Fractal Paths and Schrödingers Equation in an Electromagnetic Field

It is well known that the trajectories of random walks on discrete lattices have a fractal dimension of 2 when viewed on scales much greater than the lattice spacing and that these walks provide a microscopic model for the diffusion equation. Less well known is the fact that some instances of Schrödingers equation occur naturally in the description of the statistics of these same walks and, thus, have classical contexts which are distinct from their usual association with quantum mechanics. In this article, we show that Schrödingers equation for a particle in an electromagnetic field in 1+1 dimensions is one such instance.