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arXiv:2512.19051 · Physical Review A

The energy-speed relationship of quantum particles challenges Bohmian mechanics?

S. Di Matteo, C. Mazzoli

quant-phphysics.opticsSubmitted Dec 22, 2025• Announced Dec 23, 2025• 0 citations• Revised Dec 28, 2025 (v2)
v1 · Dec 22, 2025v2 · Dec 28, 2025v3 · Feb 3, 2026v4 · Mar 7, 2026v5 · Apr 6, 2026v6 · Jul 23, 2026v7 · Aug 17, 2026v8 · Oct 5, 2026
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Abstract

Recently, Sharoglazova et al. claimed to have proven a violation of the basic tenet of Bohmian mechanics, namely the phase-speed relation $\vec{v}(\vec{r},t)=\frac{\hbar}{m}\vec{\nabla}S(\vec{r},t)$. Here, $S(\vec{r},t)$ is the (real) phase of the wave function $ψ(\vec{r},t)=ρ^{\frac{1}{2}}(\vec{r},t)e^{iS(\vec{r},t)}$. In a nutshell, they have measured the speed of a claimed evanescent wave, which is real and therefore must have $\vec{\nabla}S=\vec{0}$. However, Fig. 2 clearly shows a density motion from one waveguide to the other, implying a nonzero density current, $\vec{j}(\vec{r},t)=\frac{\hbar}{m}\Im(ψ^*\vec{\nabla}ψ)$. If we combine this evidence with the mathematical identity $\vec{\nabla}S=\frac{m}ρ\vec{j}$, we should instead conclude that $\vec{\nabla}S\neq\vec{0}$. So, where does this apparent inconsistency come from?

