  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 $\sqrt{\eta}\lesssim1\
\mathrm{meV/c}$, 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, specially a correction to the presently accepted value
of Planck's constant to 1 part in $10^{24}$.
