On the Extinction Paradox

The extinction paradox, the difference of classical and quantum scattering cross-sections for the scattering of particles by a rigid sphere (σ$\text{}^{Q}$=2π a$\text{}^{2}$=2σ$\text{}^{C}$ for ka ≫ 1), is analyzed in a simpler 2D model of a rigid cylindrical potential. Rigorous solutions of the Schrödinger equation for particle beams, including also finite width beams, are derived and employed in the description of the scattering process. The scattering particle fluxes, with a thorough treatment of the forward directions, are being studied. It is pointed out that for wide beams (w ≫ a) the scattered flux can reach the value determined by the quantum theory, provided that it is measured at distances R ≫ waλ. Moderately narrow beams (1≪ w≪ a) behave as classical trajectories, and their scattering can be described in classical terms. Thus, the classical limit of quantum scattering requires not only that the de Broglie wavelength λ$\text{}_{B}$ is much smaller than the size of the scatterer (a ≫ λ$\text{}_{B}$), but also that the transverse width of beams of de Broglie's waves is small, w≪ a.