  We work out an exactly solvable hamiltonian model which retains all the
features of realistic quantum measurements. In order to use an interaction
process involving a system and an apparatus as a measurement, it is necessary
that the apparatus is macroscopic. This implies to treat it with quantum
statistical mechanics. The relevant time scales of the process are exhibited.
It begins with a very rapid disappearance of the off-diagonal blocks of the
overall density matrix of the tested system and the apparatus. Possible
recurrences are hindered by the large size of the latter. On a much larger time
scale the apparatus registers the outcome: Correlations are established between
the final values of the pointer and the initial diagonal blocks of the density
matrix of the tested system. We thus derive Born's rule and von Neumann's
reduction of the state from the dynamical process.
