Solving Schrödinger Equation for Finite Potential Well Using the Iterative Method

Abstract

The quantum Finite Square Well (FSW) model is a well-known topic in most quantum mechanics (QM) books. A couple of equations can be derived from one dimensional Schrodinger equation for a finite potential well for describing the bound Eigen states within the well. Sometimes the FSW problem does not have an exact solution, yet, there are in fact exact solutions. In this work a computational techniques is adopted to find the exact solution for FSW. To achieve this computational solution, a computer program has been written in Basic language for calculating the Eigen state energy for a particle confined in finite potential well using the iterative method (IM). Six values of potential width have been studied with potential depth. The results showed that wider potential width led us to more bound states, while narrower potential width led us to less bound states. Six values of potential depth have been studied with potential width. The results showed that larger potential depth led us to more bound states, while smaller potential width led us to less bound states. A Comparison Between Finite and Infinite Potential Well has been also presented. The result of comparison showed that energy levels of an infinite well are much higher than that the corresponding energy levels for finite potential well. In general, the matching between the results of the iterative method and the graphical method (GM) proving that the iteration method can be regarded as a useful tool for describing the solutions of the 1-dimensional FSW problem.