Stiff three-frequency orbit of the hydrogen atom

Abstract

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.

Double-Slit and Electromagnetic Models to Complete Quantum Mechanics

“…The collision is not necessarily a small perturbation on the bound electron's dynamics, and the denominator on the right-hand-side of Eq. (18) can be large, as for example when v 1− performs fast solenoidal ping-pong motions 16 . Notice that Eq.…”

“…where the x k (t The fields of the heavy positive charges are essential to cancel the short-range Coulomb interaction and might participate to create vanishing far fields 16 . In turn, the motion of the bound electrons, henceforth assumed periodic, is essentially responsible to create a vanishing far-field that leaves the scattered electron undisturbed at large distances.…”

“…Equations (17) and (16) are written in a unit system where the speed of light is c and the electronic charge is e, in order to make dimensional dependences explicit. In the following we use discontinuous trajectories as an intuitive approximation tool to estimate the dynamics without having to solve the detailed state-dependent neutral differential delay equations.…”

“…Together with the estimates of the spectroscopic lines 16 , estimate (22) of the De Broglie length suggests that Wheeler-Feynman electrodynamics generalized to discontinuous velocities can offer an alternative to complete quantum mechanics. The derivation of a fundamental formula like Eq.…”

Solutions of the Wheeler-Feynman equations with discontinuous velocities

“…Despite this numerical instability, a physically sensible estimate for the hydrogenoid lines are obtained in Ref. 9 by combining a linearized version of the Lorentz-Dirac equation with a condition of vanishing far fields.…”