Past exam of the theoretical physics II course of the University of Cambridge / 2012 / Paper 1 / 6 / a

A scattering phase shift measures the change of a partial wave's outgoing phase relative to the free radial solution. For a real central potential and elastic scattering, probability conservation gives a unit-modulus Partial-wave S-matrix $S_l=e^{2i{\delta }_l}$. Indeedso the incoming amplitude is unchanged and the outgoing one acquires $S_l$.

Expanding an incident plane wave in Legendre polynomials and comparing the outgoing spherical wave givesUse $d\sigma /d\Omega =|f{|}^2$ and $\int d\Omega P_l{P}_{l^{\prime }}=4\pi {\delta }_{l{l}^{\prime }}/(2l+1)$ to remove cross terms:This is the elastic total scattering cross-section for the one-particle central-potential problem, with no identical-particle symmetrization factor.