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Physics-Uspekhi

Orbits of the hydrogen electron in a uniform electric field

Yu. M. Kopnin, M.Yu. Kopnin

Published Nov 30, 1993• 1 citation
Publisher

Abstract

The linear Stark effect is investigated, using the old quantum theory and the equations of motion of the electron in orbital variables. In contrast to the well-known solution in terms of parabolic coordinates, the new form has a simple geometric interpretation, namely, perturbation by the uniform electric field leads in the first approximation to a slow regular precession of the electron orbit around the direction of the field (subject to a single additional condition). This geometric picture outwardly resembles electron orbits precessing around a uniform magnetic field in the normal Zeeman effect.

DOI
10.1070/pu1993v036n11abeh002203

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The linear Stark effect is investigated, using the old quantum theory and the equations of motion of the electron in orbital variables. In contrast to the well-known solution in terms of parabolic coordinates, the new form has a simple geometric interpretation, namely, perturbation by the uniform electric field leads in the first approximation to a slow regular precession of the electron orbit around the direction of the field (subject to a single additional condition). This geometric picture outwardly resembles electron orbits precessing around a uniform magnetic field in the normal Zeeman effect.
A classical model of the hydrogen atom in a static electric field is studied, basing upon the work [ Hooker A. et al, {\it Phys. Rev. A}, 55 (1997) 4609 ]. In that work the electrons are supposed to move along Kepler orbits around the nucleus, while interacting with the external field. That classical model reproduces very well the true energy shift of the linear Stark effect. The agreement with the second order effect is poor. It is shown here that the results for the quadratic Stark effect may be considerably improved if the electrons are still allowed to move along classical trajectories, but whose initial conditions are statistically sampled from a distribution resembling the quantum mechanical one.
The explanation of the first-order Stark effect in hydrogen by Epstein and Schwarzschild in 1916 was seen as a great success for the old quantum theory. Yet, it also revealed some serious limitations of the theory. To recover the experimentally found line splittings, one had to make some arbitrary assumptions in addition to the basic quantum conditions to rule out certain orbits. The calculation of intensities of lines on the basis of Bohr's correspondence principle likewise required arbitrary additional assumptions. Finally, the actual orbits predicted by the old quantum theory depend on the coordinates chosen to impose the quantum conditions. Both Sommerfeld and Epstein recognized this problem but offered no solution for it. All these problems were solved in 1926 when Schrödinger and Epstein explained the Stark effect on the basis of the new wave mechanics. The calculations in the two theories are similar. In particular, both the Schrödinger equation in the new theory and the Hamilton-Jacobi equation in the old theory are separated in parabolic coordinates. The new quantum mechanics determines all allowed states and transitions without any additional assumptions. It also replaced the ambiguous guidelines based on the correspondence principle for calculating intensities by the straightforward prescription that intensities are given by the squares of the matrix elements of position, leading to results that agreed much better with the experimental data. Finally, the embarrassing non-uniqueness of orbits in the old quantum theory turned into the innocuous non-uniqueness of bases of eigenfunctions in wave mechanics. To this day, the Stark effect is remembered as one of the few qualified successes of the old quantum theory. We suspect that this is largely because after 1926 it became just one of the many unqualified successes of the new quantum theory.
By means of multiple exposures with a stereoscopic camera of a rotating wire frame representing the curved quadrangle formed by the intersection of the parabolic coordinates of the Epstein-Schwarzchild treatment of the Stark effect, within which quadrangle the electron is progressively advanced along a properly curved wire, the orbital motion within the anchor-ring is very clearly brought out. The pictures when viewed with a stereoscope show the anchor-ring as a wire cage within which the precessing orbit appears as a white dotted line. The change in orientation and eccentricity of the orbit, with the constancy of the distance of the electric center from the YZ\mathrm{YZ}YZ plane (field parallel to XXX) is easily seen. Photographs are shown illustrating various assignments of the quantum numbers n1{n}_{1}n1​ and n2{n}_{2}n2​.
The Stark effect of the fine structure of hydrogenic atoms is discussed in the light of the quantum mechanics. Explicit formulas are given for the energy levels in either a weak or a strong electric field, including in each case both the relativity corrections and the Stark effect. (The conventional treatment neglects either one or the other.) The results are, except in form, similar to those recently given by Schlapp. In the new mechanics, in contradistinction to the old Bohr theory, there is a linear Stark effect even in weak fields, because of the identity of energy for the ns\mathrm{ns}ns and np1n{p}_{1}np1​ levels in the absence of external fields. This degeneracy is shown to account for the results of the recent experiments of Ornstein, Zernike, and Snoek, who found that the 2s2s2s level in HHH is not metastable.The coefficients are found for the development of the "parabolic" eigenfunctions in terms of the "polar" ones, and a set of numerical values of these coefficients is tabulated.Methods are given for the calculation of the relative intensities in the Stark effect in terms of the already available theoretical fine structure intensities in the absence of fields.
Due to spin-orbit coupling, the adiabatic perturbation of an electron's orbital motion induced by a revolving external electric field lead to the electron spin-precession. The obtained results describe both transverse and longitudinal dynamics of an effective spin, for conical and figure-8 trajectories of a driving electric field, and elucidate the link between the quantum-mechanical, geometrical, and classical pictures of spin rotation. Within the limit of adiabatic approximation the derived formulas are valid regardless of whether eigenvectors of the total Hamiltonian of the problem are explicitly available or not, and are convenient for approximate calculations. The main expression for the time evolution of the Bloch vector has pure geometric character and is independent of the physical context.
The unique properties of the hydrogen atom, especially those that result from the dynamic symmetry of the 1/r potential, are presented from both classical and quantal points of view. Using these properties as a starting point, the response of a classical hydrogen atom, an electron executing a Keplerian ellipse about a proton, to a weak electric field is described. Because the field is weak, the elliptical orbit may be treated as a dynamical element itself, variations in its shape and orientation depicting the classical Stark effect. The behavior of this orbit is then correlated with the quantum mechanical description of the first-order Stark effect in hydrogen, a subject that is included in most introductory courses in quantum mechanics.