Abstract
Proceedings of the Royal Society London 459, 2297-2316 (2003) The energy-based stochastic extension of the Schrodinger equation is perhaps
the simplest mathematically rigourous 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 formulae for 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 timescale of relaxation
is derived. We also consider the small perturbation limit, and discuss the
relation between the stochastic framework and the quantum adiabatic theorem.