Trajectories in coupled waveguides: an application to a recent experiment and Hiley's lessons on the falsification of the Bohmian model

A. Matzkin; F. Daem; T. Durt

arxiv: 2510.21504 · v1 · submitted 2025-10-24 · 🪐 quant-ph

Trajectories in coupled waveguides: an application to a recent experiment and Hiley's lessons on the falsification of the Bohmian model

F. Daem , T. Durt , A. Matzkin This is my paper

Pith reviewed 2026-05-18 04:37 UTC · model grok-4.3

classification 🪐 quant-ph

keywords Bohmian mechanicsde Broglie-Bohm trajectoriescoupled waveguidesquantum tunnelingquantum interpretationcontextualityfalsification of interpretations

0 comments

The pith

When applied correctly, the Bohmian model produces the same results as standard quantum mechanics in coupled waveguide tunneling.

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

This paper shows that the Bohmian interpretation of quantum mechanics, when used properly, gives identical results to the usual quantum mechanical calculations for particles tunneling through coupled waveguides. It does this by explicitly computing the trajectories in both a simple one-dimensional case and the full two-dimensional waveguide setup. The work responds to a recent experiment that claimed to falsify the Bohmian model by demonstrating that the apparent contradiction disappears with correct application. It also points out that Bohmian trajectories are contextual, changing depending on whether or not a measurement apparatus is interacting with the system. Readers interested in quantum foundations would care because this resolves a potential refutation and illustrates the importance of careful modeling in tests of alternative interpretations.

Core claim

The authors establish that de Broglie-Bohm trajectories for a particle in coupled waveguides match the predictions of standard quantum theory. Calculations in a one-dimensional model and in the two-dimensional geometry of the waveguides confirm this agreement. The paper recalls that the trajectories of a closed system differ from those observed when the system interacts with a measurement apparatus, due to the contextual character of the Bohmian model. This shows that the recent experiment does not falsify the model when it is applied correctly.

What carries the argument

De Broglie-Bohm trajectories computed in the coupled-waveguide geometry. These trajectories serve as the explicit paths that the particle follows according to the Bohmian model, and the paper uses them to demonstrate equivalence with quantum predictions when the full context is considered.

If this is right

The recent experiment claiming to challenge the Bohmian model does not succeed when the model is correctly applied.
Trajectories in the one-dimensional tunneling model align with standard quantum results.
Full two-dimensional computations in the waveguide setup also agree with quantum mechanics.
Accounting for contextuality in measurements is essential for any valid test of the Bohmian interpretation.

Where Pith is reading between the lines

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

Future experiments testing Bohmian trajectories should model the complete detection process to avoid misinterpretation.
This approach may apply to other claimed falsifications of Bohmian mechanics in similar tunneling or interference setups.
Simulating the interaction with which-way detectors within the Bohmian framework could provide further confirmation.

Load-bearing premise

That the claimed falsification in the recent experiment can be resolved just by recalculating the trajectories in the waveguide setup without needing to model the detection process in detail.

What would settle it

A full Bohmian simulation of particle paths that includes the specific interaction with the measurement apparatus from the recent experiment and finds disagreement with the observed data would undermine the conclusion that correct application removes any discrepancy.

Figures

Figures reproduced from arXiv: 2510.21504 by A. Matzkin, F. Daem, T. Durt.

**Figure 2.** Figure 2: FIG. 2. Schematics showing the two lowest energy levels in a double well represented with the [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Time evolution of the probability density [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Bohmian trajectories computed for the 2D potential of Eq. (2). Two sets of trajectories are [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

read the original abstract

From "surreal" trajectories to which-way measurements, Basil Hiley had a lesson: claims of falsifying the Bohmian model do not withstand scrutiny provided the model is applied correctly. In this work we compute de Broglie-Bohm trajectories for particles tunneling in coupled waveguides relevant to a recent experiment having claimed to challenge the Bohmian model. We show that the Bohmian model - correctly applied - gives results identical to the standard quantum approach, first by working out a simple one-dimensional model, and then by computing Bohmian trajectories for the full two-dimensional problem representing a quantum particle propagating inside coupled waveguides. We further recall the contextual nature of the Bohmian trajectories whereby the trajectories of a closed system differ from the ones observed when an interaction with a measurement apparatus takes places.

