Schrödinger's Equation and Discrete Random Walks in a Potential Field

Wolfgang Wagner

Alexander Mikhailovich Chebotarev, A. V. Polyakov

Thomas L. Beck, Joel H. Dedrick

Garnet N. Ord, A. S. Deakin

By considering the simple binary symmetric random walk on a discrete lattice in 1+1 dimensions, we show that the discrete analog of Schr\"odinger's equation describes a simple counting problem involving the sample paths on the lattice. Schr\"odinger's equation is obtained in the continuum limit with the result that this equation is confirmed to have a classical as well as a quantum context. \textcopyright{} 1996 The American Physical Society.

M. Faber

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 Schrödinger 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.

Alán Aspuru‐Guzik