The Kronig-Penney model extended to arbitrary potentials via numerical matrix mechanics
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Bibliographic record
Abstract
The Kronig-Penney model is a common starting point for studying the quantum mechanics of electrons in a confining periodic potential. This model uses a square-well potential; the energies and eigenstates can be obtained analytically for a single well, and then Bloch's theorem allows one to extend these solutions to the periodically repeating potential. In this work, we describe how to obtain simple numerical solutions for the eigenvalues and eigenstates for any confining one-dimensional potential within a unit cell and then extend this procedure, with virtually no extra effort, to the case of arbitrary periodically repeating potentials. In this way, one can study the band structure effects that arise from differently shaped potentials. One of these effects is the electron-hole mass asymmetry; more realistic unit cell potentials generally give rise to higher electron-hole mass asymmetries.
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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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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 it