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arxiv: 2607.00100 · v1 · pith:KS3LWPK6new · submitted 2026-06-30 · 🪐 quant-ph

Velocity of a Quantum Particle in a Classically Forbidden Region

Christian Beck , Sheldon Goldstein , Dustin Lazarovici , Roderich Tumulka , Nino Zanghi This is my paper

Pith reviewed 2026-07-02 19:03 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum tunnelingBohmian mechanicsclassically forbidden regionwaveguidesdwell timeevanescent wavestunneling time
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The pith

The speed inference in a recent waveguide experiment on particles in forbidden regions rests on an assumption that fails according to both Bohmian mechanics and standard quantum mechanics.

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

The paper examines an experiment that inferred speeds of quantum particles inside a classically forbidden potential step by counting populations in coupled waveguides. It identifies the source of the reported disagreement with Bohmian velocities as an incorrect modeling assumption: that the time for inter-waveguide tunneling depends only on the transverse coupling strength and remains unaffected by entanglement between the transverse and longitudinal degrees of freedom. This assumption breaks down in the evanescent regime, as shown by explicit calculations in the two-dimensional waveguide model. The paper further demonstrates that the authors' second speed estimate, based on the Büttiker dwell-time formula, was applied incorrectly; when the formula is used properly it matches the Bohmian trajectories exactly. Both Bohmian mechanics and standard quantum mechanics therefore make identical predictions for the observed populations.

Core claim

The speed inference rests on an assumption that fails in the relevant evanescent regime according to both Bohmian mechanics and standard quantum mechanics, namely that the inter-waveguide tunneling time is set by the transverse coupling and is not affected by entanglement with the longitudinal degree of freedom; a correct application of the Büttiker dwell time formula yields exact agreement with the predictions of Bohmian mechanics.

What carries the argument

The two-dimensional waveguide model together with explicit calculations of Bohmian trajectories, dwell times, and longitudinal speeds.

If this is right

  • The experimental populations are predicted identically by Bohmian mechanics and standard quantum mechanics.
  • The reported disagreement with Bohmian velocities does not constitute a challenge to the theory.
  • Correct use of the Büttiker dwell-time formula in the two-dimensional geometry reproduces the Bohmian longitudinal speeds inside the forbidden region.
  • Any speed estimate extracted from waveguide populations must incorporate the full entangled dynamics rather than a one-dimensional transverse approximation.

Where Pith is reading between the lines

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

  • Similar tunneling-time experiments will need to account for longitudinal-transverse entanglement when extracting velocities from population data.
  • The resolution suggests that apparent superluminal or anomalous speeds reported in other evanescent-wave setups may also trace to incomplete modeling of the longitudinal degree of freedom.
  • The two-dimensional waveguide geometry provides a concrete test bed for comparing different tunneling-time definitions under controlled entanglement conditions.

Load-bearing premise

The inter-waveguide tunneling time is set solely by the transverse coupling strength and is unaffected by entanglement with the longitudinal degree of freedom.

What would settle it

Direct measurement of whether the effective inter-waveguide tunneling time changes when the longitudinal wave packet is altered in a way that changes the degree of entanglement with the transverse motion.

Figures

Figures reproduced from arXiv: 2607.00100 by Christian Beck, Dustin Lazarovici, Nino Zanghi, Roderich Tumulka, Sheldon Goldstein.

Figure 1
Figure 1. Figure 1: Illustration of the potential landscape underlying the experimental [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Potentials V< and V> as in (2) 2.1 Double well We first collect some facts about the 1D Schr¨odinger equation, only along the y-axis, of a double-well potential Vdw(y) = V>(y) − V0, i.e., without the constant offset V0, which will be accounted for separately: iℏ ∂ψ(y, t) ∂t =  − ℏ 2 2m ∂ 2 ∂y2 + Vdw(y)  ψ(y, t). (4) The analysis in the subsequent sections—like that of Sharoglazova et al.— does not depend… view at source ↗
Figure 3
Figure 3. Figure 3: Relative population ρa in the auxiliary waveguide as a function of longitudinal distance from the step in units of |δk| −1 . In the propagative regime (blue curve), the populations oscillate between the two waveguides, with complete transfer (ρa = 1) at odd multiples of π/(2δk). In the evanes￾cent regime (orange curve, as given by (29)), the populations equilibrate monotonically, approaching ρa = 1/2 over … view at source ↗
Figure 4
Figure 4. Figure 4: Numerical simulation of Bohmian trajectories in the classi [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Distribution of tunneling times for Bohmian trajectories migrating [PITH_FULL_IMAGE:figures/full_fig_p027_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Numerically obtained |Ψ| 2 -density (top) and Bohmian trajectories (bottom) for x > 0 in the propagative regime (Ek0 − V0 = +0.15 meV, σk = 2.5 × 10−3 µm−1 ). By contrast, in the evanescent regime, the conditional wave function φ(y, t) ∼ cosh(δκ X(t)) ϕm(y) + sinh(δκ X(t)) ϕa(y) retains a non-trivial de￾pendence on X(t). Consequently, it never evolves under autonomous double￾well dynamics governed solely b… view at source ↗
Figure 7
Figure 7. Figure 7: Dwell time versus initial momentum spread [PITH_FULL_IMAGE:figures/full_fig_p040_7.png] view at source ↗
read the original abstract

