Numerical and perturbation solutions for the gauss potential

Saad Naji Abood Saad Naji Abood,Narjis Zamil Abdulzahra Narjis Zamil Abdulzahra

doi:10.30574/gjeta.2022.10.2.0033

Saad Naji Abood Saad Naji Abood, Narjis Zamil Abdulzahra Narjis Zamil Abdulzahra

Open Access

https://doi.org/10.30574/gjeta.2022.10.2.0033

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Abstract

We will focus on the Schrodinger eigenvalue problem for a Gauss potential in this study. When high and relatively large values of the coupling constant g2 are involved, we will compare eigenvalues E determined numerically with those obtained using the asymptotic series. However, we were interested in the mathematical elements of this comparison throughout the course of this work and explored it for considerably larger, albeit no longer physically plausible, values of g2. Even for power potentials where the Gaussian is a common example, Muller's perturbation method shows some fascinating mathematical characteristics of the Schrodinger equation. The solution's overall analytic features are very similar to well-known periodic differential equations like the Mathieu equation.

Highlights

The theoretical laboratory of the Schrodinger equation and its solutions, both for the scattering as well as the bound states, have served an important purpose to shed some light on the much more complex phenomena of high energy particle collisions various potential functions were used in this equation where these potentials vary in their degree of singularities at the origin as well as at large distances
Computational Details We summarize the computational procedures used to solve the Schrodinger equation for a Gauss potential
E COGI 0 and computed, respectively, by the numerical method of section 2 and the perturbation method of section 3 which correspond to the energy levels associated with a bound system where centrifugal force does not act on the particles

Summary

Introduction

The theoretical laboratory of the Schrodinger equation and its solutions, both for the scattering as well as the bound states, have served an important purpose to shed some light on the much more complex phenomena of high energy particle collisions various potential functions were used in this equation where these potentials vary in their degree of singularities at the origin as well as at large distances. Ever since the field of potential scattering was recognized as a helpful testing ground for various particles physics model, one basic property of the solution was realized early. This was that very few potential models can have analytic form. Muller uses expansions in the domain of large coupling constants g 2 He obtains asymptotic solutions of the S-wave radial Schrodinger equation for these potentials. The overall analytic characteristics of the solution are quite analogous to well-known periodic differential equations such as the Mathieu equation

Numerical Solution for the Schrodinger Equation for a Gauss Potential

Formulation of the COGI Method

Asymptotic Solution near the origin

E COGI 0 and

Eigenvalues corresponding to very large g2

Concluding Remarks