  In this paper a quantum-mechanical behaviour of neutrons in gravitational
fields is considered. A first estimation is made using the semiclassical
approximation, neglecting General Relativity, magnetic and rotation effects,
for neutrons in weakly bound states in the weak gravitational field of the
Earth. This result was generalized for a case, in which the Randall - Sundrum
correction to Newton's gravitational law on the small scales was applied.
Application of the results to Neutron Star physics is considered and further
possible perspectives are discussed.
