New degeneracies and modification of Landau levels in the presence of a parallel linear electric field
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Bibliographic record
Abstract
We consider a three-dimensional system where an electron moves under a constant magnetic field (in the z-direction) and a \textit{linear} electric field parallel to the magnetic field above the z=0 plane and anti-parallel below the plane. The linear electric field leads to harmonic oscillations along the z-direction. There are therefore two frequencies characterizing the system: the usual cyclotron frequency $\omega_c$ corresponding to motion along the x-y plane and associated with Landau levels and a second frequency $\omega_z$ for motion along the z-direction. Most importantly, when the ratio $W=\omega_c/\omega_z$ is a rational number, the degeneracy of the energy levels does not remain always constant as the energy increases. At some energies, the degeneracy jumps i.e. it increases. In particular, when the two frequencies are equal, the degeneracy increases with each energy level. This is in stark contrast to the usual Landau levels where the degeneracy is independent of the energy. We derive compact analytical formulas for the degeneracy. We also obtain an analytical formula for the energy levels and plot them as a function of $W$. The increase in degeneracy can readily be seen in the plot at points where lines intersect. For concreteness, we consider the electric field produced by a uniformly charged ring. Besides a linear electric field in the z direction the ring produces an extra electric field in the xy plane which we treat via perturbation theory. The Landau degeneracy is now lifted and replaced by tightly spaced levels that come in "bands". The plot of the energy levels shows that there is still a degeneracy where the bands intersect.
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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 | 0.000 | 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