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

Bohmian trajectories match standard QM in the waveguide propagation but the paper stops short of modeling the detector interaction that drives the experimental claim.

read the letter

The main takeaway is that this paper runs de Broglie-Bohm trajectories through both a simple 1D model and the full 2D coupled-waveguide geometry from the recent experiment, and the paths line up with ordinary quantum predictions for tunneling and probability current. The numerical integration of the guidance equation is the concrete step they add. They also restate the contextual point that trajectories in the closed system will not match those seen after interaction with a measurement apparatus. That part draws directly from Hiley and keeps the argument from overreaching on the propagation stage alone. The 2D simulation is the part that earns credit; it moves beyond the 1D toy case and applies the method to the actual potential used in the experiment. The calculations appear straightforward and reproducible from the description, with no obvious free parameters or circular fitting. The soft spot is exactly the one the stress-test note flags. The experiment's falsification claim centers on detector outcomes, yet the work computes trajectories only inside the waveguides and then recalls, without simulating, that measurement changes the picture. This leaves the rebuttal indirect. It is not a load-bearing contradiction with the paper's own claims, but it does mean the consistency check covers only part of the setup the experiment actually reports. The paper is aimed at readers who already follow Bohmian mechanics and specific tunneling experiments. Someone working on waveguide implementations or interpretive disputes would find the numerical check useful for reference. I would send it to peer review. The calculations are specific enough that referees could examine the boundary conditions and 2D numerics, and the contextual reminder is worth having on record even if more detector modeling would strengthen the response.

Referee Report

2 major / 1 minor

Summary. The manuscript computes de Broglie-Bohm trajectories for a one-dimensional tunneling model and the full two-dimensional coupled-waveguide geometry relevant to a recent experiment. It claims that, when applied correctly, the Bohmian model produces results identical to standard quantum mechanics for propagation and tunneling, while recalling the contextual dependence of trajectories on measurement interactions to argue that prior falsification claims do not withstand scrutiny.

Significance. If the central consistency result holds, the work supplies explicit numerical support for agreement between Bohmian trajectories and the probability current in waveguide geometries, together with a clear reminder of contextuality. This strengthens methodological lessons from Hiley on proper application of the model and could help clarify apparent experimental tensions, provided the detection stage is addressed.

major comments (2)

[Abstract and Introduction] Abstract and Introduction: the claim that waveguide-only trajectory computations suffice to refute the recent experiment's falsification rests on the assumption that closed-system propagation matches observed detector outcomes; the paper recalls contextual dependence but does not model the which-way detector coupling or post-interaction trajectories, leaving the link to experimental data unverified.
[Section describing the two-dimensional simulation] Section describing the two-dimensional simulation: while the 2D integration of the guidance equation demonstrates consistency with standard QM for particle propagation inside the waveguides, the absence of an explicit detector model means the trajectories remain those of the closed system; a direct comparison to the experiment requires extending the computation to include measurement-apparatus interaction to test whether the contextual shift preserves agreement.

minor comments (1)

[Numerical methods] Clarify the precise numerical scheme, grid resolution, and boundary conditions used for integrating the guidance equation in both the 1D and 2D cases to facilitate independent verification.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. Our work focuses on demonstrating the consistency of correctly computed Bohmian trajectories with standard quantum mechanics in coupled waveguides and on recalling Hiley's lessons regarding contextuality to address claims of falsification. We respond to each major comment below.

read point-by-point responses

Referee: [Abstract and Introduction] Abstract and Introduction: the claim that waveguide-only trajectory computations suffice to refute the recent experiment's falsification rests on the assumption that closed-system propagation matches observed detector outcomes; the paper recalls contextual dependence but does not model the which-way detector coupling or post-interaction trajectories, leaving the link to experimental data unverified.

Authors: We appreciate this point. Our manuscript does not claim to simulate the full experimental apparatus including the which-way detector. Instead, we compute trajectories for the propagation stage in the waveguides (both 1D tunneling model and full 2D geometry) and show they reproduce the standard quantum probability current. The recent experiment's falsification claim rested on an incorrect application of the Bohmian model that neglected contextuality. By explicitly recalling that trajectories are contextual and depend on the full measurement interaction, we argue that the claimed falsification does not hold when the model is applied correctly to the closed-system propagation relevant to the experiment. We agree that a complete detector model would strengthen the link to raw data and will revise the abstract and introduction to more precisely delimit the scope of our refutation to the propagation phase and the methodological lesson from Hiley. revision: partial
Referee: [Section describing the two-dimensional simulation] Section describing the two-dimensional simulation: while the 2D integration of the guidance equation demonstrates consistency with standard QM for particle propagation inside the waveguides, the absence of an explicit detector model means the trajectories remain those of the closed system; a direct comparison to the experiment requires extending the computation to include measurement-apparatus interaction to test whether the contextual shift preserves agreement.

