Solving the radial Schrödinger equation by using cubic-spline basis functions

Abstract

This paper demonstrates the use of piecewise continuous (class C²) polynomial basis functions (B splines or hill functions) in solving the l=0 radial Schrödinger equation, with examples of scattering from Eckart, exponential, and static hydrogen potentials, and eigenvalues for Coulomb, harmonic oscillator, and Morse potentials. Simple nonlinear placement of spline centroids can improve accuracy by orders of magnitude. Comparisons demonstrate the greater accuracy of the Galerkin method, compared with collocation, simple finite difference, and Numerov methods.