In 1966, Edward Nelson presented an interesting derivation of the Schrödinger equation using Brownian motion. Recently, this derivation is linked to the theory of optimal transport, which shows that the Schrödinger 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 Schrödinger equation on a finite graph. The proposed system is quite different from the commonly referred discretized Schrödinger equations. It is a system of nonlinear ordinary differential equations (ODEs) with many desirable properties. Several numerical examples are presented to illustrate the properties. R d W(x, y)ρ(t, y)dy = 0 ,