An efficient approximation for accelerating convergence of the numerical power series. Results for the 1D Schr\"odinger's equation

TL;DR

This paper introduces an efficient approximation method to accelerate the convergence of the Numerov algorithm when solving the Schrödinger equation, improving accuracy for excited states and simplifying calculations across hydrogen-like ions.

Contribution

The study proposes a novel approximation technique that enhances the convergence speed of the Numerov method for the Schrödinger equation, especially for excited states and hydrogen-like ions.

Findings

The method reduces errors in energy eigenvalue calculations.

Grid-size optimization improves convergence and accuracy.

The approach simplifies calculations for hydrogen iso-electronic series.

Abstract

The numerical matrix Numerov algorithm is used to solve the stationary Schr\"odinger equation for central Coulomb potentials. An efficient approximation for accelerating the convergence is proposed. The Numerov method is error-prone if the magnitude of grid$-$size is not chosen properly. A number of rules so far, have been devised. The effectiveness of these rules decrease for more complicated equations. Efficiency of the technique used for accelerating the convergence is tested by allowing the grid-sizes to have variationally optimum values. The method presented in this study eliminates the increased margin of error while calculating the excited states. The results obtained for energy eigenvalues are compared with the literature. It is observed that, once the values of grid-sizes for hydrogen energy eigenvalues are obtained, they can simply be determined for the hydrogen iso-electronic series as, $h_{\varepsilon}(Z)=h_{\varepsilon}(1)/Z$.