The quantum measurement process in an exactly solvable model

TL;DR

This paper presents an exactly solvable quantum measurement model demonstrating how measurement induces a transition from quantum superposition to a classical state, deriving the collapse and Born rule from dynamics.

Contribution

It introduces a solvable model showing measurement as a dynamical process, deriving collapse and Born rule without postulates.

Findings

Off-diagonal elements vanish rapidly due to unitary evolution.

Measurement drives the apparatus into a definite ferromagnetic state.

The collapse and Born rule are derived from the model's dynamics.

Abstract

An exactly solvable model for a quantum measurement is discussed which is governed by hamiltonian quantum dynamics. The $z$-component $\hat s_z$ 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 $1/\sqrt{N}$ 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.