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

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.