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arXiv · 2001.01581

Homogeneous and Inhomogeneous Formulations of Nonrelativistic Potential Scattering

Jan A. Grzesik
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Abstract

Advantage is taken of the arbitrariness in energy reference to consider anew integral transcriptions of Schrodinger's equation in the presence of potentials which at infinity acquire constant, nonvanishing values. It is found possible to present for the probability amplitude $\psi$ a linear integral equation which is entirely devoid of explicit reference to the wave function incident from infinity, and thus differs markedly from the prevailing inhomogeneous formulation. Identity of the homogeneous equation with an inhomogeneous statement which is at the same time available is affirmed in general terms with the aid of the Fourier transformation, and is then still further reinforced by application of both formalisms to the particular example of a spherical potential barrier/well. Identical, closed-form outcomes are gotten in each case for wave function eigenmode expansion coefficients on both scatterer interior and exterior. Admittedly, the solution procedure is far simpler in the inhomogeneous setting, wherein it exhibits the aspect of a direct, leapfrog advance, unburdened by any implicit algebraic entanglement. By contrast, the homogeneous path, of considerably greater length, insists, at each mode index, upon an exterior/interior coefficient entanglement, an entanglement which, happily, is no more severe than that of a non-singular two-by-two linear system. Each such two-by-two linear system reproduces of course the output already gotten under the inhomogeneous route, and is indeed identical to the two-by-two system encountered during the routine procedure wherein continuity is demanded at the barrier/well interface of both $\psi$ and its radial derivative.

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BibTeXRIS

Jan A. Grzesik. 2019-12-30. Homogeneous and Inhomogeneous Formulations of Nonrelativistic Potential Scattering. https://arxiv.org/abs/2001.01581

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