Toroidal configuration of the orbit of the electron of the hydrogen ...

Energy E, a.u.-4-5-6-70 200 400 600 800 1000bFigure 12: Variational ground state energy of the hydrogen atom in themagnetic field, Eq. (3.38), as a function of the magnetic field intensity;b = B/B 0 , B 0 = 2.4 · 10 9 Gauss, 1 a.u. = 27.2 eV.4 DiscussionIn a physical context, the used adiabatic approximation implies that theposition of the nucleus is not ”fixed” in the (r, ϕ) plane. Indeed, there is noreason for the nucleus to stay ”exactly” at the center of the orbit when theCoulomb force in the transverse plane is totally ignored.In contrast, in the z direction, we have more complicated situation,namely, the magnetic field leads to the effective interaction potential C(z),which is finite at z = 0 and is Coulomb-like in the long-range asymptotic.Hence, the position of the nucleus remains to be fixed in the z directionrelative to the orbit of the electron (bound state).Due to the (crude) Coulomb approximation of Sect. 3.1.1, the electronprobability density of the hydrogen atom under very intense magnetic field,at n = 1, appears to be of rather nontrivial form schematically depicted inFig. 6.More accurate approach may yield some different electron charge distributionsbut we expect that the symmetry requirements (namely, the axialsymmetry and the (anti)symmetry under inversion z → −z) would lead againto a toroidal orbit, which is splitted into two identical coaxial orbits (the toriiare separated) similar to that shown in Fig. 6, or to an unsplitted one (thetwo torii are superimposed) as shown in Fig. 8.Accurate analytic calculation of the ground and excited hydrogen wave25