Stiff three-frequency orbit of the hydrogen atom.
We study a stiff quasiperiodic orbit of the electromagnetic two-body problem of Dirac's electrodynamics of point charges. The delay equations of motion are expanded about circular orbits to obtain the variational equations up to nonlinear terms. The three-frequency orbit involves two harmonic modes of the variational dynamics with a period of the order of the time for light to travel the interparticle distance. In the atomic magnitude, these harmonic modes have a frequency that is fast compared with the circular rotation. The quasiperiodic orbit has three frequencies: the frequency of circular rotation (slow) and the two fast frequencies of two mutually orthogonal harmonic modes. Poynting's theorem gives a mechanism for a beat of the mutually orthogonal fast modes to cancel the radiation of the unperturbed circular motion by interference. The nonradiation condition for this destructive interference is that the two fast frequencies beat at the circular frequency. The resonant orbits have magnitudes in qualitative and quantitative agreement with quantum electrodynamics (QED), as follows: (i) the orbital angular momenta are integer multiples of Planck's constant to a good approximation, (ii) the orbital frequencies agree with a corresponding emission line of QED within a few percent on average, (iii) the orbital frequencies are given by a difference of two linear eigenvalues, viz., the frequencies of the mutually orthogonal fast modes, and (iv) the angular momentum of gyration of the variational motion about a resonant circular orbit is of the order of Planck's constant.
De Luca
Jayme
J
Universidade Federal de São Carlos, Departamento de Física, Rodovia Washington Luis, km 235, Caixa Postal 676, São Carlos, São Paulo 13565-905, Brazil. deluca@df.ufscar.br