Remarks on the Noncommutative Gravitational Quantum Well

TL;DR

This paper investigates the effects of position and momentum noncommutativity on a gravitational quantum well, providing analytic insights and comparing theoretical predictions with experimental data to constrain noncommutative parameters.

Contribution

It introduces a more inclusive noncommutative phase space framework and analyzes its impact on quantum well dynamics using analytical methods.

Findings

Consistent results between commutative and noncommutative descriptions.

Upper bound on noncommutative parameter from experimental data.

Comparison with numerical analysis supports the theoretical approach.

Abstract

A planar phase space having both position and momentum noncommutativity is defined in a more inclusive setting than that considered elsewhere. The dynamics of a particle in a gravitational quantum well in this space is studied. The use of the WKB approximation and the virial theorem enable analytic discussions on the effect of noncommutativity. Consistent results are obtained following either commutative space or noncommutative space descriptions. Comparison with recent experimental data with cold neutrons at Grenoble imposes an upper bound on the noncommutative parameter. Also, our results are compared with a recent numerical analysis of a similar problem.