Finite-Dimensional Calculus

TL;DR

This paper explores finite-dimensional calculus within quantum mechanics, focusing on matrix representations of the Heisenberg-Weyl algebra, Krawtchouk polynomials, and finite operator calculus, providing theoretical insights and practical examples.

Contribution

It introduces a matrix approach to finite-dimensional quantum calculus, details Krawtchouk polynomials, and discusses finite operator calculus implementation.

Findings

Matrix representations of Heisenberg-Weyl algebra demonstrated

Krawtchouk polynomials analyzed with illustrative properties

Approach to finite operator calculus outlined

Abstract

We discuss topics related to finite-dimensional calculus in the context of finite-dimensional quantum mechanics. The truncated Heisenberg-Weyl algebra is called a TAA algebra after Tekin, Aydin, and Arik who formulated it in terms of orthofermions. It is shown how to use a matrix approach to implement analytic representations of the Heisenberg-Weyl algebra in univariate and multivariate settings. We provide examples for the univariate case. Krawtchouk polynomials are presented in detail, including a review of Krawtchouk polynomials that illustrates some curious properties of the Heisenberg-Weyl algebra, as well as presenting an approach to computing Krawtchouk expansions. From a mathematical perspective, we are providing indications as to how to implement in finite terms Rota's "finite operator calculus".