Landau levels, edge states, and gauge choice in 2D quantum dots
Bibliographic record
Abstract
We examine the behavior of a charged particle in a two dimensional quantum dot in the presence of a magnetic field. Emphasis is placed on the high magnetic field regime. Compared to free space geometry, confinement in a dot geometry provides a more realistic system where edge effects arise naturally. It also serves to remove the otherwise infinite degeneracy due to the magnetic field; nonetheless, as described in this paper, additional ingredients are required to produce sensible results. We treat both circular and square geometries, and in the latter, we explicitly demonstrate the gauge invariance of the energy levels and wave function amplitudes. The characteristics of bulk states closely resemble those of free space states. For edge states, with sufficiently high quantum numbers, we achieve significant differences in the square and circular geometries. Both circular and square geometries are shown to exhibit level crossing phenomena, similar to parabolic dots, where the confining potential is a parabolic trap. Confinement effects on the probability current are also analyzed; it is the edge states that contribute non-zero current to the system. The results are achieved using straightforward matrix mechanics, in a manner that is accessible to novices in the field. On a more pedagogical note, we also provide a thorough review of the theory of single electron Landau levels in free space and illustrate how the introduction of surfaces naturally leads to a more physically transparent description of a charged particle in a magnetic field.
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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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.001 | 0.000 |
| 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".