Lorentz atom revisited by solving Abraham-Lorentz equation of motion

TL;DR

This paper revisits the classical Lorentz atom model by analyzing the Abraham-Lorentz equation, revealing its limitations and deriving classical polarizability and absorption properties that align with quantum results, thus deepening understanding of Lorentz's early model.

Contribution

It demonstrates the inappropriateness of the Abraham-Lorentz equation for steady-state Lorentz atom modeling and establishes classical-quantum parameter equivalence for oscillator characteristics.

Findings

AL equation lacks steady-state solutions for Lorentz atom.

Classical polarizability violates Kramers-Kronig relations near resonance.

Derived classical polarizability and absorption cross section agree with quantum calculations.

Abstract

By solving the non-relativistic Abraham-Lorentz (AL) equation, I demonstrate that AL equation of motion is not suited for treating the Lorentz atom, because a steady-state solution does not exist. The AL equation serves as a tool, however, for deducing appropriate parameters $\Omega,\Gamma$ to be used with the equation of forced oscillations in modelling the Lorentz atom. The electric polarizability, which many authors "derived" from AL equation in recent years, is found to violate Kramers-Kronig relations rendering obsolete the extracted photon-absorption rate, for example. Fortunately, errors turn out to be small quantitatively, as long as light frequency $\omega$ is neither too close to nor too far from resonance frequency $\Omega$. Polarizability and absorption cross section are derived for the Lorentz atom by purely classical reasoning and shown to agree with quantum-mechanical calculations of the same quantities. In particular, oscillator parameters $\Omega,\,\Gamma$ deduced by treating the atom as a quantum oscillator are found equivalent to those derived from classical AL equation. The instructive comparison provides a deep insight into understanding the great success of Lorentz's model which was suggested long before the advent of quantum theory.