⚛️ quantum physics

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

This paper argues that a recent experiment claiming to challenge Bohmian mechanics by observing nonzero motion in stationary states is flawed due to misinterpretation of time-averaged data and invalid measurement methods, while acknowledging that the experiment's underlying concept of a nonzero "osmotic" velocity, though absent from standard Bohmian mechanics, can be physically meaningful within a generalized Madelung fluid model.

Original authors: Mordecai Waegell

Published 2026-02-18

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Mordecai Waegell

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: A "Ghost" in the Machine?

Imagine you are watching a movie of a ball bouncing back and forth inside a box. In the real world, the ball is always moving. But in Bohmian Mechanics (a specific way of understanding quantum physics), if the ball is in a "stationary state" (a perfect, unchanging pattern), the theory says the ball is actually frozen in place. It has zero speed, even though it has energy.

Recently, a group of scientists ran an experiment using light waves (which act like quantum particles) to test this. They claimed to see "ghostly motion" in these stationary states. They argued that while the main wave looks still, there is a hidden, invisible speed flowing through it that Bohmian mechanics missed.

This paper argues that the experiment didn't actually find a ghost. Instead, the scientists saw a "movie" of a moving wave, but because they looked at it for a long time, the motion blurred into a still picture. They mistakenly thought the blur was a stationary ghost, and their math tricked them into thinking they measured a new kind of speed.

The Three Main Problems with the Experiment

The author, Mordecai Waegell, breaks down why the experiment's conclusion is wrong using three main points:

1. The "Long Exposure" Mistake

The Analogy: Imagine taking a photo of a race car driving back and forth very quickly. If you leave your camera shutter open for a long time (a "long exposure"), the car will look like a blurry, stationary streak across the track. If you didn't know it was a long exposure, you might think the car is just a static, blurry object sitting there.

The Reality: The experimenters claimed they were looking at "stationary states" (frozen waves). But in reality, they were sending pulses (like little packets of energy) down a track. These pulses hit a wall, bounced back, and moved around. The "stationary" patterns they saw were just the time-averaged blur of these moving pulses. They weren't looking at frozen waves; they were looking at the average of a moving wave.

2. The Broken Ruler

The Analogy: Imagine trying to measure the speed of a car by looking at how much paint it leaves on the road. If the car is actually stopped, but the paint is spreading because of wind, your "paint measurement" will give you a fake speed.

The Reality: The experimenters used a specific mathematical method to calculate speed based on how the wave "spread" into a side channel. The author ran computer simulations of true stationary states (where the wave is actually frozen). He found that this mathematical method fails completely for true stationary states. It gives the wrong answer. The method is like a broken ruler that only works if you are lying about what you are measuring.

3. The Lucky Coincidence

The Analogy: Imagine you use a broken ruler to measure a moving car. By pure luck, the broken ruler gives you a number that matches the car's actual speed. You might think, "Wow, my broken ruler works!" But it's just a coincidence.

The Reality: The experimenters used their broken ruler on the "blurry" moving pulses (from point #1). Surprisingly, the numbers they got matched the theoretical speed they were looking for. This made them think they had successfully measured the "ghost speed" in stationary states.
The Truth: It was a double error. They measured a moving wave, used a broken ruler, and got a lucky number that looked right. It wasn't a discovery of new physics; it was a perfect storm of mistakes.

The "Missing Motion": Is There Something Real?

Even though the experiment was flawed, the author admits there is an interesting idea hidden in the math.

In the equations of quantum mechanics, there are actually two types of velocity:

The Bohm Velocity: The "real" flow of the particle (which is zero in stationary states).
The Symmetric (or Osmotic) Velocity: A mathematical quantity that represents a kind of "jitter" or internal pressure. It is non-zero even when the particle is "frozen."

The Analogy: Think of a spinning top that is standing perfectly still on a table.

The Bohm velocity is the top's movement across the table (it's zero; it's not going anywhere).
The Symmetric velocity is the internal spin and vibration of the top. It has energy and "motion" inside it, even if the top isn't traveling across the room.

The experimenters were trying to measure this "internal spin" (the symmetric velocity). The author says:

Bohmian Mechanics ignores this "internal spin" and only cares about the travel across the table.
Other theories (like the "Local Many Worlds" model) suggest this "internal spin" might be real physical motion.

The Verdict: The author concludes that while this "internal spin" might be a real physical thing that Bohmian mechanics ignores, this specific experiment did not prove it. The data was just a confusing mix of moving waves and bad math.

Summary

The Claim: A recent experiment proved Bohmian mechanics is wrong because it found "motion" in stationary states.
The Reality: The experiment was looking at moving waves, not stationary ones. Their math was broken for stationary states. The results were a lucky coincidence.
The Takeaway: Bohmian mechanics is still safe from this specific challenge. However, the idea of a "hidden internal motion" (symmetric velocity) is still an interesting mystery that might need a new theory to explain, but we haven't found the proof yet.

1. Problem Statement

The paper addresses a recent experimental claim (referenced as [1]) that challenges Bohmian mechanics (de Broglie–Bohm pilot-wave theory).

