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

Abstract

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.

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

“…The results below have been obtained in paper [9]. The positive turning points x 2 and x 3 (x 3 > x 2 ) of the motion in the x-coordinate are the real roots of the cubic equation:…”

“…Following paper [9], the energy of the electron in the field of the point-like electric dipole (of the dipole moment D) located at the origin of the spherical polar coordinate reference frame (the z-axis being along the dipole) can be expressed as:…”

“…These two parts intersect at K = K max = 2/3 3/2 , where x 2 (2/3 3/2 ) = x 3 (2/3 3/2 ) = 1/3 1/2 . are the lower part and the upper part of the double-valued curve, respectively [9].…”

“…The most detailed classical analytical solution for the motion of an electron in the field of a point-like dipole was presented in paper [9]. The author of paper [9] showed that, in the general case, the oscillations in the θ-direction and the precession in the ϕ-precessions can actually happen on the same time scale-consequently, the opposite statement from paper [8] was erroneous.…”

“…The author of paper [9] obtained numerous specific analytical results for a variety of particular cases. He also deduced a condition, under which the electron motion (being in general conditionally periodic) can turn out to be periodic, manifested by a closed trajectory of the electron.…”