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Is Bohmian mechanics missing some motion? Why a recent experiment is inconclusive

Mordecai Waegell

TL;DR

This work critiques a recent claim that stationary quantum states possess nonzero motion by showing the experiment measures time-averaged pulse densities rather than true stationary eigenstates and demonstrates that the proposed diagnostic for propagation speed is invalid in the stationary-state limit. It distinguishes between the Bohm velocity $\\vec{v}_B$ and the symmetric velocity $\\vec{v}_s$ from a generalized Madelung framework, arguing that $\\vec{v}_s$ may carry physical meaning but does not affect the total density flow $\\rho = R^2$. Through a numerical 2D Schrödinger simulation with a tunable potential that mimics coupled waveguides, the authors assess stationary-state speeds and time-averaged-pulse speeds, finding good agreement between the evanescent De Broglie speed $v_{DB}$ and $\\vec{v}_s$ inside the step, but poor agreement with the original experimental method for stationary states. For time-averaged pulses, the inferred speeds approximate $v_{DB}$ near the barrier, explaining the experimental results as time-averaged Bohm-velocity effects rather than genuine stationary-state motion. The study thus tempers claims of evidence against Bohmian mechanics, while highlighting subtle connections between evanescent effects, energy eigenvalues, and Madelung-type momentum densities that merit further exploration.

Abstract

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 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 $ψ= Re^{iS/\hbar}$, the velocity of interest, $\vec{v}_s = -\frac{\hbar}{m}\frac{\vec{\nabla}R}{R}$, is different than the Bohm velocity $\vec{v}_B=\frac{1}{m}\vec{\nabla}S$, and may be nonzero for stationary states. So, even though we do not think this experiment makes a compelling case for it, if $\vec{v}_s$ 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.

Is Bohmian mechanics missing some motion? Why a recent experiment is inconclusive

TL;DR

This work critiques a recent claim that stationary quantum states possess nonzero motion by showing the experiment measures time-averaged pulse densities rather than true stationary eigenstates and demonstrates that the proposed diagnostic for propagation speed is invalid in the stationary-state limit. It distinguishes between the Bohm velocity and the symmetric velocity from a generalized Madelung framework, arguing that may carry physical meaning but does not affect the total density flow . Through a numerical 2D Schrödinger simulation with a tunable potential that mimics coupled waveguides, the authors assess stationary-state speeds and time-averaged-pulse speeds, finding good agreement between the evanescent De Broglie speed and inside the step, but poor agreement with the original experimental method for stationary states. For time-averaged pulses, the inferred speeds approximate near the barrier, explaining the experimental results as time-averaged Bohm-velocity effects rather than genuine stationary-state motion. The study thus tempers claims of evidence against Bohmian mechanics, while highlighting subtle connections between evanescent effects, energy eigenvalues, and Madelung-type momentum densities that merit further exploration.

Abstract

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 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 , the velocity of interest, , is different than the Bohm velocity , and may be nonzero for stationary states. So, even though we do not think this experiment makes a compelling case for it, if 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.

Paper Structure

This paper contains 7 sections, 2 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Bohm velocity and the symmetric velocity
  3. Numerical Simulation
  4. The Potential Step and Coupling Constant
  5. Simulated Speeds for Stationary Energy Eigenstates
  6. Simulated Speeds for Time-Averaged Pulse Densities
  7. Conclusions

Figures (3)

  • Figure 1: The potential used for our simulation, which is designed to closely mimic the one used in the experiment, shown roughly to scale. We cut off the highest values of the potential here so that the color map provide a more visible contrast. We used a similar color map to the one used in Fig. 1b of the article to aid comparison, but it is not quite a perfect match.
  • Figure 2: 3D image of the smoothed potential energy surface used in the simulation.
  • Figure 3: Evanescent speeds, just inside the step, using an effective 1D treatment for comparison with sharoglazova2025energy. $v_{DB}$ is the so-called De Broglie speed for a particle with $x$-energy $E_x$. For the 72 eigenstates with $n_y=0$, the of symmetric speed $\vec{v}_s = -\frac{\hbar}{m}\frac{\vec{\nabla} R}{R}$ is computed and evaluated at the center of the main waveguide ($y = 8$$\mu$m). For the same 72 eigenstates, the relative centerline population of the auxiliary waveguide, $\rho_a(x) \equiv |\psi_a(x)|^2/[|\psi_m(x)|^2 + |\psi_a(x)|^2]$$= |\psi(x,-8$$\mu$m$)|^2/[|\psi(x,8$$\mu$m$) + |\psi(x,-8$$\mu$m$)|^2]$ for $0<x<10.5$$\mu$m, was fit to the function $Cx^2$, from which the inferred propagation speed is $v_{\textrm{eigenstate}}^{\textrm{fit}} = J_0/\sqrt{C}$ (the $n_x = 65$ and $n_x=70$ eigenstates are poorly approximated as separable, and do not fit the same pattern as the other $n_x$ values). Wave pulse reflections were simulated at 42 different average energies, separated by roughly $0.01$ meV, and time-averaged densities $\bar{\rho}(x,y) = \frac{1}{T}\int_T |\psi(x,y,t)|^2 dt$ were computed. For these 42 cases, the relative centerline population of the auxiliary waveguide, $\rho_{\bar{a}}(x) \equiv \bar{\rho}(x,-8$$\mu$m$)/[\bar{\rho}(x,8$$\mu$m$) + \bar{\rho}(x,-8$$\mu$m$)]$ was fit to function $Cx^2$, from which the inferred propagation speed is $v_{\bar{\rho}}^{\textrm{fit}} = J_0/\sqrt{C}$.