Quantum Mechanics in a Space with Finite Number of Points

TL;DR

This paper develops a quantum mechanical model for a finite discrete space, defining a deformed kinetic energy operator and analyzing its properties, including energy spectra, with implications for boundary conditions and continuum limits.

Contribution

It introduces a novel deformed kinetic energy operator for finite discrete spaces with both periodic and nonperiodic boundaries, and derives their energy eigenvalues and wave functions.

Findings

Nonunitary translation operator in nonperiodic case

Energy spectra match known models in continuum limit

Distinct algebraic structure for finite discrete space

Abstract

We define a deformed kinetic energy operator for a discrete position space with a finite number of points. The structure may be either periodic or nonperiodic with well-defined end points. It is shown that for the nonperiodic case the translation operator becomes nonunitary due to the end points. This uniquely defines an algebra which has the desired unique representation. Energy eigenvalues and energy wave functions for both cases are found. As expected, in the continuum limit the solution for the nonperiodic case becomes the same as the solution of an infinite one dimensional square well and the periodic case solution becomes the same as the solution of a particle in a box with periodic boundary conditions.