DOI
10.1103/7swm-z666

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Recently, Sharoglazova et al. claimed to have proven a violation of the basic tenet of Bohmian mechanics, namely the phase-speed relation v⃗(r⃗,t)=ℏm∇⃗S(r⃗,t)\vec{v}(\vec{r},t)=\frac{\hbar}{m}\vec{\nabla}S(\vec{r},t)v(r,t)=mℏ​∇S(r,t). Here, S(r⃗,t)S(\vec{r},t)S(r,t) is the (real) phase of the wave function ψ(r⃗,t)=ρ12(r⃗,t)eiS(r⃗,t)ψ(\vec{r},t)=ρ^{\frac{1}{2}}(\vec{r},t)e^{iS(\vec{r},t)}ψ(r,t)=ρ21​(r,t)eiS(r,t). In a nutshell, they have measured the speed of a claimed evanescent wave, which is real and therefore must have ∇⃗S=0⃗\vec{\nabla}S=\vec{0}∇S=0. However, Fig. 2 clearly shows a density motion from one waveguide to the other, implying a nonzero density current, j⃗(r⃗,t)=ℏmℑ(ψ∗∇⃗ψ)\vec{j}(\vec{r},t)=\frac{\hbar}{m}\Im(ψ^*\vec{\nabla}ψ)j​(r,t)=mℏ​ℑ(ψ∗∇ψ). If we combine this evidence with the mathematical identity ∇⃗S=mρj⃗\vec{\nabla}S=\frac{m}ρ\vec{j}∇S=ρm​j​, we should instead conclude that ∇⃗S≠0⃗\vec{\nabla}S\neq\vec{0}∇S=0. So, where does this apparent inconsistency come from?
A recent experiment raises a supposed challenge to Bohmian mechanics, claiming to observe stationary states, which should have zero Bohm velocity, while indirectly measuring a nonzero speed based on how an evanescent wavefunction spreads from one waveguide to another coupled waveguide. There were numerous problems with this experiment and how it was interpreted. First, the experiment is not observing stationary states as claimed, but rather the time-averaged density of wave pulses which reflect off the potential step. Second, the proposed method for measuring a propagation speed is shown to be invalid for true stationary states. Third, the invalid method was misapplied to the time-averaged density, and this is shown to have created the false impression that it yields correct speed values for stationary states. These issues notwithstanding, for a wavefunction ψ=ReiS/ℏψ= Re^{iS/\hbar}ψ=ReiS/ℏ, the velocity of interest, v⃗s=−ℏm∇⃗RR\vec{v}_s = -\frac{\hbar}{m}\frac{\vec{\nabla}R}{R}vs​=−mℏ​R∇R​, is different from the Bohm velocity v⃗B=1m∇⃗S\vec{v}_B=\frac{1}{m}\vec{\nabla}SvB​=m1​∇S, and may be nonzero for stationary states. This quantity has been called the \textit{symmetric} or \textit{osmotic} velocity, and while it emerges naturally as an imaginary component of the velocity in the derivation of the Madelung/Bohm description, it is usually disregarded. So, even though we do not think this experiment makes a compelling case for it, if v⃗s\vec{v}_svs​ is somehow associated with real physical motion, then this motion is indeed absent from Bohmian mechanics, as the authors contend. We discuss a generalized Madelung fluid model where this velocity is given physical meaning, and show how it roughly agrees with the authors' concept of an evanescent De Broglie speed.
A recent article claims to measure the speed of quantum particles in the classically forbidden regime where the energy of the particles is lower than the local potential, and further claims that the results of this experiment challenge Bohmian mechanics. But this interpretation of the experiment is incorrect (and dubious even in the context of ordinary quantum mechanics). A proper analysis of the system from a pilot-wave perspective shows that it predicts the same distribution of particle positions (and so the same experimental results) as ordinary quantum theory. The speed measured by the experiment in this regime is fictitious.
In our work [Sharoglazova et al., Nature 643, 67-72 (2025)], we reported the first measurement of the speed of tunnelling particles by comparing two orthogonal motions in a coupled-waveguide system. Here we respond to objections raised in comments, referee reports, preprints, and articles. In particular, we distinguish two questions that are often conflated: whether Bohmian mechanics reproduces the measured density, and whether the standard guiding equation assigns the correct state of motion to the particles. The first point follows under the usual quantum equilibrium assumptions. By contrast, identifying the guiding velocity with particle motion is an additional physical postulate, since the standard guiding equation does not follow from the Schrödinger equation alone. We argue that, in the evanescent regime, the state of motion assigned by the standard guiding equation is in disagreement with the speed following from our experiments. To make the distinction explicit, we also present a bidirectional Bohmian model that reproduces the same stationary density while assigning finite speeds compatible with the speed inferred in the evanescent regime.
Abstract In a recent paper, Englert et al. reject Bohm's reinterpretation of quantum mechanics as "surrealistic." Responses to this by Dewdney et al. and Durr et al. have generally missed the point of this somewhat strange characterization of the interpretation. In this paper I explain the experiment which is supposed to illustrate the surrealism. I argue that what is really being objected to is the non-locality of the interpretation. While this is an undesirable feature of Bohm's interpretation, it is not a feature of which the standard interpretation gives a more satisfactory account.
Since its inception Bohmian mechanics has been generally regarded as a hidden-variable theory aimed at providing an objective description of quantum phenomena. To date, this rather narrow conception of Bohm’s proposal has caused it more rejection than acceptance. Now, after 65 years of Bohmian mechanics, should still be such an interpretational aspect the prevailing appraisal? Why not favoring a more pragmatic view, as a legitimate picture of quantum mechanics, on equal footing in all respects with any other more conventional quantum picture? These questions are used here to introduce a discussion on an alternative way to deal with Bohmian mechanics at present, enhancing its aspect as an efficient and useful picture or formulation to tackle, explore, describe and explain quantum phenomena where phase and correlation ( entanglement ) are key elements. This discussion is presented through two complementary blocks. The first block is aimed at briefly revisiting the historical context that gave rise to the appearance of Bohmian mechanics, and how this approach or analogous ones have been used in different physical contexts. This discussion is used to emphasize a more pragmatic view to the detriment of the more conventional hidden-variable (ontological) approach that has been a leitmotif within the quantum foundations. The second block focuses on some particular formal aspects of Bohmian mechanics supporting the view presented here, with special emphasis on the physical meaning of the local phase field and the associated velocity field encoded within the wave function. As an illustration, a simple model of Young’s two-slit experiment is considered. The simplicity of this model allows to understand in an easy manner how the information conveyed by the Bohmian formulation relates to other more conventional concepts in quantum mechanics. This sort of pedagogical application is also aimed at showing the potential interest to introduce Bohmian mechanics in undergraduate quantum mechanics courses as a working tool rather than merely an alternative interpretation.