Towards a direct numerical solution of Schrodinger's equation for (e, 2e) reactions

The finite-difference method for electron-hydrogen scattering is presented in a simple, easily understood form for a model collision problem in which all angular momentum is neglected. The model Schrodinger equation is integr ated outwards from the atomic centre on a grid of fixed spacing h. The numb er of difference equations is reduced each step outwards using an algorithm due to Poet, resulting in a propagating solution of the partial-differenti al equation. By imposing correct asymptotic boundary conditions on this gen eral, propagating solution, the particular solution that physically corresp onds to scattering is obtained along with the scattering amplitudes. Previo us works using finite differences (and finite elements) have extracted scat tering amplitudes only for low-level transitions (elastic scattering and n = 2 excitation). If we are to eventually extract ionisation amplitudes, how ever, the numerical method must remain stable for higher-level transitions. Here we report converged cross sections for transitions up to n = 8, as a first step towards obtaining ionisation (e; 2e) results.