Bound states in a semi-infinite square potential well

TL;DR

This paper explores the quantum mechanics of a particle in a semi-infinite square potential well, providing methods to determine energy eigenvalues, eigenfunctions, and probabilities, along with correcting previous flawed simplifications.

Contribution

It introduces a new accurate graphical method for finding energy levels and corrects a flawed simplification in existing literature.

Findings

Developed a standard graphical method for energy eigenvalues

Provided a simplified transcendental equation for better approximations

Constructed exact solutions and computed particle probabilities

Abstract

The finite square potential well is a staple problem in introductory quantum mechanics. There is an extensive literature on the determination of the allowed energies, which requires the solution of a transcendental equation by numerical, graphical or approximate analytic methods. Here we investigate the less explored problem of a particle in a semi-infinite potential well. The energy eigenvalues, which are also determined by a transcendental equation, are found by a standard graphical method, and a simple rule that yields the number of stationary states is provided. Next a simplification of the aforementioned transcendental equation is attempted. During the process pitfalls are encountered and a purportedly simpler graphical treatment of the problem given in the solutions manual to a fine textbook is shown to be flawed. A more careful analysis brings forth the correct simplification, which is shown to be particularly suitable for finding highly accurate approximations to the energy levels. Finally, a class of exact solutions is produced, the associated normalized eigenfunctions are constructed and the probability of finding the particle inside the well is computed.