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College Physics

Solution of the particles bound in one-dimensional square potential well

Hao Zheng-tong

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

With the methods of the matrix representation,the wave functions and energy levels are carried out for the particle in one-dimensional square potential well,then the graphical representation of eigenvalue equation,eigenfunction and eigenvalue are presented by using Wolfram Mathematica.

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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.
The one-dimensional finite-depth potential well problem under certain boundary conditions discussed in this paper is mainly the eigenvalue problem in quantum mechanics. The basis of this article is the solution of Schrodinger equation and the treatment of transcendental equation. In many quantum mechanics textbooks, the one-dimensional infinite potential well problem is regarded as the foundation of quantum mechanics because it is easy to handle and understand. On this basis, this paper discusses the more challenging finite-depth potential well situation. In this paper, the transcendental equation is obtained by analogy between the wave function with time and the continuity of the boundary conditions. In order to solve the problem that transcendental equation is difficult to solve in this process, this paper adopts the method of solving numerical solutions by using images, which includes two different cases of odd and even functions. This paper reveals the possibility of particles appearing outside the well in this problem, and on this basis, some discussions and analysis are made on the one-dimensional asymmetric finite-depth potential well problem.
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.
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.