Propagation matrix formalism and efficient linear potential solution to Schrödinger’s equation
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
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.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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