  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.
