The Schrödinger equation for the continuum states of a hydrogenic system in front of a metal surface described by a step potential is solved using prolate spheroidal coordinates. The potential considered includes the classical core image on the metal side.

The boundary conditions used describe an electron incident on the surface from the metal side, and consist of incident and reflected plane waves in this sub-space, and a transmitted plane wave in vacuum. The presence of the nuclear charge on the vacuum side and the polarization it induces on the surface, described by an image charge, act as scattering centres that produce an outgoing spherical wave in the asymptotic region.

We present the exact continuum wavefunction for different values of the energy and the remaining quantum numbers. Using the condition of continuity of the wavefunction and its normal derivative on the surface plane we obtain the solution in analytical form. Efficient algorithms, easily implemented on any microcomputer, allow for the calculation of the separation constants of the wave equation, the amplitudes of the waves scattered from the charge and its image, its phase shifts and from them the continuum wavefunction.