Delta-like singularity in the Reduction of the Laplace Equation for Spherical Coordinates and the Validity of Radial Schrodinger Equation

TL;DR

This paper investigates the delta-like singularity in the Laplace equation in spherical coordinates, revealing how its elimination imposes boundary conditions on the radial wave function in the Schrödinger equation.

Contribution

It identifies the delta-like singularity in the Laplacian and clarifies its impact on the boundary conditions of the radial Schrödinger equation.

Findings

Elimination of the singularity imposes boundary conditions at the origin.

The singularity affects the validity of the radial Schrödinger equation.

Provides a detailed derivation of the boundary condition from the Laplacian.

Abstract

By careful exploration of separation of variables into the Laplacian in spherical coordinates, we obtain the extra delta-like singularity, elimination of which restricts the radial wave function at the origin. This constraint has the form of boundary condition for the radial Schrodinger equation.