Bound states in a semi-infinite square potential well

• URL: http://arxiv.org/abs/2603.08394v1
• Date: Mon, 09 Mar 2026 13:52:32 GMT
• Title: Bound states in a semi-infinite square potential well
• Abstract summary: The finite square potential well is a staple problem in introductory quantum mechanics.<n>The energy eigenvalues, which are also determined by a transcendental equation, are found by a standard graphical method.<n>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.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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