Relativistic linear response wave function of the lowest <i>ns</i> <sub>1/2</sub> states in hydrogen-like atoms. New analytic results
Bibliographic record
Abstract
A novel closed-form analytic representation for the linear response relativistic wave function of the hydrogenic ns 1/2 level that is exposed to a uniform dipole radiation of frequency ω is reported. This result is derived using a relativistic version of the PodolskySternheimerDalgarnoLewis method, that is, by means of direct analytical solving of the appropriate inhomogeneous ω-dependent Dirac equation, thus requiring no prior knowledge of the relativistic Coulomb Green's function. The utility of the formulas obtained for numerous potential applications is illustrated by new results of analytic and numerical calculations of the relativistic dipole dynamic polarizabilities of the 1s 1/2 and 2s 1/2 levels in hydrogen-like atoms. PACS Nos.: 31.15Pf, 31.30Jv, 32.10Hq
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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 | 0.001 |
| 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".