Noncommutative gravitational quantum well

We study noncommutative geometry at the quantum mechanics level by means of a model where noncommutativity of both configuration and momentum spaces is considered. We analyze how this model affects the problem of the two-dimensional gravitational quantum well and use the latest experimental results for the two lowest energy states of neutrons in the Earth's gravitational field to establish an upper bound on the fundamental momentum scale introduced by noncommutativity, namely, $\sqrt{\ensuremath{\eta}}\ensuremath{\lesssim}1\text{ }\mathrm{meV}/c$, a value that can be improved in the future by up to $3$ orders of magnitude. We show that the configuration space noncommutativity has, in leading order, no effect on the problem. We also analyze some features introduced by the model, especially a correction to the presently accepted value of Planck's constant to $1$ part in ${10}^{24}$.