Quantum mechanics and umbral calculus

TL;DR

This paper introduces a novel approach to discretize quantum mechanics using umbral calculus, aiming to preserve properties like integrability and symmetries of the continuous Schrödinger equation.

Contribution

It presents the first method to discretize quantum mechanics via umbral calculus, maintaining key properties and enabling easy recovery of continuous solutions.

Findings

Discretization preserves integrability and symmetries.

Solutions of continuous equations can be recovered discretely.

Method applied to Schrödinger equation with space-dependent potential.

Abstract

In this paper we present the first steps for obtaining a discrete Quantum Mechanics making use of the Umbral Calculus. The idea is to discretize the continuous Schroedinger equation substituting the continuous derivatives by discrete ones and the space-time continuous variables by well determined operators that verify some Umbral Calculus conditions. In this way we assure that some properties of integrability and symmetries of the continuous equation are preserved and also the solutions of the continuous case can be recovered discretized in a simple way. The case of the Schroedinger equation with a potential depending only in the space variable is discussed.