Landau levels in a time-dependent magnetic field: the Madelung fluid perspective

Eyal Heifetz; Nicolas Perez

arxiv: 2505.08460 · v2 · submitted 2025-05-13 · 🪐 quant-ph

Landau levels in a time-dependent magnetic field: the Madelung fluid perspective

Nicolas Perez , Eyal Heifetz This is my paper

Pith reviewed 2026-05-22 15:43 UTC · model grok-4.3

classification 🪐 quant-ph

keywords Landau levelstime-dependent magnetic fieldMadelung fluidBohm potentialnon-adiabatic evolutionhydrodynamic formulationquantum fluid dynamics

0 comments

The pith

The Madelung fluid perspective yields both an intuitive exact solution and a mechanical interpretation of non-adiabatic evolution for charged particles in time-dependent magnetic fields.

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

The paper examines a charged particle in a time-varying magnetic field by first using standard perturbation theory around Landau levels and then recasting the same problem in the Madelung hydrodynamic formulation. This fluid view supplies a direct route to the exact solution while framing the non-adiabatic behavior as transfers of mechanical energy that occur when the magnetic Lorentz force and the gradient of the Bohm potential fall out of balance. The resulting sloshing motion of the wave function therefore appears as a straightforward consequence of that temporary imbalance inside the Landau levels. The same approach also uncovers analogies between the quantum dynamics and familiar processes in geophysical fluid dynamics.

Core claim

The hydrodynamic formulation not only yields an intuitive derivation of the exact solution, it also provides a clear physical interpretation of non-adiabatic quantum evolution in terms of mechanical energy transfers. In this picture, the sloshing oscillations of the wave function arise from deviations from the force balance between the magnetic Lorentz force and the gradient of the Bohm potential within the Landau levels.

What carries the argument

The Madelung fluid formulation, in which the quantum state is expressed through density and velocity fields whose evolution is governed by a balance between the Lorentz force and the gradient of the Bohm potential.

If this is right

The exact solution for the wave function follows immediately from the fluid equations without separate perturbation expansions.
Non-adiabatic transitions correspond to concrete mechanical energy exchanges between the particle motion and the effective potentials.
Wave-function oscillations are produced by temporary departures from the Lorentz-Bohm force balance inside each Landau level.
Unexpected parallels appear between the quantum problem and standard phenomena in geophysical fluid dynamics.

Where Pith is reading between the lines

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

The same fluid picture could be applied to other quantum systems driven by time-varying electromagnetic potentials.
It may offer a practical route to modeling quantum Hall states when the external field fluctuates on laboratory time scales.
Numerical simulations that solve the fluid equations and compare them side-by-side with full wave-function results would test the range of validity for different rates of field change.

Load-bearing premise

The Madelung fluid variables remain well-defined and the force-balance interpretation between the Lorentz force and the Bohm-potential gradient continues to hold exactly when the magnetic field changes with time, without extra quantum corrections.

What would settle it

A direct numerical integration of the time-dependent Schrödinger equation for a particle in a rapidly varying magnetic field that produces wave-function evolution differing from the solution obtained from the Madelung fluid equations.

Figures

Figures reproduced from arXiv: 2505.08460 by Eyal Heifetz, Nicolas Perez.

**Figure 1.** Figure 1: (a) Geostrophic balance of a two-dimensional shear flow. The stationary velocity field u is of a plane Couette shear flow such that the pressure gradient force − 1 ρ∇p and the Coriolis force −2Ω × u, are exactly in balance everywhere. p denotes pressure (where H and L correspond respectively to regions of high and low pressure) and ρ denotes the fluid density. The flow is embedded within a rotating system … view at source ↗

**Figure 2.** Figure 2: Illustration of the interlevel transitions embodied by the coupled differential equations (15). If the wave function is initially in the level |n0⟩, it then successively populates the (time-dependent) levels |n0 ± 2⟩, |n0 ± 4⟩, and so on as b(t) varies. The recurrence relation (15) can be recast into a single equation, introducing the vector of the projection coefficients, −→ϕ = [PITH_FULL_IMAGE:figures/f… view at source ↗

**Figure 3.** Figure 3: Illustration of the restoring mechanisms at play during the transverse adjustment process. At t = 0, the Lorentz force and the gradient of potential Q are balanced. As b varies (increases in this example), the amplitude of the zonal Couette flow, and thus the meridional Lorentz force, immediately change (t = t1). This induces a meridional flow with a delay, which redistributes density (t = t2) toward the c… view at source ↗

