Unexpected Delta-Function Term in the Radial Schrodinger Equation

TL;DR

This paper investigates the emergence of a delta-function term in the radial Schrödinger equation, revealing a boundary condition necessary for both regular and singular potentials, which clarifies the equation's compatibility with the full Schrödinger equation.

Contribution

It identifies a delta-function term in the radial Schrödinger equation and derives a universal boundary condition, enhancing the understanding of radial equations in quantum mechanics.

Findings

Delta-function term appears in the radial Schrödinger equation.

A boundary condition for the radial wave function is established.

The boundary condition applies to both regular and singular potentials.

Abstract

Careful exploration of the idea that equation for radial wave function must be compatible with the full Schrodinger equation shows appearance of the delta-function while reduction of full Schrodinger equation in spherical coordinates. Elimination of this extra term produces a boundary condition for the radial wave function, which is the same both for regular and singular potentials.