Gravitational Quantum Well

TL;DR

This paper explores how noncommutative phase space affects the gravitational quantum well, deriving bounds on noncommutativity parameters and discussing potential experimental tests with atoms and molecules.

Contribution

It extends noncommutative quantum mechanics to phase space and analyzes its effects on gravitational quantum wells, providing bounds and experimental implications.

Findings

Noncommutativity in configuration space has no leading order effect.

A correction to Planck's constant of 1 part in 10^24 is derived.

Transition to classical behavior occurs with increasing particle mass.

Abstract

We discuss the implications of a model of noncommutative Quantum Mechanics where noncommutativity is extended to the phase space. We analyze how this model affects the problem of the two-dimensional gravitational quantum well and use the latest experimental results for the energy states of neutrons in the Earth's gravitational field to establish an upper bound on the fundamental momentum scale introduced by noncommutativity. We show that the configuration space noncommutativity has, in leading order, no effect on the problem and that in the context of the model, a correction to the presently accepted value of Planck's constant to 1 part in $10^{24}$ arises. We also study the transition between quantum and classical behaviour of particles in a gravitational quantum well and analyze how an increase in the particles mass turns the energy spectrum into a continuous one. We consider these effects and argue that they could be tested by through experiments with atoms and fullerene-type molecules.