What is the boundary condition for radial wave function of the Schr\"odinger equation ?

TL;DR

This paper clarifies the boundary condition for the radial wave function in the Schrödinger equation, showing that a delta function term enforces the wave function to vanish at the origin, resolving ambiguities for singular potentials.

Contribution

It provides a rigorous derivation demonstrating that the radial wave function must vanish at the origin due to the delta function term in the Schrödinger equation.

Findings

Radial Schrödinger equation includes a delta function term.

Radial wave function must vanish at the origin.

Boundary condition applies regardless of potential singularity.

Abstract

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 Schr\"odinger 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.