Energy corrections due to the Non-commutative Phase-Space of the Charged Harmonic Oscillator in a constant magnetic field in 3D

TL;DR

This paper explores how noncommutative quantum mechanics in three dimensions affects the energy levels of a charged harmonic oscillator in a magnetic field, revealing negative energy corrections that grow with quantum numbers and field strength.

Contribution

It provides the first closed-form first-order energy corrections for a 3D charged harmonic oscillator in a magnetic field under weak noncommutativity.

Findings

Energy corrections are negative.

Corrections increase with quantum numbers.

Corrections grow with magnetic field strength.

Abstract

In this study, we investigate the effects of noncommutative Quantum Mechanics in three dimensions on the energy levels of a charged isotropic harmonic oscillator in the presence of a uniform magnetic field in the z-direction. The extension of this problem to three dimensions proves to be non-trivial. We obtain the first-order corrections to the energy-levels in closed form in the low energy limit of weak noncommutativity. The most important result we can note is that all energy corrections due to noncommutativity are negative and their magnitude increase with increasing Quantum numbers and magnetic field.