Calculation and Conservation of Probability and Energy in the Numerical Solution of the Schrödinger Equation With the Finite- Difference Time-Domain Method
Bibliographic record
Abstract
The finite-difference time-domain (FDTD) method is a widely used numerical technique for solving Maxwell’s equations. FDTD has also been applied to numerically solve the Schrödinger equation and coupled systems of Maxwell and Schrödinger equations. In this work, we study the properties of the FDTD method for the Schrödinger equation, specifically the conservation of probability and energy. We propose accurate expressions for the total numerical probability and energy contained in a region and for the flux of probability current and power through its boundary. We show that the proposed expressions satisfy the conservation laws under suitable conditions and demonstrate their connection to the Courant–Friedrichs–Lewy stability limit. We discuss how these findings can be used to create new stable FDTD algorithms for the Schrödinger equation in an intuitive and modular fashion.
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 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) | 0.001 | 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 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".