In quantum mechanics a localized interaction scatters an incident wave into an outgoing spherical wave. Far from a short-range target, a stationary scattering state has the asymptotic form $$\psi(\mathbf r)\sim e^{ikz}+f(\theta,\phi)\frac{e^{ikr}}{r}.$$ The first term is the incident plane wave. The second is the outgoing scattered wave. The complex scattering amplitude $f$ determines the angular probability flux, and $$\frac{d\sigma}{d\Omega}=|f(\theta,\phi)|^2.$$

For a central potential the amplitude depends only on the polar scattering angle. Rotational symmetry makes angular-momentum eigenfunctions the natural basis, so the incident wave can be decomposed into partial waves labeled by orbital quantum number $\ell$.

Each partial wave acquires a phase shift $\delta_\ell$ relative to free propagation. The total cross section is $$\sigma=\frac{4\pi}{k^2}\sum_{\ell=0}^{\infty}(2\ell+1)\sin^2\delta_\ell.$$ At sufficiently low energy, the wavelength is large compared with the interaction range and the $\ell=0$ s-wave often dominates.

Resonant scattering occurs when an interaction temporarily supports a state that strongly changes a phase shift, producing a large energy-dependent cross section. This is why scattering experiments can reveal bound or quasi-bound structure that cannot be imaged directly.

Quantum scattering replaces deterministic impact-parameter trajectories by amplitudes whose phases interfere. The experimentally observed cross section is still the same rate-per-flux object used in classical and nuclear scattering, but its microscopic prediction comes from wave dynamics.