(9260) Phase shifts for a sphere

Consider scattering of a particle by a sphere of radius \(a\). The potential \(V\) of the sphere might be either positive or negative. The units of length are chosen such that the momentum of the incident particles is \(k=1\). The units of time are adjusted such that the mass of the particle is \(M=1\). Thus the dimensionless parameters of the problem are \((a,V)\). The objective is to produce a diagram that shows the cross section \(\sigma\) as a function of \((a,V)\).

(1) Write explicit expressions for the determination of the logarithmic derivative \(k_{\ell}\), and provide formulas for the phase shift \(\delta_{\ell}\) and for the partial cross sections \(\sigma_{\ell}\).

(2) Produce two plots, one for "small" and one for "large" value of \(a\). In each plot display \(k_{\ell}\) versus \(V\) for \(\ell=0,1,2,3,4\).

(3) Produce a 2D image-plot of \(\sigma/(\pi a^2)\) as a function \((a,V)\).

(4) The effect of a shield is \(k_{\ell} \mapsto k_{\ell} + 2Mu\). Choose a large value for \(2Mu\) and plot again the 2D image plot. The objective is to demonstrate that the resonances become narrow.

Please provide a Matlab(?) file that produces the results.

Based on lecture notes [33.7], and on [C04.3]