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arXiv:quant-ph/0409212

An experiential formula for the energy eigenvalues of a particle in a one-dimension finite-deep square well potential

Zhiming Zhang, Chun-Hua Yuan

Published Sep 29, 2004• 0 citations
arXivHTMLPDF

Abstract

We propose an experiential formula for the calculation of the energy eigenvalues of a particle moving in a one-dimension finite-deep square well potential after some physical considerations. This formula shows a simple relation between the energy eigenvalues and the potential papameters, and can be used to estimate the energy eigenvalues in a very simple way.

DOI
10.48550/arxiv.quant-ph/0409212

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We propose an experiential formula for the calculation of the energy eigenvalues of a particle moving in a one-dimension finite-deep square well potential after some physical considerations. This formula shows a simple relation between the energy eigenvalues and the potential papameters, and can be used to estimate the energy eigenvalues in a very simple way.
The computation of allowed energy levels for a particle bounded to a finite square well potential is ubiquitous in modern physics and introductory quantum mechanics textbooks. For stationary bound states, the matching conditions for the wave functions lead to a pair of transcendental equations whose roots correspond to the energy eigenvalues. However, the graphical solutions available do not make clear the dependence of the energies on the well potential parameters. In this note, I present a simpler graphical solution involving only one dimensionless parameter that determines a straight-line crossing identical sinusoidal curves. I then reduce this solution to a single cosine curve, and from a three-point interpolation, I derive an approximate formula for all energy levels valid for any square quantum well and that demands only a pocket calculator.
Three methods-a documental analysis, a textual method, a comprehensive analysis- are used in this article to analyze the energy levels of motional particles and their wave functions of one-dimensional finite deep symmetrical square potential wells, thus solving the eigen-values and eigen-functions at their corresponding energetic levels. Due to the three following advantages of a one-dimensional problem-its simpler mathematical processing, and the possible detailed-discussed results, and the display of the many features of quantum systems, so in the paper, the eigen-values and the exact formulas of the wave functions of moving particles in a one-dimensional finite deep symmetric square potential wells, which are not specified in the textbooks, are deviated according to the Schrodinger equation of stationary state through a case study of moving particles in a one-dimensional finite deep symmetric square potential wells.
The three principal graphical methods for obtaining the energy eigenvalues of the finite square well potential are presented. The simplest method, with sine, cosine and straight line, is provided with an easily applied rule, based on a physically obvious argument, for eliminating the unacceptable roots. The forms of the wavefunctions within the well, and the corresponding linear probability densities, are derived directly from the method. A simple extension of the method allows the energy level spectrum to be obtained directly on a linear energy scale. The variations of the energy eigenvalues with well depth and width are separately and jointly displayed, and explicit corresponding functional relationships are derived. Finally, two universal graphs are deduced, each of which allows the rapid appreciation and calculation of the dependence of the energy levels on the depth and width of the well and on the mass of the particle.