Variational electrodynamics of Atoms

TL;DR

This paper extends Wheeler-Feynman electrodynamics using a variational approach, deriving equations with velocity discontinuities, and applies it to atomic models, achieving results consistent with quantum mechanics and experiments.

Contribution

It introduces a variational formulation of electrodynamics with boundary conditions, handling velocity discontinuities, and applies it to atomic models to reproduce quantum-like spectral properties.

Findings

Discrete orbit neighborhoods with vanishing far-fields predicted

Angular momentum spacings match Planck's constant within 30%

Orbital frequencies agree within a few percent with spectroscopy

Abstract

We generalize Wheeler-Feynman electrodynamics by the minimization of a finite action functional defined for variational trajectories that are required to merge continuously into given past and future boundary segments. We prove that the boundary-value problem is well-posed for two classes of boundary data and show that the well-posed solution in general has velocity discontinuities, henceforth broken extrema. Along regular segments, broken extrema satisfy the Euler-Lagrange neutral differential delay equations with state-dependent deviating arguments. At points where velocities are discontinuous, broken extrema satisfy the Weierstrass-Erdmann conditions that energies and momenta are continuous. The electromagnetic fields of the variational trajectories are derived quantities that can be extended only to a bounded region B of space-time. For extrema with a finite number of velocity discontinuities, extended fields are defined for all point in B with the exception of sets of zero measure. The extended fields satisfy the integral laws of classical electrodynamics for most surfaces and curves inside B. As an application, we study globally bounded trajectories with vanishing far-fields for the hydrogenoid atomic models of hydrogen, muonium and positronium. Our model uses solutions of the neutral differential delay equations along regular segments and a variational approximation for the collisional segments. Each hydrogenoid model predicts a discrete set of finitely measured neighbourhoods of orbits with vanishing far-fields at the correct atomic magnitude and in quantitative and qualitative agreement with experiment and quantum mechanics, i.e., the spacings between consecutive discrete angular momenta agree with Planck's constant within thirty-percent, while orbital frequencies agree with a corresponding spectroscopic line within a few percent.