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 rela
ted 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.
