  The time evolution of a bounded quantum system is considered in the framework
of the orthogonal, unitary and symplectic circular ensembles of random matrix
theory. For an $N$ dimensional Hilbert space we prove that in the large $N$
limit the return amplitude to the initial state and the transition amplitude to
any other state of Hilbert space are Gaussian distributed. We further compute
the exact first and second moments of the distributions. The return and
transition probabilities turn out to be non self-averaging quantities with a
Poisson distribution. Departures from this universal behaviour are also
discussed.
