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Physics of Particles and Nuclei Letters

The Boundary Condition for Reduced Radial Wave Function in Multi-Dimensional Schrodinger Equation

Anzor Khelashvili, T. Nadareishvili

Published Aug 1, 2024• 0 citations

Abstract

Abstract We study the behavior of reduced radial wave function at the origin for multidimensional Schrodinger equation, where the angular variables are separated by using a hyperspherical formalism and the overall potential is chosen symmetric under rotations in full Euclidean space. It is shown that the rigorous restriction at the origin—Dirichlet boundary condition follows only in three-dimensional space, whereas in other dimensions (more than three) some physical reasonings are necessary in addition. According to our previous investigation the most appropriate is the Hermiticity of Hamiltonian or, equivalently, the conservation of particle number. In this case the preferable is a Dirichlet condition again for regular potentials, but for singular potentials (not soft) other conditions are also allowed together with it. In this meaning the three dimensions is a peculiar one.

DOI: 10.1134/s1547477124701474

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Abstract We study the behavior of reduced radial wave function at the origin for multidimensional Schrodinger equation, where the angular variables are separated by using a hyperspherical formalism and the overall potential is chosen symmetric under rotations in full Euclidean space. It is shown that the rigorous restriction at the origin—Dirichlet boundary condition follows only in three-dimensional space, whereas in other dimensions (more than three) some physical reasonings are necessary in addition. According to our previous investigation the most appropriate is the Hermiticity of Hamiltonian or, equivalently, the conservation of particle number. In this case the preferable is a Dirichlet condition again for regular potentials, but for singular potentials (not soft) other conditions are also allowed together with it. In this meaning the three dimensions is a peculiar one.

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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.

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