  The existence of a minimum observable length and/or a maximum observable
momentum is in agreement with various candidates of quantum gravity such as
string theory, loop quantum gravity, doubly special relativity and black hole
physics. In this scenario, the Heisenberg uncertainty principle is changed to
the so-called Generalized (Gravitational) Uncertainty Principle (GUP) which
results in modification of all Hamiltonians in quantum mechanics. In this
paper, following a recently proposed GUP which is consistent with quantum
gravity theories, we study the quantum mechanical systems in the presence of
both a minimum length and a maximum momentum. The generalized Hamiltonian
contains two additional terms which are proportional to $\alpha p^3$ and
$\alpha^2 p^4$ where $\alpha \sim 1/M_{Pl}c$ is the GUP parameter. For the case
of a quantum bouncer, we solve the generalized Schrodinger equation in the
momentum space and find the modified energy eigenvalues and eigenfunctions up
to the second-order in GUP parameter. The effects of the GUP on the transition
rate of ultra cold neutrons in gravitational spectrometers are discussed
finally.
