Scattering 1 Classical scattering of a charged particle (Rutherford ...

sClosestApproachθxθθFigure 2: The geometry to convert the angle used to solve the equation of motion to thescattering angle2 Solution to the scattering equationIn the above section we considered scattering of a particle from another particle when theinteraction between them was the Coulomb force (represented as will be seen later, bythe exchange of a virtual photon). In this section we develop a general description of thescattering of a classical, real photon from a charge.This scattering can be considered as a solution to the wave equation. Although we will usethe scalar wave equation, recognize that a vector such as an EM field can be considered asa set of scalar components. The wave equation has the form;[∇ 2 − (1/c 2 ) ∂2∂t 2 ]ψ = S(⃗r, t)In the above, ψ is the wave amplitude and S the scattering center, or in this case the sourceof the scattered wave. In the case of EM scattering, the source of the scattered wave dependson the strength of the incident wave so we rewrite this term as S → Sψ. In the scatteringregion (ie as r → ∞) the solution, ψ, consists of an incident wave plus an outgoing sphericalwave. Scattering assumes that the the source term is localized so that sufficiently far awaythis term → 0 and waves in this region are solutions to the homogeneous wave equation.Now we also assume that the source ands solution are harmonic in time, ψ(⃗r, t) → ψ(⃗r)e iωtwhich removes the time dependence of the equation. The full wave equation is then;[∇ 2 − k 2 ]ψ = S(⃗r)ψThe incident wave has the form of a plane wave solution;3