Quantum harmonic oscillator in a time dependent noncommutative background

TL;DR

This paper investigates the dynamics of a quantum harmonic oscillator in a time-dependent noncommutative space, deriving analytical solutions for eigenfunctions, energy expectations, and uncertainty relations, extending previous work with new exact solutions.

Contribution

It introduces a generalized Bopp-shift framework and provides exact analytical solutions for the time-dependent noncommutative harmonic oscillator using Lewis invariants.

Findings

Analytical eigenfunctions derived using Lewis invariants.

Exact solutions for the Ermakov-Pinney equation obtained.

Energy expectation values and uncertainty relations analyzed.

Abstract

This work explores the behaviour of a noncommutative harmonic oscillator in a time-dependent background, as previously investigated in [1]. Specifically, we examine the system when expressed in terms of commutative variables, utilizing a generalized form of the standard Bopp-shift relations recently introduced in [2]. We solved the time dependent system and obtained the analytical form of the eigenfunction using the method of Lewis invariants, which is associated with the Ermakov-Pinney equation, a non-linear differential equation. We then obtain exact analytical solution set for the Ermakov-Pinney equation. With these solutions in place, we move on to compute the dynamics of the energy expectation value analytically and explore their graphical representations for various solution sets of the Ermakov-Pinney equation, associated with a particular choice of quantum number. Finally, we determined the generalized form of the uncertainty equality relations among the operators for both commutative and noncommutative cases. Expectedly, our study is consistent with the findings in [1], specifically in a particular limit where the coordinate mapping relations reduce to the standard Bopp-shift relations.