Classical and quantum study of the motion of a particle in a gravitational field

The classical and the quantum mechanical description of a one-dimensional motion of a particle in the presence of a gravitational field is thoroughly discussed. The attention is centered on the evolution of classical and quantum mechanical position probability distribution function. The classical case has been compared with three different quantum cases: (a) a quantum stationary case, (b) a quantum non-stationary zero approximation case, where the wave packet has the shape of the first eigenfunction, and (c) a quantum non-stationary general case, where the wave packet is a superposition of stationary states.