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 αp 3 and α 2 p 4 where α ∼ 1/M P l c is the GUP parameter. For the case of a quantum bouncer, we solve the generalized Schrödinger equation in the momentum space and find the modified energy eigenvalues and eigenfunctions up to the secondorder in GUP parameter. The effects of the GUP on the transition rate of ultra cold neutrons in gravitational spectrometers are discussed finally.

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