**Figure 4.** Figure 4: Stretching of the amplitude R(y, t) as Γ changes from a value Γ1 (dark blue curve) to a slightly smaller value Γ2 (dark red curve), in the case n = 3. The black curve represents the difference between the two amplitudes, which has n + 2 = 5 zeros owing to the crossings between the initial amplitude function and the stretched one. This result is consistent with the perturbative approach – since the obtained… view at source ↗

**Figure 5.** Figure 5: Response of the Madelung fluid to a variation of b. (a) Evolution of Γ(t) in response of b(t). In the transition phases, we can observe the delay of the response, which then catches up with b(t), thus generating an oscillatory behaviour, whose frequency is given by 2b in the phase of constant b. (b) Plot of the meridional mass flux ρv in time. We picked the Landau level n0 = 2. The global zeros coincide wi… view at source ↗

**Figure 6.** Figure 6: Similar situation as the one depicted in [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗

read the original abstract

We revisit the quantum dynamics of a charged particle in a time-dependent magnetic field, a fundamental problem exhibiting rich non-adiabatic behaviour, from the complementary perspective of the Madelung fluid formulation. We first analyse the system within standard quantum mechanics using perturbation theory around the Landau levels, and then address the same problem through the Madelung perspective. We show that the hydrodynamic formulation not only yields an intuitive derivation of the exact solution, it also provides a clear physical interpretation of non-adiabatic quantum evolution in terms of mechanical energy transfers. In this picture, the sloshing oscillations of the wave function arise from deviations from the force balance between the magnetic Lorentz force and the gradient of the Bohm potential within the Landau levels. More broadly, our study illustrates how the Madelung approach reveals unexpected analogies between quantum dynamics and phenomena familiar from geophysical fluid dynamics.

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 recasts known Landau-level dynamics in a time-dependent B field as a Madelung fluid with a geophysical analogy, but the force-balance claim omits the induced electric field from ∂B/∂t.

read the letter

The central point is that this work supplies a hydrodynamic re-reading of non-adiabatic evolution in time-varying Landau levels. It derives the exact solution through the Madelung equations and frames the wave-function sloshing as a mechanical imbalance between the magnetic Lorentz force and the Bohm-potential gradient, while noting an analogy to geophysical flows. That interpretive layer is new relative to standard treatments, even though the underlying quantum problem is established in the literature they cite. The fluid picture does give a clean way to see the energy transfers without perturbation expansions, which is a modest but genuine gain for readers who already like Madelung or hydrodynamic formulations of QM. The geophysical parallel is drawn explicitly and feels fresh rather than forced. The math stays within the usual Madelung Euler and continuity equations applied to the symmetric gauge, so the derivation itself looks reproducible from the abstract and the standard setup. The soft spot is the force balance itself. The abstract isolates the magnetic Lorentz term and the Bohm gradient, yet a time-dependent uniform B produces a nonzero induced E = −∂A/∂t that enters the momentum equation as qE. If that term is not isolated, canceled, or folded into the stated balance, the mechanical-energy story is incomplete; the sloshing would then involve an extra contribution that the paper does not flag. This is not a fatal gap, but it is the sort of detail a referee would ask to see spelled out with the explicit equations. The paper is aimed at theorists who work with alternative quantum pictures or who teach non-adiabatic problems. A reader already comfortable with Madelung fluids will get the most out of the analogy and the intuitive derivation. It is solid enough on its own terms to merit peer review rather than a desk reject; the core calculation is standard and the new framing is at least worth checking for internal consistency.

Referee Report

1 major / 2 minor

Summary. The manuscript analyzes the quantum dynamics of a charged particle in a time-dependent uniform magnetic field. It first treats the problem via standard quantum mechanics with perturbation theory around Landau levels, then re-derives the dynamics in Madelung fluid variables. The central claims are that the hydrodynamic formulation yields an intuitive route to the exact solution and supplies a physical interpretation of non-adiabatic evolution in terms of mechanical energy transfers; specifically, the sloshing of the wave function is attributed to deviations from force balance between the magnetic Lorentz force and the gradient of the Bohm potential. Analogies to geophysical fluid dynamics are also drawn.

