A Quantum Particle Undergoing Continuous Observation

A stochastic model for the continuous nondemolition ohservation of the position of a quantum particle in a potential field and a bo-son reservoir is given. lt is shown that any Gaussian wave function evolving according to the posterior wave equation with a quadratic potential collapses to a Gaussian wave packet given by the stationary solution of this equation.. The recently developed methods of quantum stochastic calculus [7, ?] can serve for the description of the time-development of continuously observed quantum systems [1, 4, 5]. We apply this approach to describe the time-behaviour of a one-dimensional quantum particle in the field of the linear force F = kx + mg. The effect of coupling of the particle to a measuring apparatus is represented by an extra stochastic force. Assuming a non-ideal indirect observation of the particle position one can select such an observation channel that the observation is nondemolition [4, 5], i.e. it does not affect the actual as well as the future states of the perturbed particle. The posterior dynamics of the observed particle is then given by the nonlinear stochastic wave equations rigorously derived by quantum filtering method in [4, 3].