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.