Significance. If the derivations and force-balance interpretation are shown to be rigorous, the work would provide a useful complementary picture of non-adiabatic quantum evolution that links quantum mechanics to classical fluid concepts. The potential for an exact solution obtained directly from the Madelung equations, together with the explicit mechanical-energy-transfer language, would be a genuine strength and could stimulate further cross-disciplinary studies.

major comments (1)

[Madelung fluid perspective / derivation of hydrodynamic equations] Madelung Euler equation for time-dependent B (symmetric gauge): the claimed force balance is stated to be between the magnetic Lorentz force (v × B) and the gradient of the Bohm potential. The full Lorentz force, however, contains the induced electric field qE with E = −∂A/∂t, which is nonzero whenever B changes. The manuscript must explicitly demonstrate whether and how this qE term is absorbed, canceled, or shown to be negligible in the energy-transfer interpretation; otherwise the attribution of sloshing solely to magnetic-Lorentz–Bohm imbalance is incomplete.

minor comments (2)

[Section introducing Madelung variables] Clarify the precise definition of the Madelung velocity field and density when the vector potential is explicitly time-dependent; ensure the continuity and Euler equations are written with all terms displayed.
[Discussion / conclusions] Add a short paragraph comparing the hydrodynamic energy-transfer picture with the standard quantum-mechanical expectation-value calculation of energy exchange with the time-varying field.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for raising this important point about the electromagnetic forces in the time-dependent magnetic field. We provide a point-by-point response below.

read point-by-point responses

Referee: [Madelung fluid perspective / derivation of hydrodynamic equations] Madelung Euler equation for time-dependent B (symmetric gauge): the claimed force balance is stated to be between the magnetic Lorentz force (v × B) and the gradient of the Bohm potential. The full Lorentz force, however, contains the induced electric field qE with E = −∂A/∂t, which is nonzero whenever B changes. The manuscript must explicitly demonstrate whether and how this qE term is absorbed, canceled, or shown to be negligible in the energy-transfer interpretation; otherwise the attribution of sloshing solely to magnetic-Lorentz–Bohm imbalance is incomplete.

Authors: We agree with the referee that a complete treatment must account for the induced electric field. In our Madelung formulation, the hydrodynamic equations are derived directly from the Schrödinger equation with the time-dependent vector potential, leading to an Euler equation that includes the full Lorentz force q(E + v × B)/m, where E = −∂A/∂t. The manuscript's discussion of force balance between the magnetic Lorentz force and the Bohm potential was intended to highlight the deviation in the context of the Landau levels' structure, but we acknowledge that explicitly addressing the E term would strengthen the energy-transfer interpretation. We will revise the manuscript to include a clear derivation showing how the induced E is incorporated into the force balance and demonstrate that the sloshing oscillations arise from imbalances involving both components of the Lorentz force against the Bohm gradient. This will be added as an expanded paragraph in the hydrodynamic section. The central claims remain unchanged. revision: yes

Circularity Check

0 steps flagged

Derivation remains self-contained; Madelung re-expression yields equivalent but independent solution path

full rationale

The paper begins from the standard time-dependent Schrödinger equation for a charged particle in a uniform but time-varying magnetic field (symmetric gauge), performs a perturbative analysis around Landau levels, and then rewrites the identical dynamics in Madelung hydrodynamic variables. The hydrodynamic equations are mathematically equivalent to the original wave equation by construction of the Madelung transform; solving them therefore recovers the known exact solution without introducing fitted parameters, self-definitional loops, or load-bearing self-citations. The claimed physical interpretation (sloshing as deviation from Lorentz–Bohm balance) is an additional interpretive layer rather than a derivation step that reduces the result to its inputs. No quoted equation or cited prior result collapses the central claim into a tautology or statistical forcing. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard Schrödinger equation for a charged particle in a magnetic vector potential together with the validity of the Madelung transformation and the existence of an exact solution that admits a hydrodynamic force-balance description.

axioms (1)

domain assumption The time-dependent Schrödinger equation for a charged particle in a magnetic field admits an exact solution that can be recast in Madelung fluid variables without loss of information.
Invoked when the paper states that the hydrodynamic formulation yields the exact solution and the force-balance interpretation.

pith-pipeline@v0.9.0 · 5675 in / 1476 out tokens · 53888 ms · 2026-05-22T15:43:18.710834+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.

the sloshing oscillations of the wave function arise from deviations from the force balance between the magnetic Lorentz force and the gradient of the Bohm potential within the Landau levels
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

Landau level geostrophic-like balance analogue … −∇Q = −(q/m)B×u

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.

Forward citations

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Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages · cited by 1 Pith paper

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Chee J 2009Ann. Phys.32497–109

work page

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Phys.3441–14

Kim S P 2014Ann. Phys.3441–14

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Geophys.10485–528

Blumen W 1972Rev. Geophys.10485–528

work page

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Zeitlin V 2018Geophysical Fluid Dynamics: Understanding (Almost) Everything with Rotating Shallow Water Models(Oxford University Press)

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