The solution to the Schrodinger equation for a spherically symmetric potential, will have the form $$ \psi_k(r) = e^{ikr} + f(\theta)\frac{e^{ikr}}{r} $$ where \( f(\theta) \) is the scattering amplitude, and related to the differential cross section as $$ \frac{d\sigma}{d\Omega} = |f(\theta)|^2 $$ Using the expansion of a plane wave in spherical waves, we can relate the scattering amplitude \( f(\theta) \) with the partial wave phase shifts \( \delta_l \) by identifying the outgoing wave $$ \psi_k(r) = e^{ikr} + \left[\frac{1}{2ik}\sum_l i^l (2l+1) (S_l(k)-1)P_l(\cos(\theta))e^{-il\pi/2}\right] \frac{e^{ikr}}{r} $$ which can be simplified further by cancelling \( i^l \) with \( e^{-il\pi/2} \)