The study of a continuously observed quantum particle

We analyze the time-behaviour of the Gaussian wave packet of a quantum part icle moving in the field of a linear force, undergoing a continuous nondemo lition observation of its position. In particular, the time-dependence of p osition and momentum dispersions, as well as position-momentum uncertainty is investigated. The discussion includes the case of a free particle and in itially coherent state of a harmonic oscillator. It is found that in the be ginning of observation the momentum dispersion and position-momentum uncert ainty always increase in contrast to the position dispersion. It is also pr oved that all considered functions asymptotically take the form of harmonic oscillations, decaying to finite values. For initially coherent states, th e asymptotic bounds for all considered functions are given.