Scattering Theory: Differential Cross Section and Partial Waves

An interactive quantum elastic-scattering playground. The partial-wave expansion $f(\theta) = \frac{1}{k}\sum_l (2l+1)\,e^{i\delta_l}\sin\delta_l\,P_l(\cos\theta)$ gives the differential cross section $d\sigma/d\Omega = |f|^2$ (drawn as a rotating-probe surface of revolution about the beam axis, with the incident plane wave and the outgoing spherical wave), the partial-wave phase shifts $\delta_l$ (hard sphere: $\tan\delta_l = j_l(ka)/n_l(ka)$, on the principal branch so $\delta_l \to 0$ as $l \to \infty$), and the total cross section $\sigma_{\mathrm{tot}} = \frac{4\pi}{k^2}\sum (2l+1)\sin^2\delta_l = \frac{4\pi}{k}\,\mathrm{Im}\,f(0)$ (the optical theorem). Yukawa and square-well targets use the Born approximation, the Fourier transform of the potential $f_B(q) = -\frac{1}{q}\int r\,V(r)\sin(qr)\,dr$ with $q = 2k\sin(\theta/2)$, and the panel then shows $V(r)$. Watching the phase shifts and the angular pattern shows the physics: a hard sphere's cross section tends to $4\pi a^2$ at low energy (four times the geometric area) and $2\pi a^2$ at high energy, a pure $s$-wave scatters isotropically, low-energy scattering is dominated by the $l = 0$ term, and the optical theorem ties the forward amplitude to the total cross section through the removal of flux from the beam.