First and second-order relativistic corrections to the two and higher-dimensional isotropic harmonic oscillator obeying the spinless Salpeter equation
Bibliographic record
Abstract
We study the relativistic version of the d -dimensional isotropic quantum harmonic oscillator based on the spinless Salpeter equation. This has no exact analytical solutions. We use perturbation theory to obtain compact formulas for the first and second-order relativistic corrections; they are expressed in terms of two quantum numbers and the spatial dimension d . The formula for the first-order correction is obtained using two different methods and we illustrate how this correction splits the original energy into a number of distinct levels each with their own degeneracy. Previous authors obtained results in one and three dimensions and our general formulas reduce to them when d = 1 and d = 3 respectively. Our two-dimensional results are novel and we provide an example that illustrates why two dimensions is of physical interest. We also obtain results for the two-dimensional case using a completely independent method that employs ladder operators in polar coordinates. In total, three methods are used in this work and the results all agree.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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 source (direct Gemma or distilled Codex), 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".