Cross sections in quantum mechanics

The definition of scattering cross sections requires an averaging over wavepackets with random impact parameters ρ; this leads to an integral of the scattering probability over all ρ in a plane perpendicular to the incident beam. We show that, for scattering off a potential which is O (1/rβ) as r→∞, the scattering probability is O (1/ρ2β−4) as ρ→∞. Thus for any β ≳ 3, the integral over impact parameters is well‐defined and convergent.