Exact Finite Differences for Quantum Mechanics

Abstract

We introduce a finite difference derivative, on a non-uniform partition, with the characteristic that the derivative of the exponential function is the exponential function itself, times a constant, which is similar to what happens in the continuous variable case. Aside from its application to perform numerical computations, this is particularly useful in defining a quantum mechanical discrete momentum operator.

Alternative momentum concept for a quantum mechanical particle in a box

“…On the lattice the momentum operator should be represented by a discretized first derivative. It is necessary to distinguish forward and backward derivatives [17][18][19]…”