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The hydrogen atom in electric and magnetic fields : Pauli\'s 1926 article

Galliano Valent A

Published Dec 2, 2002• 14 citations
OpenAlex

Abstract

The results obtained by Pauli, in his 1926 article on the hydrogen atom, made essential use of the dynamical so(4) symmetry of the bound states. Pauli used this symmetry to compute the perturbed energy levels of an hydrogen atom in a uniform electric field (Stark effect) and in uniform electric and magnetic fields. Although the experimental check of the single Stark effect on the hydrogen atom has been studied experimentally, Pauli\'s results in mixed fields have been studied only for Rydberg states of rubidium atoms in crossedfields and lithium atoms in parallel fields.

Related papers

The hydrogen atom in electric and magnetic fields: Pauli’s 1926 article

Galliano Valent A

The results obtained by Pauli in his 1926 article on the hydrogen atom made essential use of the dynamical so(4) symmetry of the bound states. Pauli used this symmetry to calculate the perturbed energy levels of an hydrogen atom in a uniform electric field (Stark effect) and in uniform electric and magnetic fields. Although the Stark effect in hydrogen has been studied experimentally, Pauli’s results in mixed fields have been studied only for Rydberg states of rubidium atoms in crossed fields and for lithium atoms in parallel fields.

arXiv:quant-ph/0212010•American Journal of Physics•Jan 10, 2003•20 citations

Rydberg states of rubidium in crossed electric and magnetic fields

Francis Jean Penent, Dominique Delande, J. C. Gay

The results of an experimental study of the Rydberg spectrum of rubidium in crossed electric and magnetic fields are reported and compared with theoretical predictions. The present investigations have been conducted in the low-field regime in which the paramagnetic and linear Stark interactions are perturbations to the Coulomb energy. When quantum-defect corrections are negligible, the SO(4) symmetry of the Coulomb interaction organizes the hydrogenic energy spectrum according to the law ${\mathit{E}}_{\mathit{n}}$,k=-R/${\mathit{n}}^{2}$+\ensuremath{\Elzxh}k(${\mathrm{\ensuremath{\omega}}}_{\mathit{L}}^{2}$+${\mathrm{\ensuremath{\omega}}}_{\mathit{S}}^{2}$${)}^{1/2}$, where k is an integer such that -(n-1)\ensuremath{\le}k\ensuremath{\le}(n-1), and ${\mathrm{\ensuremath{\omega}}}_{\mathit{L}}$,${\mathrm{\ensuremath{\omega}}}_{\mathit{S}}$, are, respectively, the Larmor and linear Stark frequencies. The experimental evidence for such a quantization is presented in rubidium Rydberg series using Doppler-free two-photon spectroscopy. The role of quantum defects is incorporated into the theory and experimentally demonstrated.

Physical Review A•Jun 1, 1988•48 citations

Rydberg Atoms In Parallel Electric And Magnetic Fields

Robert L. Waterland

I have calculated the energy spectrum of a highly excited atom which lies in parallel, static electric and magnetic fields. In parallel fields the Coulomb quantum numbers n and m are still "good" quantum numbers but 1 is not: the calculation is for n = 30, m = 1 atoms.;The eigenvalues were obtained by semi-classical quantisation of first-order classical perturbation theory and have been calculated for a large range of electric and magnetic field strengths. The results are in good agreement with those found from first-order degenerate quantum perturbation theory.;The semi-classical analysis provides a correlation diagram connecting the Stark effect states to those of the diamagnetic effect.

•Jan 1, 1986•0 citations

Classical examination of the Stark effect in hydrogen

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

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.

Physical Review A•Jun 1, 1997•14 citations

Rydberg atoms in parallel magnetic and electric fields. II. Theoretical analysis of the Stark structure of the diamagnetic manifold of hydrogen

Patrice Cacciani, Eliane Luc-Koenig, Jacques Pinard +2 more

For pt. I see ibid., vol.21, no.21, p.3473-98 (1988). The semiclassical approach based on the secular perturbation method, primarily introduced by Solov'ev (1982) to study the pure diamagnetism of hydrogen in the weak-field limit, is extended to the case where the atom is subjected to electric and magnetic fields both parallel to the 0z direction. The structure of the manifold of states with principal quantum number n is governed by the existence of an approximate constant of motion and the states can be classified into three groups. The semiclassical Bohr-Sommerfeld quantisation rule allows the energy of the states to be calculated. The main properties of the spectra and their evolution when one of the fields increases are analysed. An original comparison between the structure of the pure diamagnetic n multiplet with azimuthal quantum number M not=0 and the structure of the n, M=0 multiplet with parallel fields is proposed.