Authors: We agree that the trajectories we compute are those of the closed system. The central methodological point, drawn from Hiley, is precisely that one cannot extract trajectories from the closed-system wave function and apply them unchanged once a measurement interaction occurs. The experiment's claim of falsification appears to have done exactly that. Our 2D computation establishes what the correct closed-system trajectories are and confirms agreement with standard QM; the contextual reminder then explains why a direct, non-contextual comparison to detector outcomes is invalid. Extending the simulation to include an explicit detector model is a natural next step but lies outside the present scope, which centers on waveguide propagation and the lessons for proper application of the Bohmian model. We will add a clarifying paragraph in the relevant section to emphasize this distinction and the limits of the current calculation. revision: partial

Circularity Check

0 steps flagged

No significant circularity; results from explicit numerical integration of guidance equation

full rationale

The paper's central derivation applies the standard de Broglie-Bohm guidance equation to the wave function obtained from the Schrödinger equation in a 1D model and then the full 2D coupled-waveguide geometry. These steps consist of direct numerical computation of trajectories for the given potential, which reproduces standard quantum statistics by the design of the theory but does so through explicit integration rather than by redefining inputs or fitting parameters. The recall of contextual dependence is a standard feature of Bohmian mechanics and does not serve as a load-bearing self-citation that forces the result. No self-definitional loops, fitted inputs renamed as predictions, or ansatz smuggling via citation are present in the described chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The calculation rests on the standard de Broglie-Bohm guidance equation and the Schrödinger equation for the waveguide potential; no new free parameters or invented entities are introduced.

axioms (2)

standard math The guidance equation v = (ħ/m) Im(ψ* ∇ψ / |ψ|^2) determines particle trajectories from the wave function.
Invoked throughout the computation of trajectories in both the 1D and 2D models.
domain assumption The wave function evolves according to the time-dependent Schrödinger equation in the coupled-waveguide potential.
Used to obtain the probability current that is shown to match the Bohmian velocity field.

pith-pipeline@v0.9.0 · 5679 in / 1232 out tokens · 27880 ms · 2026-05-18T04:37:17.765144+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages · 3 internal anchors

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left” and “right

Tunnel oscillations Let us now focus on the region between the two wells where the trapped particle exhibits tunnel oscillations. Let us define the “left” and “right” wave functions as follows: 6 0-8 8 16 -16 0 0.2 -0.2 y FIG. 2. Schematics showing the two lowest energy levels in a double well represented with the dashed lines: fundamental level in blue, ...

work page
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[1] [1]

Lowest energy levels. As the potential is symmetric around the origin of theyaxis, one can predict that the lowest energy level (of energyE0) is associated in the position representation with an even function of the position and the first excited level (of energyE1) with an odd function (Fig. 2). These levels are determined, as is usually done for simple ...

work page

[2] [2]

left” and “right

Tunnel oscillations Let us now focus on the region between the two wells where the trapped particle exhibits tunnel oscillations. Let us define the “left” and “right” wave functions as follows: 6 0-8 8 16 -16 0 0.2 -0.2 y FIG. 2. Schematics showing the two lowest energy levels in a double well represented with the dashed lines: fundamental level in blue, ...

work page

[3] [3]

(6) does not vanish in the tunnel region in between the two wells, which is the main result of this section

Tunneling current density and non-zero Bohmian velocities The state|ψ(t)⟩fulfills Schrödinger’s equation in between the two wells so that the Born density|ψ(y, t)| 2 obeys the conservation equation∂ ∂t |ψ(y, t)|2+ ∂ ∂y Jy(y, t) = 0, where J(y, t) = ℏ m Im [ψ∗(y, t)∂yψ(y, t)].(5) Starting from|ψ(y, t)| 2 =|ψ(0,0)| 2 · |ch y L0 tunnel e−i E0 ℏ t +sh y L1 tu...

work page

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De Broglie, J

L. De Broglie, J. Phys. Radium, 8, 225 (1927)

work page 1927

[5] [5]

Bohm, Phys

D. Bohm, Phys. Rev. 85, 166 (1952) ; Phys. Rev. 85, 180 (1952)

work page 1952

[6] [6]

B. J. Hiley, Some Personal Reflections on Quantum Non-locality and the Contributions of John Bell, arXiv:1412.0594v1 [physics.hist-ph], 2014

work page internal anchor Pith review Pith/arXiv arXiv 2014

[7] [7]

