The movie below shows the evolution of a quantum-mechanical wavepacket bouncing on a hard surface under the influence of gravity. This sort of thing might actually be observed in the laboratory someday, by dropping ultracold atoms onto an atomic mirror.
The red dot in the picture shows the corresponding motion for a classical object. The number at the top of the picture is the number of classical bounces. You may notice that the quantum motion is not periodic and that after a while it becomes seemingly random. However, after about twenty classical bounces the wavepacket seems to collect itself and starts to bounce fairly regularly again, only it is a half-cycle out of phase with the classical "ball."
If your computer doesn't let you see the movie, you can try clicking here to download it as an animated GIF instead.
This behavior is explained in my article "A quantum bouncing ball," The American Journal of Physics, 67, 776-782 (1999) which also discusses the classical limit of this system.
The figure below shows the expectation value of the vertical position variable as a function of time, for the particle shown above.
If you are interested in atomic mirrors, you may also want to check out my review article,``Evanescent Light-Wave Atom Mirrors, Resonators, Waveguides, and Traps,'' by J. P. Dowling and J. Gea-Banacloche, in the series of books Advances in Atomic, Molecular and Optical Physics, vol. 37, p. 1 (1996).