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arXiv:1810.11682 · Results in Physics

ERS approximation for solving Schrödinger’s equation and applications

Hichem Eleuch, Michael Hilke

Published Nov 13, 2018• 8 citations
arXivHTMLPDF

Abstract

A new technique was recently developed to approximate the solution of the Schrödinger equation. This approximation (dubbed ERS) is shown to yield a better accuracy than the WKB-approximation. Here, we review the ERS approximation and its application to one and three-dimensional systems. In particular, we treat bound state solutions. We further focus on random potentials in a quantum wire and discuss the solution in the context of Anderson localization.

DOI
10.1016/j.rinp.2018.11.004

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A new technique was recently developed to approximate the solution of the Schrödinger equation. This approximation (dubbed ERS) is shown to yield a better accuracy than the WKB-approximation. Here, we review the ERS approximation and its application to one and three-dimensional systems. In particular, we treat bound state solutions. We further focus on random potentials in a quantum wire and discuss the solution in the context of Anderson localization.
A method of approximating solutions of the one-dimensional Schr\"odinger equation is presented in this paper. The method closely resembles the usual WKB approximation. Whereas in the ordinary WKB method the exponential function is used as the basis of the approximation, in this paper the solutions of an arbitrary Schr\"odinger equation are used. The general advantage is that by proper choice of the arbitrary equation an improved approximation can be obtained. The method is illustrated by treating the potential well and potential barrier problems when there are two turning points. The approximations to the wave functions are continuous even across the turning points. The barrier transmission problem is treated uniformly for energies above and below the peak of the barrier.
In this chapter we study another very important and powerful approximation method, the so-called eikonal approximation or Wentzel-Kramer-Brillouin (WKB) method, which is used in studying quantum mechanical systems that are subjected to complicated potentials. This method can be used to find approximate solutions to differential equations where the highest derivative is multiplied by a small parameter. The WKB method gives approximate solution to the Schrödinger equation irrespective of its complicated potential and is mostly used to study one-dimensional systems and also the radial equations for higher-dimensional systems with rotational symmetry.
Abstract This paper introduces a widely used approximation method, the Wentzel-Kramers-Brillouin (WKB) method, for solving the Schrodinger equation, sets up a theoretical framework to derive the general solutions of the method, and carries out the application of the approximation method in specific cases to verify the validity and practicality of the method. Throughout the study, numerical methods, such as the Simpsons method, and programming tools, such as Python, are used to process sophisticated calculations. The final approximated solutions for the Schrodinger equation are able to demonstrate the state of a particle in an isolated quantum system. All the presented studies are based upon the knowledge and skillsets of a high school student.
In the previous chapters, we have shown how to solve the Schrodinger equation for several systems as a particle in an infinite and finite potential well, scattering, harmonic oscillator, and hydrogen atom. For all these systems, we were able to solve Schrodinger's equation exactly. However, there are complicated potentials for which Schrodinger's equation cannot be solved exactly. For such situations, we resort to approximation methods for finding approximate solutions to the Schrodinger equation. In this chapter, we discuss several approximate methods for solving Schrodinger's equation.