# Séminaire de Mécanique d'Orsay

## Le Jeudi 8 décembre à 14h30 - Nouveau bâtiment FAST : http://www.fast.u-psud.fr/fr/plan

### A classical derivation of Hydrogen spectrum

## Arnaud Lazarus

Institut Jean Le Rond d'Alembert, CNRS & UMPC

In 1911, in light of new ground breaking experiments, Rutherford suggested that atoms could be modeled by electrons orbiting a solar nucleus. Although naturally tempting, this planetary-like model had a major technical difficulty: the laws of classical electromagnetism predict that the atom would be unstable because of electron emission radiation. To overcome this difficulty, Bohr proposed, in 1913, that the laws of classical mechanics apply to the motion of the electron only when restricted by a quantum rule. He encountered immediate success since he was able to retrieve, in a simple manner, the Rydberg formula predicting the experimental spectral emission lines of atomic hydrogen. Bohr's semi-classical model is now superseded by the precise simulation of atomic orbitals through Schrödinger's famous equation but it helped to lead the way to the development of Today's powerful quantum mechanics. However, since the birth of Bohr's inspiration, the quantum and classic vision of the world coexist but almost never merge.
In this talk, Bohr model of the hydrogen atom is revisited in a purely Newtonian framework. We suggest that the electron is collapsing because of the nucleus attraction, therefore emitting light, but that in return, some incoming light (because of reflections or other atoms) periodically sustains the rotation of the free-spinning atom. The electrodynamics of the emitting and absorbing electromagnetic wave is modeled in a naive fashion: incoming light is harmonically modulating the electrostatic forces applying on the rotating falling electron. Doing so, a 1D time-varying linear equation arises that governs the linear stability of the orbiting particle. Surprisingly, this fundamental equation has been only partially studied to the best of our knowledge. By way of numerical computations, scaling laws and asymptotic analysis, we are able to analytically compute the quantized stability regions of this modulated oscillator. Not only, our seemingly simple dynamical system predicts the experimental spectral lines of the hydrogen atom but also the computed stable modes of vibration and their associated energy are asymptotic solutions of a 1D Schrödinger's equation with a harmonic potential. This model suggests new physical insights in the physics of light and matter and could eventually offers new tools to reconcile the quantum and classic realm.