An efficient approximation for accelerating convergence of the numerical power series. Results for the 1D Schrödinger's equation

Abstract

The numerical matrix Numerov method is used to solve the stationary Schrödinger 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. The grid−size that is used for discretization of the Schrödinger's equation is allowed to be variationally for the given boundary conditions. In order to test efficiency of the technique used for accelerating the convergence, the grid-sizes are determined via the optimization procedure performed for each energy state. The results obtained for energy eigenvalues and corresponding eigen−vectors 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, hε(Z) = hε(1)/Z.