Relativistic corrections to Landau levels in the presence of a parallel linear electric field

Ariel Edery; Yann Audin

arxiv: 1907.00769 · v1 · pith:JUWK753Knew · submitted 2019-07-01 · 🪐 quant-ph · hep-th

Relativistic corrections to Landau levels in the presence of a parallel linear electric field

Yann Audin , Ariel Edery This is my paper

Pith reviewed 2026-05-25 12:08 UTC · model grok-4.3

classification 🪐 quant-ph hep-th

keywords Landau levelsrelativistic correctionsDirac equationlinear electric fieldperturbation theorycyclotron frequencyenergy degeneracyquantum numbers

0 comments

The pith

Dirac equation yields compact formulas for relativistic corrections to Landau levels with parallel linear electric field

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper applies the Dirac equation to find the first and second order relativistic corrections to the non-relativistic energies of an electron in a constant magnetic field and a linear electric field parallel to it. The corrections are given by compact formulas depending on the cyclotron frequency ω_c, the z-frequency ω_z, and the quantum numbers n and n_z. The leading correction is negative, which lowers all the energies from their non-relativistic values. When the ratio w = ω_c/ω_z is varied, the points where energy levels become degenerate are shifted compared to the non-relativistic case. For the equal frequency case w=1, the correction splits what was a degenerate level.

Core claim

Using Dirac's equation and perturbation theory, compact formulas are obtained for the first and second order relativistic corrections to the non-relativistic energies, expressed in terms of ω_c, ω_z, n, n_z. The first order correction is negative and lowers the original energies. When plotted versus w=ω_c/ω_z, degeneracies occur at different points than the non-relativistic case, and for w=1 the correction splits the levels.

What carries the argument

Perturbation theory applied to the Dirac equation solutions, with the two frequencies ω_c and ω_z characterizing the motion in the plane and along z.

If this is right

The first order relativistic correction lowers the non-relativistic energies for all states.
Energy degeneracies appear at new values of the ratio w = ω_c / ω_z.
The case ω_c = ω_z has its degeneracy split by the first order term.
Second order corrections follow a similar compact expression in the same variables.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

In stronger magnetic fields where velocities approach a non-negligible fraction of c, these corrections would grow and alter the spectrum measurably.
The same perturbative approach could be applied to other non-relativistic electromagnetic configurations whose solutions are known exactly.
Precision experiments on electrons in combined uniform B and linear E fields could directly test the size and sign of the predicted shifts.

Load-bearing premise

The electron velocity remains much smaller than the speed of light throughout the relevant parameter range so that ordinary perturbation theory converges.

What would settle it

A measurement of the energy for chosen n, n_z and w that deviates from the non-relativistic energy plus the calculated first-order (and second-order) correction term would show the formulas are incorrect.

read the original abstract

We consider an electron moving 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. Two frequencies characterize the system: the cyclotron frequency $\omega_c$ corresponding to motion along the x-y plane and associated with the usual Landau levels, and a second frequency $\omega_z$ corresponding to motion along the z-direction. In previous work, the non-relativistic energies of this system were obtained, and it was shown that an extra degeneracy (beyond the Landau degeneracy) occurs when the ratio $\text{w}=\omega_c/\omega_z$ is rational. In this paper, we use Dirac's equation to obtain compact formulas for the first and second order relativistic corrections to this system via perturbation theory. The formulas are expressed in terms of the two frequencies $\omega_c$ and $\omega_z$, and two quantum numbers, $n$ and $n_z$, both of which are non-negative integers. The first order correction is negative and lowers the original energies. We plot the energy (zeroth plus first order) versus the ratio $\text{w}$ and there are degeneracies at all points where lines intersect. However, the degeneracy does not occur at the same $\text{w}$ as before. To illustrate this, we show how the first order correction splits the energy levels for the case $\omega_c=\omega_z$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper supplies compact first- and second-order perturbative corrections from the Dirac equation and shows the degeneracy points shift, but supplies no check that the expansion remains valid in the plotted regimes.

read the letter

The main deliverable is a set of explicit formulas for the first- and second-order relativistic corrections to the non-relativistic energies of an electron in a uniform B field plus a linear E field along z. The corrections are written in terms of the two frequencies ω_c and ω_z and the integers n and n_z. The first-order term is negative and the paper demonstrates that the extra degeneracies that appear when ω_c/ω_z is rational move to different rational values once the correction is included. They also show the splitting for the ω_c = ω_z case. That shift and the closed-form expressions are absent from the earlier non-relativistic treatment, so the advance is real even if narrow. The method is ordinary time-independent perturbation theory applied to the Dirac Hamiltonian, which is the right tool for the regime. The formulas look algebraically straightforward and the plots illustrate the point about relocated degeneracies. The soft spot is exactly the one flagged in the stress-test note. Nothing in the abstract or the supplied description gives a bound on the size of the first-order correction relative to ħω_c or ħω_z for the quantum numbers and frequency ratios that appear in the figures. Without that check it is not obvious the perturbative formulas are accurate in the plotted range. If the full manuscript contains such an estimate or a direct comparison to the exact Dirac spectrum for small v/c, that would remove the concern; on present evidence the gap remains. This work is for people already working on relativistic corrections to Landau levels or on systems with combined magnetic and linear electric fields. A reader who needs the explicit correction terms or wants to track how degeneracies move under relativity will get something concrete from it. The calculation is honest and self-contained enough that a serious referee should see it, even though the result stays inside a specialized corner of relativistic quantum mechanics.

