The hydrogen atom according to wave mechanics - III. ellipsoidal coordinates
Bibliographic record
Abstract
Schroedinger's temporally independent partial-differential equation is directly solvable in ellipsoidal coordinates to yield three ordinary-differential equations; with a common factor in equatorial angular coordinate φ as in spherical polar and paraboloidal coordinates, the product of their solutions contains confluent Heun functions in coordinates ξ and η that impede further calculations at present. To provide plots of these functions, we apply published solutions from Kereselidze et al. in series to illustrate the dependence of the shape of the amplitude functions on distance d between the foci of the ellipsoids, between limiting cases of amplitude functions in spherical polar coordinates as d → 0 and in paraboloidal coordinates as d → ∞. These ellipsoidal coordinates are most appropriate for a treatment of a hydrogen atom in a diatomic-molecular context.
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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 | 0.003 | 0.001 |
| Scholarly communication | 0.003 | 0.002 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 0.001 |
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".