Noncommutative Simple Harmonic Oscillator in Quantum Gravitational Well

TL;DR

This paper explores the effects of noncommutative geometry on a quantum harmonic oscillator within a gravitational well, revealing subtle differences in energy levels compared to the commutative case.

Contribution

It introduces a theoretical analysis of a noncommutative harmonic oscillator in a quantum gravitational well, highlighting differences in energy spectra due to noncommutativity.

Findings

Similarities between free and oscillator potentials in commutative space

Energy level differences in noncommutative space and momentum

Potential implications for quantum gravity models

Abstract

This work is mainly based on some theoretical surveys on two dimensional quantum gravitational well, considering harmonic oscillator potential causes an effective plank constant. We find that there is a similarity between two different cases, potential free and oscillator potential in noncommutative phase space in commutative momentum, but in noncommutative, non zero space and momentum The Hamiltonian would a little bit different in energy levels.