Quantum Topology / Hopf Algebra Seminar
Arkady Kholodenko
Clemson University
How the modified Bertrand theorem explains regularities of the periodic table
Abstract: The Bertrand theorem permits closed orbits in 3 dimensional Euclidean space only for 2 centrally symmetric potentials: that of the harmonic oscillator and that of the Coulombic field. Since it works perfectly well for the classical model of hydrogen atom, quantization of this model produces the observable spectrum of hydrogen atom. According to Bohr, the spectra of other elements could be obtained by readjusting the charge of the hydrogen nucleus. At the more advanced level, the Hartree-Fock variational method produces a centrally symmetric correction to the Coulombic potential of the nucleus so that the combined potential for multielectron atoms is also centrally symmetric. Since any correction to the Coulombic potential leads to violation of the Bertrand theorem, the question arises: What is the classical (semiclassical) model for the multielectron atoms? At the quantum level directly linked with this question is the question about how electronic orbitals are being filled. Demkov and Ostrovsky (D-O) found that the observed filling patterns are explained by the phenomenological Madelung rule. To obtain this rule quantum mechanically, D-O did 2 things. 1st, at the classical level they replaced the Coulombic potential by the Maxwell’s fish-eye potential and demonstrated that classically the hydrogen atom model with the Coulombic potential can be converted into model with the fish-eye potential. 2nd to bypass the restrictions of the Bertrand theorem for multielectron atoms, they proposed to replace the Maxwell potential by the conformally deformed fish-eye potential. Although such a replacement was made ad hock, the obtained results do reproduce the Madelung rule. The D-O work does not reproduce the spectrum of hydrogen atom though. The extent of conformal deformation was left completely disconnected from the Hartree-Fock calculations . D-O believed that the conformally deformed fish-eye potential removes limitations of the Betrand theorem. This issue was studied later by J.A.Wheeler who found that the classical trajectories associated with conformally deformed fish-eye potential are closed but involve self -intersections. D-O believed (incorrectly) that such selfintersections are not in contradiction with the Bertrand theorem.
In our talk we argue that these self-intersections are explainable if the content of the Bertrand theorem is transferred to curved spaces. We do reproduce the hydrogen spectrum with the fish -eye potential and we do reinterpret the multielectron effects in terms of the motion of individual electrons in conformally flat spaces taking the spaces of constant positive curvature (associated with models reproducing the hydrogen atom spectrum) as the reference spaces. The spin effects in our formalism are taken into consideration automatically. Unlike the Hartree-Fock method, the developed formalism permits us to study the semiclassical and classical limits of quantum mechanical treatments of multielectron atoms.
Tuesday March 19, 2019 at 3:00 PM in 612 SEO