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Proceedings of the Samahang Pisika ng Pilipinas

Quantum scattering of the wave-packet confined in anisotropic paraboloidal shell potential

Philip Jordan D. Blancas, Eric A. Galapon

Published Oct 22, 2012• 0 citations
OpenAlex

Abstract

The differential scattering cross sections of the wave-packet incident to the convex and concave paraboiloidal shells, each of which has an anisotropic factor, are calculated by solving the Schrodinger equation in paraboloidal coordinate system.

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Parabolic coordinates allow Schrödinger’s equation to be solved analytically for a particle confined to the xy plane and incident with velocity v on a potential C/r. In sharp contrast to the three-dimensional case, the cross section assumes its classical value only in the rationally foreseeable classical regime ‖C‖/ℏv≫1, but not otherwise; and in this low-velocity regime the enhancement factor (probability density at the origin) for an attractive potential tends to 2 (independently of the value of ‖C‖), instead of diverging as in three dimensions like some inverse power of v.
A wave-packet formalism is presented for obtaining the differential cross section for the elastic scattering of a single particle by a potential. The method avoids difficulties inherent in the more common formulations such as interference between incident and scattered waves, box normalization, and density of final states. The time sequence of events in the scattering process is pictured in an intuitively appealing manner. The initial form of the wave packet is specified explicitly, and the time development of the wave packet is determined from the time-translation operator. The asymptotic form of the wave packet is investigated after a sufficiently long time lapse, and the cross section is obtained from a calculation of particle flux densities.