Calculation and Conservation of Probability and Energy in the Numerical Solution of the Schrödinger Equation With the Finite- Difference Time-Domain Method
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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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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) | 0.000 | 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