In continuous Brownian motion - while it is completely unpredictable - there are nevertheless many theorems about its behavior. For instance, continuous sample Brownian paths are almost surely nowhere differentiable. Since the Schrodinger equation for a free particle is the heat equation with a factor of i tacked on, one might ask what the paths of free particles look like. The experiment would maybe be to break up a fixed time interval into n increments and at each successive increment measure the position of the particle with a detecting screen. This will give a piecewise linear path with n segments. Now let the increments become increasingly smaller and take the limit. What are the properties of these paths? One might expect some interesting statistics since amplitudes follow a Markov like process. Instead of conditional probabilities, one has conditional amplitudes. But otherwise, it is the same. For a path following a Brownian motion, one can not predict exactly where it will be at a future time, but one can describe its probabilities. So then why can't one think of a quantum mechanical particle as following some path but whose statistics are determined by the Schrodinger equation rather than the Heat equation?