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arXiv:physics/9807036

On the classical treatment of the Stark effect in hydrogen atoms

F. Sattin

Published Jul 21, 1998• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.physics/9807036

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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 response of a hydrogen atom to a perturbative electric field is presented from a classical viewpoint. We find classical energy shifts in the hydrogen atom in situations that correspond to both the linear and quadratic Stark effects. We then compare these energy shifts to the exact quantum-mechanical values for the Stark effect. We find that the linear Stark effect can be accurately reproduced classically (given the correct initial conditions). In contrast, quantum-mechanical results for the quadratic Stark effect cannot be accurately reproduced classically, except in the correspondence principle limit of large principal quantum numbers n.
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.
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.
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.