The quantum Gaussian well

TL;DR

This paper explores the quantum Gaussian well using variational, WKB, and numerical diagonalization methods, providing insights into its physical properties and applications in quantum mechanics.

Contribution

It introduces a self-contained numerical diagonalization approach for solving the Schrödinger equation with a Gaussian well potential, complementing traditional analytical methods.

Findings

Numerical solutions align with variational and WKB approximations.

The Gaussian well potential exhibits unique bound state features.

Applications in quantum systems are discussed.

Abstract

Different features of a potential in the form of a Gaussian well have been discussed extensively. Although the details of the calculation are involved, the general approach uses a variational method and WKB approximation, techniques which should be familiar to advanced undergraduates. A numerical solution of the Schr\"odinger equation through diagonalization has been developed in a self-contained way, and physical applications of the potential are mentioned.