On the Extinction Paradox |

W. ¯akowiczInstitute of Physics, Polish Academy of Sciences, al. Lotników 32/46, Warsaw 02-668, Poland |

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The extinction paradox, the difference of classical and quantum
scattering cross-sections for the scattering of particles by a rigid
sphere (σ^{Q}=2πa^{2}=2σ^{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_{B} is much smaller than the size of the scatterer (a>>λ_{B}), but also that the transverse width of beams of de Broglie's waves is small, w<<a. |

PACS numbers: 32.80.Cy, 42.25.Fx, 61.10.Dp |