Propagation matrix formalism and efficient linear potential solution to Schrödinger’s equation
Bibliographic record
Abstract
The one-dimensional Schrödinger equation for an arbitrary potential with position-dependent mass is often solved by the transfer-matrix method. While the usual definition relates wave-function coefficients on two sides of an interface, this article presents an alternative approach, in which a propagation matrix evolves the wave function and its derivative between a pair of points. The formalism is developed without an a priori commitment to a breakdown of the potential into a series of flat, linear, or other types of segments. We obtain a Wick-expansion form for the matrix and also provide a geometrical interpretation based on the SL(2,R) group. Turning to a variably spaced discretized potential we show how this approach can be flexibly applied to any potential segments. We discuss explicitly the case of constant potential and the Wentzel–Kramers–Brillouin approximation, as well as the linear potential segment. For the latter, the obtained propagation matrix has definite advantages, from both speed and robustness standpoints. Applications to transport in the ballistic regime are discussed and explicit results are presented for a InP–InGaAs junction.
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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 | 0.000 | 0.001 |
| 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 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 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 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".