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arXiv · 1203.0551

Solutions of the Schrödinger equation, boundary condition at the origin, and theory of distributions

Y. C. Cantelaube
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Abstract

In a central potential the usual resolution of the Schrödinger equation in spherical coordinates consists in determining the solutions R(r) or u(r) of the radial equations considered as the radial parts of the Schrödinger equation. However, the solutions must be supplemented with the boundary condition u(0) = 0 in order to rule out singular solutions. There is still no consensus to justify this condition, with good reason. It is based on a misunderstanding that comes from the fact that the radial equation in terms of R(r) is derived from the Schrödinger equation, and the radial equation in terms of u(r) from the former, by taking the Laplacians in the sense of the functions. By taking these Laplacians in the sense of the distributions, as it is required, we show that the radial equations are derived from the Schrodinger equation when their solutions are regular, but not when they are singular, so that the equations need not be supplemented with any supplementary condition such as u(0) = 0.

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Y. C. Cantelaube. 2012-03-02. Solutions of the Schrödinger equation, boundary condition at the origin, and theory of distributions. https://arxiv.org/abs/1203.0551

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