Procedure for finding the scattering solutions of the Schrödinger equation

Abstract

We present a convenient numerical method for the calculation of scattering phase shifts and wave functions. In this procedure the wave function is chosen to be an expansion in a convenient basis set for radial distances within the range of the short-range interaction. Outside this range, the wave function is a linear combination of the correct asymptotic solutions. Matching these two solutions determines the phase shift and overall normalization of the wave function. This method is an application of the theory first proposed by Bloch. The method is similar in some respects to $R$-matrix theory. The computational procedure can also be shown to be equivalent to the Kohn variational principle. In this method the choice of basis states is not as restricted as in previous applications and, in particular, the artificial boundary condition of the usual $R$-matrix approach is discarded. The freedom thus obtained in the choice of basis states leads to a rapidly convergent expansion for the wave function and the phase shifts. Furthermore, this freedom enables the method to be easily applied to a wide variety of problems including the case where there is a long-range Coulomb interaction in addition to short-range interactions. Several numerical examples are presented to illustrate the simplicity, flexibility, accuracy, and rate of convergence of the method.