Monitoring the states of single quantum systems.
Continuous weak measurement provide a convenient way to gather information about a quantum system without the need to prepare huge ensembles of identical systems as required by standard quantum measurement theory. Even though weak measurement alter the dynamics of the wave function slightly, they nevertheless are a good tool to monitor the dynamics of the wave function in real time in the presence of certain perturbations, for example, sudden momentum kicks due to collisions with particles of a surrounding gas. With weak measurement it is possible to monitor the dynamics of the wave function without knowing it initially. The continuous monitoring can be employed to influence the dynamics by means of feedback. This thesis focuses on the numeric simulation of the continuous monitoring of the position of a free massive particle as well as a particle bound in the following one-dimensional potentials: harmonic and double well. The monitoring scheme involves estimating the wave function of the hydrogen atom initially and then applying the results of the weak measurement its position to update the estimate through a numerically simulated stochastic evolution. We also simulate evolution of the true wave function. The key highlights of this thesis include: discussion of an alternative way to derive the stochastic differential equations that govern the evolution of the true and estimated wave functions of the system, as well as the explanation of the second order numerical scheme.