Classical analytical solution for Rydberg states of muonic–electronic helium and helium-like ions
Bibliographic record
Abstract
In our previous papers (Can. J. Phys. 91 (2013) 715; 92 (2014) 1405), we studied Rydberg states of systems consisting of a nucleus of charge Z, a muon, and an electron, both the muon and electron being in circular states. The studies of such quasimolecules μZe were motivated by numerous applications of muonic atoms and molecules, where one of the electrons is substituted by the heavier lepton μ – . We demonstrated that the muonic motion can represent a rapid subsystem, while the electronic motion can represent a slow subsystem. We showed that the spectral lines emitted by the muon in such systems experience a red shift compared to the corresponding spectral lines that would have been emitted by the muon in a muonic hydrogenic atom/ion. In the present paper, we also consider Rydberg states of quasimolecules μZe with Z > 1 (i.e., Rydberg states of muonic–electronic helium and helium-like ions). However, our current approach has important distinctions from our previous papers. The systems considered here are truly stable and the electron orbit is generally elliptical (although the relatively small influence of the electron on the muon is neglected). In our previous papers, the influence of the electron on the muon was taken into account; however, in the rotating frame used in our previous papers, the motion of the muon was only metastable (not truly stable), and furthermore, only circular orbits of the electron were considered in our previous paper. In the present paper, we show that the effective potential energy of the Rydberg electron is mathematically equivalent to the potential energy of a satellite moving around an oblate planet. Based on this, we demonstrate that the unperturbed orbital plane of the Rydberg electron undergoes simultaneously two different precessions: precession within the orbital plane and precession of the orbital plane around the axis of the muonic circular orbit. We provide analytical expressions for the frequencies of both precessions. The shape of the elliptical orbit of the Rydberg electron is not affected by the perturbation, which is the manifestation of the (approximate) conservation of the square of the angular momentum of the Rydberg electron. This means that the above physical systems have a higher than geometrical symmetry (also known as a hidden symmetry) which is a counterintuitive result of general physical interest. We note that the above problem of the motion of the Rydberg electron in muonic–electronic helium atoms or helium-like ions is mathematically equivalent to another problem from atomic physics: a hydrogen Rydberg atom in a linearly-polarized electric field of a high-frequency laser radiation.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".