Lorentz atom revisited by solving Abraham-Lorentz equation of motion

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.