Scattering in 3D

Abstract

This section deals with the expansion of a free plane wave moving in the z direction into spherical harmonics. In stationary scattering theory one considers the free Schrodinger equation in the form $$ ({\nabla^2} + {k^2})u(r,\theta, \phi ) = 0 $$ , with k2 =2mE/ħ2 determined by the known collision energy E. For the problem of scattering from a potential the Schrodinger equation is to be solved with boundary conditions representing an incident free beam and an outgoing spherical wave modulated by the scattering amplitude f(θ, φ) that carries the scattering information. For central scattering potentials V(r) this complex-valued amplitude does not depend on the azimuthal angle φ due to angular momentum conservation. The differential scattering cross section is related to the scattering amplitude by $$ \frac{{d\sigma }}{{d\theta }} = {\left| {f(\theta )} \right|^2} $$ , and represents the generalization of reflection and transmission coefficients from the ID to the 3D situation. The partial wave expansion discussed for free plane waves in this section provides the foundation for the separation-of-variables method to attack the scattering-state Schrodinger equation, i.e., to rewrite the partial differential equation into an infinite set of ordinary differential equations.