Harmonic oscillator in a background magnetic field in noncommutative quantum phase-space

TL;DR

This paper explicitly solves the 2D harmonic oscillator in a magnetic field within noncommutative phase-space, revealing conditions under which time reversal symmetry is restored due to degeneracy in the energy spectrum.

Contribution

It provides an explicit solution to the harmonic oscillator in noncommutative phase-space without using representations, highlighting symmetry restoration at specific parameters.

Findings

Time reversal symmetry can be restored in noncommutative phase-space.

Energy spectrum degeneracy occurs at specific noncommutative parameters.

Contrasts with noncommutative configuration space where symmetry is always broken.

Abstract

We solve explicitly the two-dimensional harmonic oscillator and the harmonic oscillator in a background magnetic field in noncommutative phase-space without making use of any type of representation. A key observation that we make is that for a specific choice of the noncommutative parameters, the time reversal symmetry of the systems get restored since the energy spectrum becomes degenerate. This is in contrast to the noncommutative configuration space where the time reversal symmetry of the harmonic oscillator is always broken.