1. Introduction

Even though the calculus of finite differences is an interesting subject on its own [1–4] that scheme is mainly used to perform numerical computations with the help of a computer. Finite differences methods give approximate expressions for operators like the derivative or the integral of functions, and it is expected that we get a good approximation when the separation between the points of the partition is small; the smaller it becomes the better.

The momentum operator of Quantum Mechanics, when considering continuous variables, is related to the derivative of functions, but its form, when the variable takes discrete values, is not known yet (an approach is found in Ref. [5]); we need an exact expression for the momentum operator in discrete Quantum Mechanics. Thus, to have an expression for the quantum mechanical momentum operator on a mesh of points, we need an exact expression for the derivate on a mesh of points. In this chapter, we intend to modify the usual finite differences definition of the derivative on a partition to propose an operator that can be used as a momentum operator for discrete Quantum Mechanics.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.
