= Solution
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> $S_l=e^{2i\delta_l}$. Indeed
$$
e^{i\delta_l}\sin(z+\delta_l)
=\frac{e^{2i\delta_l}e^{iz}-e^{-iz}}{2i},
\qquad z=kr-l\pi/2,
$$
so 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> gives
$$
f(\theta)=\frac1k\sum_{l=0}^\infty
(2l+1)e^{i\delta_l}\sin\delta_l\,P_l(\cos\theta).
$$
Use $d\sigma/d\Omega=|f|^2$ and $\int d\Omega\,P_lP_{l'}=4\pi\delta_{ll'}/(2l+1)$ to remove cross terms:
$$
\boxed{\sigma=\frac{4\pi}{k^2}
\sum_{l=0}^\infty(2l+1)\sin^2\delta_l.}
$$
This is the elastic total <scattering cross-section> for the one-particle central-potential problem, with no identical-particle symmetrization factor.
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