Expansion of Positive Energy Coulomb Wave Functions in Powers of the Energy

Abstract

The non-relativistic wave function (in spherical polar coordinates) for a charged particle in a Coulomb field is expressed in a form suitable for problems in which the particle has a small positive energy. This formulation amounts to expanding the radial part of the wave function in powers of the energy $E$ and is achieved by simple algebraic manipulation of power series and recurrence formulas. The coefficients of the expansion are functions of the radial coordinate and are identified with Bessel functions of integral order.