Quantum scattering from cylindrical barriers

We solve the two-dimensional Schrödinger equation for particles with momentum $p x = ℏ k$ scattering off of a hard circular cylinder using the finite element method; we compare our results with the exact analytic solution. The quantity of interest to experimentalists is the differential cross section $σ ( ϕ ) = | f k ( ϕ ) | 2 ,$ which represents the final angular distribution of only the scattered particles. Here, we are also interested in the interference between the incident and scattered wave, which can be seen in the probability density for the total wave function, $ρ ( x , y ) = | ψ k ( x , y ) | 2$⁠. We also apply the finite element method to the problem of particles scattering off of a hard rectangular cylinder, for which there is no analytic solution.