On the appearance of a Dirac delta term at the origin in the Schroedinger radial equation

TL;DR

This paper clarifies the origin of a Dirac delta term in the Schrödinger radial equation and emphasizes the necessity of a zero boundary condition at the origin for all central potentials.

Contribution

It demonstrates that a Dirac delta term appears at the origin due to coordinate modifications, requiring a zero boundary condition in radial solutions.

Findings

Dirac delta term at the origin identified

Boundary condition at the origin must be zero

Coordinate modification reveals delta term

Abstract

We revisit a recent discussion about the boundary condition at the origin in the Schroedinger radial equation for central potentials. Using a slight modification of the usual spherical coordinates, the origin of a previously reported Dirac delta function term at the origin is clearly shown up. As a consequence, a vanishing boundary condition at the origin must be imposed in solving the radial equation, regardless the kind of potential.