It is the asymptotic form of Ψ Ψ s ( r ) that is of interest, because observations near the scattering center (e.g. an atomic nucleus) are mostly not feasible, and detection of particles takes place far away from the origin. At large distances, the particles should behave like free particles, and Ψ Ψ s ( r ) should therefore be a solution to the free Schrödinger equation. This suggests that it should have a similar form to a plane wave, omitting any physically meaningless parts. We therefore investigate the plane-wave expansion :