Comment on "Bidimensional bound states for charged polar nanoparticles''

TL;DR

This comment critiques a recent model of an electron near a charged nanostructure, pointing out errors in the angular eigenvalue solution and radial equation that invalidate the original results.

Contribution

It identifies and corrects errors in the previous model's mathematical approach, clarifying the proper solution of the Schrödinger equation for the system.

Findings

Errors in the angular eigenvalue boundary conditions.

Mistakes in eliminating the first derivative in the radial equation.

Invalidation of the original Einstein coefficient results.

Abstract

In a recent paper Castellanos-Jaramillo and Castellanos-Moreno proposed a simple quantum-mechanical model for an electron in the vicinity of an ionized nanostructure with a permanent electric dipole. They chose the interaction of the electron with the charge and the dipole in such a way that the resulting Schr\"{o}dinger equation is separable into radial and angular parts. In this comment we show that those authors did not solve the angular eigenvalue equation with proper periodic boundary conditions and that they also made a mistake in the elimination of the first derivative in the radial equation. Such errors invalidate their results of the Einstein coefficients for the $\left( GaAs\right)_{3}$ system considered.