Journal of Physics B Atomic Molecular and Optical Physics•Nov 14, 1988•40 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
The results obtained by Pauli, in his 1926 article on the hydrogen atom, made essential use of the dynamical so(4) symmetry of the bound states. Pauli used this symmetry to compute the perturbed energy levels of an hydrogen atom in a uniform electric field (Stark effect) and in uniform electric and magnetic fields. Although the experimental check of the single Stark effect on the hydrogen atom has been studied experimentally, Pauli's results in mixed fields have been studied only for Rydberg states of rubidium atoms in crossedfields and lithium atoms in parallel fields.
The results obtained by Pauli in his 1926 article on the hydrogen atom made essential use of the dynamical so(4) symmetry of the bound states. Pauli used this symmetry to calculate the perturbed energy levels of an hydrogen atom in a uniform electric field (Stark effect) and in uniform electric and magnetic fields. Although the Stark effect in hydrogen has been studied experimentally, Pauli’s results in mixed fields have been studied only for Rydberg states of rubidium atoms in crossed fields and for lithium atoms in parallel fields.
The results of an experimental study of the Rydberg spectrum of rubidium in crossed electric and magnetic fields are reported and compared with theoretical predictions. The present investigations have been conducted in the low-field regime in which the paramagnetic and linear Stark interactions are perturbations to the Coulomb energy. When quantum-defect corrections are negligible, the SO(4) symmetry of the Coulomb interaction organizes the hydrogenic energy spectrum according to the law En{\mathit{E}}_{\mathit{n}}En​,k=-R/n2{\mathit{n}}^{2}n2+\ensuremath{\Elzxh}k(${\mathrm{\ensuremath{\omega}}}_{\mathit{L}}^{2}$+${\mathrm{\ensuremath{\omega}}}_{\mathit{S}}^{2}$)1/2{)}^{1/2})1/2, where k is an integer such that -(n-1)\ensuremath{\le}k\ensuremath{\le}(n-1), and ${\mathrm{\ensuremath{\omega}}}_{\mathit{L}}$,${\mathrm{\ensuremath{\omega}}}_{\mathit{S}}$, are, respectively, the Larmor and linear Stark frequencies. The experimental evidence for such a quantization is presented in rubidium Rydberg series using Doppler-free two-photon spectroscopy. The role of quantum defects is incorporated into the theory and experimentally demonstrated.
I have calculated the energy spectrum of a highly excited atom which lies in parallel, static electric and magnetic fields. In parallel fields the Coulomb quantum numbers n and m are still "good" quantum numbers but 1 is not: the calculation is for n = 30, m = 1 atoms.;The eigenvalues were obtained by semi-classical quantisation of first-order classical perturbation theory and have been calculated for a large range of electric and magnetic field strengths. The results are in good agreement with those found from first-order degenerate quantum perturbation theory.;The semi-classical analysis provides a correlation diagram connecting the Stark effect states to those of the diamagnetic effect.
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.
For pt. I see ibid., vol.21, no.21, p.3473-98 (1988). The semiclassical approach based on the secular perturbation method, primarily introduced by Solov'ev (1982) to study the pure diamagnetism of hydrogen in the weak-field limit, is extended to the case where the atom is subjected to electric and magnetic fields both parallel to the 0z direction. The structure of the manifold of states with principal quantum number n is governed by the existence of an approximate constant of motion and the states can be classified into three groups. The semiclassical Bohr-Sommerfeld quantisation rule allows the energy of the states to be calculated. The main properties of the spectra and their evolution when one of the fields increases are analysed. An original comparison between the structure of the pure diamagnetic n multiplet with azimuthal quantum number M not=0 and the structure of the n, M=0 multiplet with parallel fields is proposed.
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.