Classical motion of an electron in an electric-dipole field. I. Finite dipole case

Review of Classical Analytical Results for the Motion of a Rydberg Electron around a Polar Molecule under Magnetic or Electric Fields of Arbitrary Strengths in Axially Symmetric Configurations

We review classical studies of the oscillatory-precessional motion of an electron in the field of an electric dipole (the latter representing the polar molecule) with or without external magnetic or electric fields. The focus is on the most recent studies. In one study (at zero external field), it was shown that, generally, the oscillations being in the meridional direction and the precession being along parallels of latitude can take place on the same time scale—contrary to the statement from the previous literature. In another study, it was shown that a magnetic field enables new ranges of the bound oscillatory-precessional motion of the Rydberg electron and that in one of the new ranges, the period of the θ-oscillations has the non-monotonic dependence on primary parameter of the system. This is a counterintuitive result. In yet another study, it was shown that under the electric field there are two equilibrium circular states of a positive energy and one equilibrium state of a negative energy. The existence of the equilibrium state of the negative energy is a counterintuitive result since at the absence of the field, the bound state was possible only for the zero energy. Thus, it is a counterintuitive result that in this case the electric field can play the role of a stabilizing factor.

Oscillatory-Precessional Motion of a Rydberg Electron Around a Polar Molecule

We provide a detailed classical description of the oscillatory-precessional motion of an electron in the field of an electric dipole. Specifically, we demonstrate that in the general case of the oscillatory-precessional motion of the electron (the oscillations being in the meridional direction (θ-direction) and the precession being along parallels of latitude (φ-direction)), both the θ-oscillations and the φ-precessions can actually occur on the same time scale—contrary to the statement from the work by another author. We obtain the dependence of φ on θ, the time evolution of the dynamical variable θ, the period Tθ of the θ-oscillations, and the change of the angular variable φ during one half-period of the θ-motion—all in the forms of one-fold integrals in the general case and illustrated it pictorially. We also produce the corresponding explicit analytical expressions for relatively small values of the projection pφ of the angular momentum on the axis of the electric dipole. We also derive a general condition for this conditionally-periodic motion to become periodic (the trajectory of the electron would become a closed curve) and then provide examples of the values of pφ for this to happen. Besides, for the particular case of pφ = 0 we produce an explicit analytical result for the dependence of the time t on θ. For the opposite particular case, where pφ is equal to its maximum possible value (consistent with the bound motion), we derive an explicit analytical result for the period of the revolution of the electron along the parallel of latitude.

Electronic trajectories in atomic physics: The chemical bond in the H2 + ion

The H₂ ⁺ ion is the simplest example in which a chemical bond exists, created by one electron between two protons. As all chemical bonds, it is usually considered inexplicable in a classical frame. Here, in view of the extremely large velocities attained by the electron near the protons, we consider a relativistic extension of the standard classical three-body model. This has a great impact since the reference unperturbed system (clamped protons) is no more integrable, and indeed by molecular dynamics simulations, we find that the modification entails the existence of a large region of strongly chaotic motions for the unperturbed system, which lead, for the full system, to a collapse of the molecule. For motions of generic type, with the electron bouncing between the protons, there exists an open region of motions regular enough for producing a bond. Such a region is characterized by the property that the electron's trajectories have an angular momentum p_φ along the inter-nuclear axis of the order of the reduced Planck's constant ℏ. Moreover, special initial data exist for which the experimental bond length and oscillation frequency of the protons (but not the dissociation energy) are well reproduced. Also, well reproduced is the quantum potential, albeit only in an extended interval about the minimum.