Stiff three-frequency orbit of the hydrogen atom

Abstract

We study a stiff quasi-periodic orbit of the electromagnetic two-body problem of Dirac's electrodynamics of point charges. We expand the delay equations of motion about circular orbits to obtain the variational equations up to nonlinear terms. We study the normal modes of the variational dynamics with period of the order of the time for light to travel the interparticle distance. In the atomic magnitude these are fast frequencies compared to the circular rotation. We construct a quasi-periodic orbit with three frequencies; the frequency of the unperturbed circular rotation (slow) and the two fast frequencies of two mutually orthogonal harmonic modes of the variational dynamics. Poynting's theorem gives a simple mechanism for a beat of two mutually orthogonal fast modes to cancel the radiation of the unperturbed circular motion by interference. This mechanism operates when the two fast frequencies beat at the circular frequency, a no-radiation condition. The resonant orbits turn out to have unperturbed orbital angular momenta that are integer multiples of Planck's constant to a good approximation. This dynamics displays many qualitative agreements with quantum electrodynamics (QED); (i) the unperturbed frequency of each resonant orbit agrees with a corresponding emission line of QED within a few percent on average (ii) the unperturbed orbital frequency of a resonant orbit is given by a difference of two linear eigenvalues (the frequencies of the mutually orthogonal fast modes) and (iii) the averaged angular momentum of gyration is of the order of Planck's constant.