What is the boundary condition for the radial wave function of the Schrödinger equation?

There is much discussion in the mathematical physics literature as well as in quantum mechanics textbooks on spherically symmetric potentials. Nevertheless, there is no consensus about the behavior of the radial function at the origin, particularly for singular potentials. A careful derivation of th...

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Veröffentlicht in:American journal of physics 2011-06, Vol.79 (6), p.668-671
Hauptverfasser: Khelashvili, Anzor A., Nadareishvili, Teimuraz P.
Format: Artikel
Sprache:eng
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Zusammenfassung:There is much discussion in the mathematical physics literature as well as in quantum mechanics textbooks on spherically symmetric potentials. Nevertheless, there is no consensus about the behavior of the radial function at the origin, particularly for singular potentials. A careful derivation of the radial Schrödinger equation leads to the appearance of a delta function term when the Laplace operator is written in spherical coordinates. As a result, regardless of the behavior of the potential, an additional constraint is imposed on the radial wave function in the form of a vanishing boundary condition at the origin.
ISSN:0002-9505
1943-2909
DOI:10.1119/1.3546099