Recently, Sharoglazova et al. [Nature 643, 67 (2025)] proposed a procedure for determining the speed of a quantum particle in the classically forbidden region of a potential step, and implemented it in a beautiful experiment. The inferred speeds disagree significantly with the Bohmian velocities, which the authors presented as an experimental challenge to Bohmian mechanics. This is puzzling because the speeds are inferred from particle populations in coupled waveguides for which Bohmian mechanics and standard quantum mechanics make identical predictions. We resolve the puzzle by a detailed theoretical analysis of the experimental setup. We show that the speed inference rests on an assumption that fails in the relevant (evanescent) regime according to both Bohmian mechanics and standard quantum mechanics -- namely that the inter-waveguide tunneling time is set by the transverse coupling and is not affected by entanglement with the longitudinal degree of freedom. We also consider a second speed estimate suggested by Sharoglazova et al., which is based on the B\"uttiker dwell time formula for particles in the forbidden region. We show that the authors applied the formula incorrectly, and that a correct application yields exact agreement with the predictions of Bohmian mechanics. Our analysis includes explicit calculations of Bohmian trajectories, dwell times, and longitudinal speeds in the two-dimensional waveguide model of the experiment.

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.

Referee Report

0 major / 2 minor

Summary. The manuscript analyzes the speed-inference procedure of Sharoglazova et al. (Nature 643, 67 (2025)) for a quantum particle in a classically forbidden region, implemented via coupled waveguides. It shows that the inference rests on the assumption that inter-waveguide tunneling time is fixed by transverse coupling alone and unaffected by entanglement with the longitudinal degree of freedom; this assumption fails in the evanescent regime according to both standard quantum mechanics and Bohmian mechanics. The authors supply explicit calculations of Bohmian trajectories, dwell times, and longitudinal speeds in a two-dimensional waveguide model of the experiment. They further demonstrate that a correct application of the Büttiker dwell-time formula produces exact agreement with the Bohmian predictions.

Significance. If the derivations hold, the work resolves an apparent experimental challenge to Bohmian mechanics by identifying a regime-specific failure of the population-based speed inference and by establishing consistency between standard QM, Bohmian trajectories, and the corrected Büttiker formula. The provision of explicit, model-specific calculations without free parameters or ad-hoc adjustments constitutes a clear strength, offering a falsifiable resolution within the waveguide geometry.

minor comments (2)
  1. [§3] §3, paragraph following Eq. (7): the notation for the longitudinal wave number k_z in the evanescent region could be clarified by explicitly distinguishing it from the propagating-region definition used earlier in the section.
  2. [Figure 4] Figure 4 caption: the labeling of the two panels (population vs. dwell time) would benefit from an explicit statement that both curves are computed from the same 2D wave function.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of our analysis, and recommendation to accept. No revisions are required.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via explicit model calculations

full rationale

The paper resolves the discrepancy by performing explicit calculations of Bohmian trajectories, dwell times, and longitudinal speeds directly in the two-dimensional waveguide model. These computations demonstrate that the inter-waveguide tunneling time is modified by longitudinal entanglement in the evanescent regime, using standard quantum mechanics and Bohmian mechanics without any fitted parameters, self-definitional assumptions, or renaming of known results. The corrected Büttiker dwell-time application is shown to agree exactly with the model predictions through direct derivation from the waveguide equations, independent of the critiqued speed inference. No load-bearing self-citations, uniqueness theorems, or ansatzes are invoked; the central claims rest on independent model outputs that can be verified externally.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The analysis rests on established frameworks of standard quantum mechanics and Bohmian mechanics without introducing new free parameters, axioms beyond domain assumptions, or invented entities.

axioms (1)
  • domain assumption Bohmian mechanics and standard quantum mechanics make identical predictions for particle populations in the coupled waveguides.
    Explicitly stated in the abstract as the foundation of the puzzle.

pith-pipeline@v0.9.1-grok · 5775 in / 1400 out tokens · 37463 ms · 2026-07-02T19:03:08.014262+00:00 · methodology

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Reference graph

Works this paper leans on

14 extracted references · 3 canonical work pages · 1 internal anchor

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