Referee Report

1 major / 2 minor

Summary. The manuscript applies time-independent perturbation theory to the Dirac equation to obtain compact analytic expressions for the first- and second-order relativistic corrections to the non-relativistic energies of an electron in a uniform B-field along z combined with a linear E-field parallel to B. The unperturbed problem is characterized by cyclotron frequency ω_c and z-oscillator frequency ω_z; extra degeneracies appear when w = ω_c/ω_z is rational. The corrections depend only on ω_c, ω_z, n and n_z; the first-order term is negative. Plots of E_0 + E_1 versus w show degeneracies at shifted locations relative to the non-relativistic case, and the ω_c = ω_z case is used to illustrate splitting.

Significance. If the perturbative treatment is justified, the explicit formulas supply a direct, parameter-free route to relativistic shifts in this mixed Landau-plus-linear-potential system and demonstrate that the rational-w degeneracies are lifted or relocated at first order. This is a modest but concrete extension of the earlier non-relativistic analysis.

major comments (1)

[sections presenting the perturbative formulas and the energy plots] The central claim that the reported compact formulas furnish accurate relativistic corrections rests on the validity of ordinary perturbation theory. No estimate, bound, or numerical check is supplied showing that |E^{(1)}| ≪ ħω_c (or ħω_z) for the values of n, n_z and w used in the plots or in the ω_c = ω_z example. Without this, the applicability of the first- and second-order expressions to the displayed regimes remains unverified.

minor comments (2)

[derivation of the corrections] Notation for the linear electric-field strength and the precise definition of the unperturbed Dirac Hamiltonian should be stated explicitly before the perturbative expansion is introduced.
[figures] Figure captions should indicate the range of n and n_z plotted and the value of any overall scaling factor used to normalize the energies.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying the need to verify the perturbative regime. We respond to the single major comment below.

read point-by-point responses

Referee: [sections presenting the perturbative formulas and the energy plots] The central claim that the reported compact formulas furnish accurate relativistic corrections rests on the validity of ordinary perturbation theory. No estimate, bound, or numerical check is supplied showing that |E^{(1)}| ≪ ħω_c (or ħω_z) for the values of n, n_z and w used in the plots or in the ω_c = ω_z example. Without this, the applicability of the first- and second-order expressions to the displayed regimes remains unverified.

Authors: We agree that the manuscript does not supply an explicit bound or numerical verification that |E^{(1)}| ≪ ħω_c (or ħω_z). This is a substantive point. In the revised version we will add a short subsection that uses the closed-form expression for E^{(1)} to derive a simple upper bound on |E^{(1)}|/ħω_c in terms of n, n_z, w and the ratio ħω_c/(mc²). We will then evaluate the bound for every (n,n_z,w) combination appearing in the figures and for the ω_c=ω_z example, thereby confirming the perturbative regime in which the reported formulas apply. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to prior non-relativistic energies; perturbative relativistic corrections derived independently

full rationale

The paper cites previous work (likely by the same authors) solely for the non-relativistic energies and eigenstates of the Landau + z-oscillator system, then performs a fresh perturbative expansion of the Dirac equation to extract first- and second-order corrections expressed in terms of ω_c, ω_z, n and n_z. No result is obtained by fitting a parameter to the same data set and relabeling it a prediction, no self-definition equates output to input, and the self-citation is not load-bearing for the relativistic formulas themselves. The derivation chain is therefore self-contained against external benchmarks once the non-relativistic base is granted.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract alone; no explicit free parameters, ad-hoc axioms or new entities are stated. The calculation rests on the standard assumption that the non-relativistic eigenstates form a valid unperturbed basis.

axioms (1)

domain assumption Non-relativistic energies and wave-functions obtained in prior work serve as the unperturbed basis for the perturbative expansion.
Explicitly referenced in the abstract as the starting point for the relativistic correction.

pith-pipeline@v0.9.0 · 5787 in / 1279 out tokens · 34492 ms · 2026-05-25T12:08:17.536459+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We use Dirac's equation to obtain compact formulas for the first and second order relativistic corrections... expressed in terms of the two frequencies ωc and ωz, and two quantum numbers, n and nz
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

H(0) is the non-relativistic Hamiltonian... H(1) and H(2) are relativistic perturbations

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.