= Solution
Assume a real, central, short-range potential and elastic <quantum scattering>. At large <radius>, normalize the stationary <wavefunction> as
$$
\psi(\mathbf r)\sim e^{ikz}+f(\theta)\frac{e^{ikr}}r,
$$
where $k=\sqrt{2mE}/\hbar$ and $\theta$ is the <scattering angle>. Equal incoming and outgoing speeds give the <differential scattering cross-section> $d\sigma/d\Omega=|f(\theta)|^2$.
The <partial-wave scattering> expansion separates the <spherical harmonics>. The incident <plane wave> has expansion $\sum_{\ell\ge0}i^\ell(2\ell+1)j_\ell(kr)P_\ell(\cos\theta)$, with <Spherical Bessel functions> $j_\ell$ and <Legendre polynomials> $P_\ell$. The regular radial solution outside the potential has asymptotic form $u_\ell\propto\sin(kr-\ell\pi/2+\delta_\ell)$. Its real <scattering phase shift> $\delta_\ell$ changes the ratio of outgoing to incoming waves to the <Partial-wave S-matrix> $S_\ell=e^{2i\delta_\ell}$. Matching outgoing coefficients yields
$$
\boxed{f(\theta)=\frac1{2ik}\sum_{\ell=0}^{\infty}(2\ell+1)(S_\ell-1)P_\ell(\cos\theta)
=\frac1k\sum_{\ell=0}^{\infty}(2\ell+1)e^{i\delta_\ell}\sin\delta_\ell P_\ell(\cos\theta).}
$$
Here $\ell$ labels the <orbital angular momentum>, with squared magnitude $\hbar^2\ell(\ell+1)$. The assumption of purely elastic scattering is what makes every $S_\ell$ have unit <modulus>.
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