Oscillatory-Precessional Motion of a Rydberg Electron Around a Polar Molecule
We provide a detailed classical description of the oscillatory-precessional motion of an electron in the field of an electric dipole. Specifically, we demonstrate that in the general case of the oscillatory-precessional motion of the electron (the oscillations being in the meridional direction (θ-di...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-08-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/12/8/1275 |
| _version_ | 1797560522341613568 |
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| author | Eugene Oks |
| author_facet | Eugene Oks |
| author_sort | Eugene Oks |
| collection | DOAJ |
| description | We provide a detailed classical description of the oscillatory-precessional motion of an electron in the field of an electric dipole. Specifically, we demonstrate that in the general case of the oscillatory-precessional motion of the electron (the oscillations being in the meridional direction (θ-direction) and the precession being along parallels of latitude (φ-direction)), both the θ-oscillations and the φ-precessions can actually occur on the same time scale—contrary to the statement from the work by another author. We obtain the dependence of φ on θ, the time evolution of the dynamical variable θ, the period T<sub>θ</sub> of the θ-oscillations, and the change of the angular variable φ during one half-period of the θ-motion—all in the forms of one-fold integrals in the general case and illustrated it pictorially. We also produce the corresponding explicit analytical expressions for relatively small values of the projection p<sub>φ</sub> of the angular momentum on the axis of the electric dipole. We also derive a general condition for this conditionally-periodic motion to become periodic (the trajectory of the electron would become a closed curve) and then provide examples of the values of p<sub>φ</sub> for this to happen. Besides, for the particular case of p<sub>φ</sub> = 0 we produce an explicit analytical result for the dependence of the time t on θ. For the opposite particular case, where p<sub>φ</sub> is equal to its maximum possible value (consistent with the bound motion), we derive an explicit analytical result for the period of the revolution of the electron along the parallel of latitude. |
| first_indexed | 2024-03-10T18:01:53Z |
| format | Article |
| id | doaj.art-2cc32c697b1249f687319e5a9839f424 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T18:01:53Z |
| publishDate | 2020-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-2cc32c697b1249f687319e5a9839f4242023-11-20T08:49:17ZengMDPI AGSymmetry2073-89942020-08-01128127510.3390/sym12081275Oscillatory-Precessional Motion of a Rydberg Electron Around a Polar MoleculeEugene Oks0Physics Department, 380 Duncan Drive, Auburn University, Auburn, AL 36849, USAWe provide a detailed classical description of the oscillatory-precessional motion of an electron in the field of an electric dipole. Specifically, we demonstrate that in the general case of the oscillatory-precessional motion of the electron (the oscillations being in the meridional direction (θ-direction) and the precession being along parallels of latitude (φ-direction)), both the θ-oscillations and the φ-precessions can actually occur on the same time scale—contrary to the statement from the work by another author. We obtain the dependence of φ on θ, the time evolution of the dynamical variable θ, the period T<sub>θ</sub> of the θ-oscillations, and the change of the angular variable φ during one half-period of the θ-motion—all in the forms of one-fold integrals in the general case and illustrated it pictorially. We also produce the corresponding explicit analytical expressions for relatively small values of the projection p<sub>φ</sub> of the angular momentum on the axis of the electric dipole. We also derive a general condition for this conditionally-periodic motion to become periodic (the trajectory of the electron would become a closed curve) and then provide examples of the values of p<sub>φ</sub> for this to happen. Besides, for the particular case of p<sub>φ</sub> = 0 we produce an explicit analytical result for the dependence of the time t on θ. For the opposite particular case, where p<sub>φ</sub> is equal to its maximum possible value (consistent with the bound motion), we derive an explicit analytical result for the period of the revolution of the electron along the parallel of latitude.https://www.mdpi.com/2073-8994/12/8/1275polar moleculeRydberg electronclassical motionperiodic orbits |
| spellingShingle | Eugene Oks Oscillatory-Precessional Motion of a Rydberg Electron Around a Polar Molecule Symmetry polar molecule Rydberg electron classical motion periodic orbits |
| title | Oscillatory-Precessional Motion of a Rydberg Electron Around a Polar Molecule |
| title_full | Oscillatory-Precessional Motion of a Rydberg Electron Around a Polar Molecule |
| title_fullStr | Oscillatory-Precessional Motion of a Rydberg Electron Around a Polar Molecule |
| title_full_unstemmed | Oscillatory-Precessional Motion of a Rydberg Electron Around a Polar Molecule |
| title_short | Oscillatory-Precessional Motion of a Rydberg Electron Around a Polar Molecule |
| title_sort | oscillatory precessional motion of a rydberg electron around a polar molecule |
| topic | polar molecule Rydberg electron classical motion periodic orbits |
| url | https://www.mdpi.com/2073-8994/12/8/1275 |
| work_keys_str_mv | AT eugeneoks oscillatoryprecessionalmotionofarydbergelectronaroundapolarmolecule |