Relaxation of Quantum States under Energy Perturbations

The energy-based stochastic extension of the Schrodinger equation is perhaps the simplest mathematically rigorous and physically plausible model for the reduction of the wave function. In this article, we apply a new simulation methodology for the stochastic framework to analyse the dynamics of a particle confined to a square-well potential. We consider the situation when the width of the well is expanded instantaneously. Through this example we are able to illustrate in detail how a quantum system responds to an energy perturbation, and the mechanism, according to the stochastic evolutionary law, by which the system relaxes spontaneously into one of the stable eigenstates of the Hamiltonian. We examine, in particular, how the expectation value of the Hamiltonian and the probability distribution for the position of the particle change in time. An analytic expression for the typical time-scale of relaxation is derived. We also consider the small perturbation limit, and discuss the relation between the stochastic framework and the quantum adiabatic theorem.