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European Journal of Physics

On a solution method for the bound energy states of a particle in a one-dimensional symmetric finite square well potential

Orion Ciftja, Brett Johnston

Published Apr 17, 2019• 6 citations
Publisher

Abstract

Abstract We present details of a new solution method for the bound energy states of a quantum particle in a one-dimensional symmetric finite square well potential. The solution is obtained in a clear way by introducing a simple method that relies on the use of two auxiliary functions. The approach is straightforward and leads to the well-known even and odd parity wave function solutions of the problem without having to do any a priori assumptions about the nature or symmetry of the quantum states.

DOI
10.1088/1361-6404/ab1a61

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Abstract We present details of a new solution method for the bound energy states of a quantum particle in a one-dimensional symmetric finite square well potential. The solution is obtained in a clear way by introducing a simple method that relies on the use of two auxiliary functions. The approach is straightforward and leads to the well-known even and odd parity wave function solutions of the problem without having to do any a priori assumptions about the nature or symmetry of the quantum states.
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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.
We present a solution of the quantum mechanics problem of the allowable energy levels of a bound particle in a one-dimensional finite square well. The method is a geometric-analytic technique utilizing the conformal mapping w -> z = we(w) between two complex domains. The solution of the finite square well problem can be seen to be described by the images of simple geometric shapes, lines, and circles, under this map and its inverse image. The technique can also be described using the Lambert W function. One can work in either of the complex domains, thereby obtaining additional insight into the finite square well problem and its bound energy states. This suggests interesting possibilities for the design of materials that are sensitive to minute changes in their environment such as nanostructures and the quantum well infrared photodetector.
A simple procedure has been found for the general solution of the time-independent Schrödinger Equation (SE) with the help of quantization of potential area in one dimension without making any approximation.Energy values are not dependent on wave functions. So, to find the energy values, it is enough to find the classic turning points of the potential function. Two different solutions were obtained, namely, symmetric and anti symmetric in bound states. These normalized wave functions are always periodic. It is enough to take the integral of the square root of the potential energy function to find the normalized wave functions. If these calculations cannot be made analytically, they should then be performed by numerical methods. The SE has been solved for a particle in many one-dimension and the spherical symmetric central potential well, the relativistic theory of Dirac as examples.
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.