  In 1966, Edward Nelson presented an interesting derivation of the Schrodinger
equation using Brownian motion. Recently, this derivation is linked to the
theory of optimal transport, which shows that the Schrodinger equation is a
Hamiltonian system on the probability density manifold equipped with the
Wasserstein metric. In this paper, we consider similar matters on a finite
graph. By using discrete optimal transport and its corresponding Nelson's
approach, we derive a discrete Schrodinger equation on a finite graph. The
proposed system is quite different from the commonly referred discretized
Schrodinger equations. It is a system of nonlinear ordinary differential
equations (ODEs) with many desirable properties. Several numerical examples are
presented to illustrate the properties.
