Electrodynamics of Helium with Retardation and Self-Interaction Effects

We show that an extra constant of motion with an analytic form can exist in the neighborhood of some discrete circular orbits of helium when one includes retardation and self-interaction effects. The energies of these discrete stable circular orbits are in the correct atomic magnitude. The highest frequency in the stable manifold of one such orbit agrees with the highest frequency sharp line of parahelium to within 2%. The generic term of the frequency in the stable manifold to higher orbits is also in agreement with the asymptotic form of quantum mechanics for helium. [S0031-9007(97)05078-3] PACS numbers: 31.15.Ct, 03.20. + i, 05.45. + b In this Letter we explore some surprising consequences of the retardation effects of Maxwell’s electrodynamics to atomic physics. We show that electrodynamics with retardation prescribes a discrete set of stable circular orbits for the helium atom. The name “circular orbit” refers here to an orbit where the two electrons are performing the same circular motion and in phase opposition, that is, along a diameter [1]. The linearized dynamics about a circular orbit has a one-dimensional stable direction, a one-dimensional unstable direction, and ten neutrally stable directions [1]. For an infinitely massive a particle, helium has a sixdegree-of-freedom Hamiltonian system with only four independent constants of motion (namely, the energy and the three components of the total angular momentum). In the neighborhood of circular orbits, even if one restricts to the center manifold [2], one still has a five-degree-offreedom system. Because there are only four constants, the dynamics in the neighborhood of a generic circular orbit can be unstable, and actually is unstable in this case [1]. Here we show that an extra complex constant can exist in the neighborhood of a discrete set of circular