  We present a phase-space noncommutative version of quantum mechanics and
apply this extension to Quantum Cosmology. We motivate this type of
noncommutative algebra through the gravitational quantum well (GQW) where the
noncommutativity between momenta is shown to be relevant. We also discuss some
qualitative features of the GQW such as the Berry phase. In the context of
quantum cosmology we consider a Kantowski-Sachs cosmological model and obtain
the Wheeler-DeWitt (WDW) equation for the noncommutative system through the ADM
formalism and a suitable Seiberg-Witten (SW) map. The WDW equation is
explicitly dependent on the noncommutative parameters, $\theta$ and $\eta$. We
obtain numerical solutions of the noncommutative WDW equation for different
values of the noncommutative parameters. We conclude that the noncommutativity
in the momenta sector leads to a damped wave function implying that this type
of noncommmutativity can be relevant for a selection of possible initial states
for the universe.
