Mathematica Eterna
Open Access

Abstract

A Solution to the Schr¨odinger Scattering Problem

Jonathan Blackledge

For a scalar plane wave characterised by wave vector k that is incident upon a scattering potential r ∈ R 3 7→ V (r), it is shown that an exact scattering solutions exist provided V (r) satisfies the equation ∇2V (r) + 2∇V (r) · ∇ ln ψi(r, k) = 0 where (∇2 + k 2 )ψi(r, k) = 0. It is further shown that when ψ ∗ i (r, k) ⊗r ψi(r, k) = δ 3 (r) where ⊗r denotes the convolution integral for r ∈ R 3 a non-iterative series solution for the scattered field can be established.