Exact Ultra Cold Neutrons' Energy Spectrum in Gravitational Quantum Mechanics

We find exact energy eigenvalues and eigenfunctions of the quantum bouncer in the presence of the minimal length uncertainty and the maximal momentum. This form of Generalized (Gravitational) Uncertainty Principle (GUP) agrees with various theories of quantum gravity and predicts a minimal length uncertainty proportional to $\hbar\sqrt{\beta}$ and a maximal momentum proportional to $1/\sqrt{\beta}$, where $\beta$ is the deformation parameter. We also find the semiclassical energy spectrum and discuss the effects of this GUP on the transition rate of the ultra cold neutrons in gravitational spectrometers. Then, based on the Nesvizhevsky's famous experiment, we obtain an upper bound on the dimensionless GUP parameter.