Efficient explicit numerical solutions of the time-dependent Schrödinger equation
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Bibliographic record
Abstract
Explicit numerical solutions of the time-dependent Schr\"odinger equation are more efficient than those obtained by commonly used implicit approaches. They are more practical, especially for a system with higher spatial dimensions. To that end, we introduce a generalization of an explicit three-level method to obtain solutions with spatial and temporal errors of the order of $O[{(\mathrm{\ensuremath{\Delta}}x)}^{2r}]$ and $O[{(\mathrm{\ensuremath{\Delta}}t)}^{2M+3}]$, where $\mathrm{\ensuremath{\Delta}}x$ and $\mathrm{\ensuremath{\Delta}}t$ are the spatial and temporal grid elements, and $r$ and $M$ are positive integers. Sample calculations illustrate the efficacy and stability of the algorithm.
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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 | 0.001 | 0.002 |
| 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 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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 it