  We describe particles in a potential by a special diffusion process, the
maximal entropy random walk (MERW) on a lattice. Since MERW originates in a
variational problem, it shares the linear algebra of Hilbert spaces with
quantum mechanics. The Born rule appears from measurements between equilibrium
states in the past and the same equilibrium states in the future. Introducing
potentials by the observation that time, in a gravitational field running in
different heights with a different speed, MERW respects the rule that all
trajectories of the same duration are counted with equal probability. In this
way, MERW allows us to derive the Schr\"odinger equation for a particle in a
potential and the Darwin term of the nonrelativistic expansion of the Dirac
equation. Finally, we discuss why quantum mechanics cannot be simply a result
of MERW, but, due to the many analogies, MERW may pave the way for further
understanding.
