The quantum measurement process in an exactly solvable model

An exactly solvable model for a quantum measurement is discussed which is governed by hamiltonian quantum dynamics. The z-component ŝ_z of a spin -1/2 is measured with an apparatus, which itself consists of magnet coupled to a bath. The initial state of the magnet is a metastable paramagnet, while the bath starts in a thermal, gibbsian state. Conditions are such that the act of measurement drives the magnet in the up or down ferromagnetic state according to the sign of s_z of the tested spin. The quantum measurement goes in two steps. On a timescale 1/√N the off-diagonal elements of the spin's density matrix vanish due to a unitary evolution of the tested spin and the N apparatus spins; on a larger but still short timescale this is made definite by the bath. Then the system is in a `classical' state, having a diagonal density matrix. The registration of that state is a quantum process which can already be understood from classical statistical mechanics. The von Neumann collapse and the Born rule are derived rather than postulated.