E Hoffmann, J

S. E Hoffmann, J. Phys. A: Math. Theor. 52 225301 (2019)

work page 2019

[8] [8]

Daem and A

F. Daem and A. Matzkin, Effects of superradiance on relativistic Foldy-Wouthuysen densities Phys. Rev. A 111, L060202, 2025 13

work page 2025

[9] [9]

Bohm and B

D. Bohm and B. J. Hiley,The Undivided Universe(Routledge, London, 1993)

work page 1993

[10] [10]

Einstein, in M

A. Einstein, in M. Born and A. Einstein, The Born-Einstein letters (Macmillan, 1971), p. 192

work page 1971

[11] [11]

Surrealistic Bohm Trajectories

Englert, J.; Scully, M.O.; Sï¿œssman, G.; Walther, H. Surrealistic Bohm Trajectories. Z. Naturforsch. 47, 1175 (1992)

work page 1992

[12] [12]

Do Bohm trajectories always provide a trustworthy physical picture of particle motion? Phys

Scully, M. Do Bohm trajectories always provide a trustworthy physical picture of particle motion? Phys. Scr. , 76, 41–46 (1998)

work page 1998

[13] [13]

Daem and A

F. Daem and A. Matzkin, Dynamics, locality and weak measurements: trajectories and which- way information in the case of a simplified double-slit setup Quantum Stud.: Math. Found. 11, 567, 2024

work page 2024

[14] [14]

Quantum trajectories, real, surreal or an approximation to a deeper process?

Hiley, B. J., Callaghan, R. E. and Maroney, O., Quantum Trajectories, Real, Surreal, or an Approximation to a Deeper Process. quant-ph/0010020, (2000)

work page internal anchor Pith review Pith/arXiv arXiv 2000

[15] [15]

B. J. Hiley and R. E. Callaghan, What is erased in the quantum erasure? Found. Phys. 36, 1869 (2006)

work page 2006

[16] [16]

B. J. Hiley, Welcher Weg Experiments from the Bohm Perspective, AIP Conf. Proc. 810, 154 (2006)

work page 2006

[17] [17]

J Hiley and P

B. J Hiley and P. Van Reeth Quantum Trajectories: Real or Surreal? Entropy, 20, 353 (2018)

work page 2018

[18] [18]

& Klaers, J

Sharoglazova, V., Puplauskis, M., Mattschas, C., Toebes, C. & Klaers, J. Energy–speed relationship of quantum particles challenges Bohmian mechanics. Nature 643, 67 (2025)

work page 2025

[19] [19]

Durt, Do (es the influence of) empty waves survive in configuration space? Found

T. Durt, Do (es the influence of) empty waves survive in configuration space? Found. Phys. 53, 13, 2023

work page 2023

[20] [20]

Muga and C.R

J.G. Muga and C.R. Leavens, Arrival time in quantum mechanics, Phys. Rep. 338, 353 (2000)

work page 2000

[21] [21]

P. R. Holland,The Quantum Theory of Motion(Cambridge University Press, Cambridge, England, 1993)

work page 1993

[22] [22]

Matzkin, Bohmian mechanics, the quantum-classical correspondence and the classical limit: the case of the square billiard, Found

A. Matzkin, Bohmian mechanics, the quantum-classical correspondence and the classical limit: the case of the square billiard, Found. Phys. 39, 903, 2009

work page 2009

[23] [23]

Matzkin and V

A. Matzkin and V. Nurock, Classical and Bohmian trajectories in semiclassical systems: Mismatch in dynamics, mismatch in reality?, Stud. Hist. Phil. Mod. Phys. 39, 17, 2008

work page 2008

[24] [24]

Tunneling photons pose no challenge to Bohmian mechanics

Y.-F. Wang, X.-Y. Wang, H. Wang and C.-Y. Lu, Tunnelling photons pose no challenge to Bohmian machanics, arXiv:2507.20101 [quant-ph], 2025

work page internal anchor Pith review Pith/arXiv arXiv 2025

[25] [25]

Energy-speedrelationshipofquantum particles challenges Bohmian mechanics

A.Drezet, D.LazaroviciandB.M.Nabet, Commenton"Energy-speedrelationshipofquantum particles challenges Bohmian mechanics", arXiv:2508.04756 [quant-ph], 2025. 14

work page arXiv 2025

[26] [26]

Durt, Testing de Broglie’s Double Solution in the Mesoscopic Regime, Found

T. Durt, Testing de Broglie’s Double Solution in the Mesoscopic Regime, Found. Phys. 53, 2, 2023. 15

work page 2023