  The experimental realization of successive non-demolition measurements on
single microscopic systems brings up the question of ergodicity in Quantum
Mechanics (QM). We investigate whether time averages over one realization of a
single system are related to QM averages over an ensemble of similarly prepared
systems. We adopt a generalization of von Neumann model of measurement,
coupling the system to $N$ "probes" --with a strength that is at our disposal--
and detecting the latter. The model parallels the procedure followed in
experiments on Quantum Electrodynamic cavities. The modification of the
probability of the observable eigenvalues due to the coupling to the probes can
be computed analytically and the results compare qualitatively well with those
obtained numerically by the experimental groups. We find that the problem is
not ergodic, except in the case of an eigenstate of the observable being
studied.
