Abstract
Explicit second and fourth algebraic order methods for the numerical solution of the Schrödinger equation are developed in this paper. The new methods have free parameters defined so that the methods are fitted to spherical Bessel and Neumann functions. Based on these new methods we obtained a variable-step algorithm. The results produced based on the numerical solution of the radial Schrödinger equation and the coupled differential equations arising from the Schrödinger equation indicate that this new approach is more efficient than other well known ones.
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