Discrete quantum harmonic oscillator and Kravchuk transform

TL;DR

This paper introduces a discretized quantum harmonic oscillator using Kravchuk functions, proving their convergence to Hermite functions and demonstrating efficient simulation methods with supporting numerical experiments.

Contribution

It presents a novel discretization approach with Kravchuk functions, proving convergence properties and providing efficient simulation techniques for quantum harmonic oscillators.

Findings

Kravchuk functions converge to Hermite functions with almost second-order accuracy.

An efficient simulation method for the discrete eigenfunctions and transformations is developed.

Numerical experiments validate the theoretical convergence and simulation efficiency.

Abstract

We consider a particular discretization of the harmonic oscillator which admits an orthogonal basis of eigenfunctions called Kravchuk functions possessing appealing properties from the numerical point of view. We analytically prove the almost second-order convergence of these discrete functions towards Hermite functions, uniformly for large numbers of modes. We then describe an efficient way to simulate these eigenfunctions and the corresponding transformation. We finally show some numerical experiments corroborating our different results.