  We study the free-fall of a quantum particle in the context of noncommutative
quantum mechanics (NCQM). Assuming noncommutativity of the canonical type
between the coordinates of a two-dimensional configuration space, we consider a
neutral particle trapped in a gravitational well and exactly solve the energy
eigenvalue problem. By resorting to experimental data from the GRANIT
experiment, in which the first energy levels of freely falling quantum
ultracold neutrons were determined, we impose an upper-bound on the
noncommutativity parameter. We also investigate the time of flight of a quantum
particle moving in a uniform gravitational field in NCQM. This is related to
the weak equivalence principle. As we consider stationary, energy eigenstates,
i.e., delocalized states, the time of flight must be measured by a quantum
clock, suitably coupled to the particle. By considering the clock as a small
perturbation, we solve the (stationary) scattering problem associated and show
that the time of flight is equal to the classical result, when the measurement
is made far from the turning point. This result is interpreted as an extension
of the equivalence principle to the realm of NCQM.
