In continuous Markov processes there is an idea of sample path. Starting at a point we follow the values of the process through time and find that almost surely the paths are continuous. In Brownian motion (Wiener process) the paths are crinkly curves, continuous paths of infinite variation. The evolution of amplitudes seems to me to be a continuous Markov like process where complex amplitudes replace real probabilities. What are the sample paths for a free particle? Are they continuous? What is their variation? If these paths exist - why can't we define them as the paths of a quantum particle's state?