Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Classical examination of the Stark effect in hydrogen

Andrew C. Hooker, Chris H. Greene, William Clark

Published Jun 1, 1997• 14 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.55.4609

Related papers

On the classical treatment of the Stark effect in hydrogen atoms

F. Sattin

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.

arXiv:physics/9807036•Jul 21, 1998•0 citations

On the Theory of the Stark Effect in Hydrogenic Atoms

Vladimir Rojansky

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 $\mathrm{ns}$ and $n{p}_{1}$ 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 $2s$ level in $H$ 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.

Physical Review•Jan 1, 1929•43 citations

Classical view of the Stark effect in hydrogen atoms

T. P. Hezel, Charles E. Burkhardt, M. Ciocca +1 more

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.

American Journal of Physics•Apr 1, 1992•31 citations

Stark effect in a hydrogen atom subjected to a strong resonant wave

Mikhail Vladimirovich Fedorov

An analysis is made of the Stark effect in a hydrogen atom subjected to a static electric field and the field of a strong resonant electromagnetic wave. It is shown that the resonant interaction modifies the linear (with respect to the static field intensity) splitting of the atomic levels and gives rise to a quadratic shift of sublevels.

Soviet Journal of Quantum Electronics•Nov 30, 1975•0 citations

The Semiclassical Origin of the 2/3 Virial Factor in the Linear Stark Effect

Leal Edgar

The perturbative response of the hydrogen atom to a constant external electric field (Linear Stark effect) is analyzed. While a classical planetary model (based on the conservation of angular momentum ) predicts an amplification of the Coulomb tension by a factor of 2, the exact quantum solution yields a factor of 2/3. We analytically demonstrate that this 2/3 factor is identical to that obtained in a classical system under a potential subjected to a rigid angular frequency restriction (constant ). This result suggests that the degenerate manifold of hydrogen behaves dynamically not as a Keplerian free particle, but as an internally driven coherent oscillator, where the effective orbital frequency is quantized and protected against adiabatic radial perturbations. We discuss the implications of this rigidity on the polarizability of Rydberg states and its application in quantum technologies.

•Dec 24, 2025•0 citations

Stark effect of Kramers-Henneberger atoms

Min Li, Qi Wei

The Electric Stark effect of a Kramers-Henneberger (KH) state of hydrogen atoms in both linearly and circularly polarized laser fields is studied. For the ground KH state of H atoms with a small quiver amplitude, the quadratic Stark effect is observed. For a large quiver amplitude, the Stark effect is quadratic only in a weak electric field and quickly changes to linear as the electric field increases. The atomic structure of the KH state is very sensitive to the electric field and can be easily polarized. The huge polarizability and induced dipole moment are comparable to those of Rydberg atoms.

The Journal of Chemical Physics•May 11, 2018•7 citations
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.
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 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 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 perturbative response of the hydrogen atom to a constant external electric field (Linear Stark effect) is analyzed. While a classical planetary model (based on the conservation of angular momentum ) predicts an amplification of the Coulomb tension by a factor of 2, the exact quantum solution yields a factor of 2/3. We analytically demonstrate that this 2/3 factor is identical to that obtained in a classical system under a potential subjected to a rigid angular frequency restriction (constant ). This result suggests that the degenerate manifold of hydrogen behaves dynamically not as a Keplerian free particle, but as an internally driven coherent oscillator, where the effective orbital frequency is quantized and protected against adiabatic radial perturbations. We discuss the implications of this rigidity on the polarizability of Rydberg states and its application in quantum technologies.
The Electric Stark effect of a Kramers-Henneberger (KH) state of hydrogen atoms in both linearly and circularly polarized laser fields is studied. For the ground KH state of H atoms with a small quiver amplitude, the quadratic Stark effect is observed. For a large quiver amplitude, the Stark effect is quadratic only in a weak electric field and quickly changes to linear as the electric field increases. The atomic structure of the KH state is very sensitive to the electric field and can be easily polarized. The huge polarizability and induced dipole moment are comparable to those of Rydberg atoms.