space quantum mechanics (a.k.a. deformation quantization) for a Morse potential. First, we find them by solving the $\ast$-eigenvalue equations, using a method that can be applied to potentials that are polynomial in an exponential. A Mellin transform converts the $\ast$-eigenvalue equations to difference equations, and factorized solutions are found directly for all values of the parameters. The symbols of both diagonal and off-diagonal density operator elements in the energy basis are found this way. The Wigner transforms of the density matrices built from the known wave functions are then shown to confirm the solutions.