Lorentz Dispersion Law from classical Hydrogen electron orbits in AC electric field via geometric algebra

TL;DR

This paper models a classical hydrogen electron's response to an AC electric field using geometric algebra, deriving a Lorentz dispersion law and analyzing energy absorption and angular momentum patterns.

Contribution

It introduces a geometric algebra approach to classical electron orbits under AC fields, linking orbital dynamics to Lorentz dispersion law without damping.

Findings

Electron absorption peaks at resonance frequency

Angular momentum traces epicyclical patterns

Electron ionization occurs at increasing oscillation amplitudes

Abstract

We studied the orbit of an electron revolving around an infinitely massive nucleus of a large classical Hydrogen atom subject to an AC electric field oscillating perpendicular to the electron's circular orbit. Using perturbation theory in geometric algebra, we show that the equation of motion of the electron perpendicular to the unperturbed orbital plane satisfies a forced simple harmonic oscillator equation found in Lorentz dispersion law in Optics. We show that even though we did not introduce a damping term, the initial orbital position and velocity of the electron results to a solution whose absorbed energies are finite at the dominant resonant frequency $\omega=\omega_0$; the electron slowly increases its amplitude of oscillation until it becomes ionized. We computed the average power absorbed by the electron both at the perturbing frequency and at the electron's orbital frequency. We graphed the trace of the angular momentum vector at different frequencies. We showed that at different perturbing frequencies, the angular momentum vector traces epicyclical patterns.