The Claim: The experiment purportedly observed nonzero particle speeds within stationary eigenstates of a 2D Schrödinger equation.
The Conflict: Standard Bohmian mechanics predicts that particles in stationary eigenstates have zero velocity ( $\vec{v}_B = 0$ ) because the phase gradient of the wavefunction is zero.
The Proposed Mechanism: The experimental authors argued they measured a "De Broglie speed" in the evanescent region (tunneling region) based on the spatial spreading rate of the wavefunction between coupled waveguides. They suggested this implies a "symmetric" or "osmotic" velocity ( $\vec{v}_s$ ) exists as a real physical motion absent from standard Bohmian mechanics.

2. Methodology

Waegell employs a combination of theoretical analysis and high-fidelity numerical simulation to deconstruct the experiment and test the validity of the proposed measurements.

Numerical Simulation of the Potential:
- The author numerically solved the 2D Schrödinger equation using a potential landscape ( $V(x,y)$ ) that closely mimics the optical setup used in the original experiment (a waveguide with a potential step and a coupled auxiliary waveguide).
- The simulation used an effective particle mass of $m = 6.95 \times 10^{-36}$ kg and a mesh resolution of $0.5 \, \mu m$ (x) and $0.1 \, \mu m$ (y).
- The potential was tuned to match the experimental coupling constant ( $J_0 \approx 2\pi(6.34)$ GHz) and barrier height ( $V_0 \approx 0.538$ meV).
Analysis of Stationary Eigenstates:
- The author computed 103 bound eigenstates ( $\psi_n$ ) of the simulated potential.
- For these true stationary states, three quantities were calculated and compared:
  1. Evanescent De Broglie Speed: $v_{DB} = \sqrt{2(V_0 - E_x)/m}$ .
  2. Symmetric (Osmotic) Velocity: $\vec{v}_s = -\frac{\hbar}{m} \frac{\nabla R}{R}$ (where $\psi = Re^{iS/\hbar}$ ).
  3. Experimental Inference Method: The method proposed by the original authors, which infers speed based on the parabolic fitting of population spreading ( $Cx^2$ ) between the main and auxiliary waveguides.
Simulation of Wave Pulses (Time-Averaged Density):
- To replicate the experimental conditions, the author simulated Gaussian wave pulses (initially at rest on a ramp) that accelerate, propagate, and reflect off the potential step.
- Time-averaged densities ( $\bar{\rho}$ ) were computed over the duration of the pulse reflection.
- The author applied the original authors' inference method to these time-averaged densities to see if it yielded "correct" speeds.

3. Key Contributions and Findings

A. Identification of Three Critical Errors in the Original Experiment

Misidentification of States: The experiment did not observe stationary eigenstates. Instead, it observed the time-averaged density of propagating wave pulses reflecting off a potential step. While the interference patterns of these pulses resemble standing waves, they are dynamic, not stationary.
Invalidity of the Measurement Method for Stationary States: When applied to true stationary eigenstates (numerically solved), the authors' method of inferring speed from spatial spreading rates yields values that do not agree with the theoretical De Broglie speed or the symmetric velocity. The method fails for the specific case it was intended to test.
Compounding Errors (The Illusion of Success): The original authors mistakenly applied their invalid method to the time-averaged density of the moving pulses. Because the pulses were moving, the Bohm velocity ( $\vec{v}_B$ ) caused actual population transfer. The resulting "measured" speeds coincidentally matched the theoretical De Broglie speed for the pulse's average energy, creating a false impression that the method works for stationary states.

B. Theoretical Clarification of Velocities

Bohm Velocity ( $\vec{v}_B$ ): Defined as $\frac{1}{m}\nabla S$ . It is zero for stationary eigenstates and drives the evolution of the probability density $\rho = R^2$ .
Symmetric/Osmotic Velocity ( $\vec{v}_s$ ): Defined as $-\frac{\hbar}{m}\frac{\nabla R}{R}$ $- \frac{ℏ}{m} \frac{\nabla R}{R}$ . It is generally nonzero in stationary states.
- It arises from the imaginary component of the momentum density in the Madelung/Bohm derivation.
- It is associated with local velocity distributions that sum to zero (hence "symmetric") or random fluctuations (hence "osmotic").
- Crucial Distinction: While $\vec{v}_s$ is nonzero, it does not contribute to the net flow of the total probability density $\rho$ . Therefore, it does not cause the spatial spreading observed in the experiment.

C. Physical Interpretation

The paper acknowledges that if $\vec{v}_s$ represents real physical motion (as suggested by time-dilation effects in muon bound states), then standard Bohmian mechanics is indeed "missing" this motion.
However, the paper argues that the Generalized Madelung Fluid Model (or Bohmian Mechanics with Complex Action) naturally incorporates $\vec{v}_s$ as a physical flow, distinct from the standard Bohm trajectory.
The experiment fails to provide evidence for this because it measured the effects of $\vec{v}_B$ (pulse motion) while misinterpreting the data as evidence for $\vec{v}_s$ in stationary states.

4. Results

Numerical Discrepancy: For true stationary eigenstates, the speed inferred via the experimental method (population spreading) showed poor agreement with both the De Broglie speed and the symmetric velocity, particularly at lower energies.
Pulse Simulation: The simulation confirmed that the "measured" speeds in the original experiment were consistent with the Bohm velocity of the moving pulses, not the symmetric velocity of a stationary state.
Coincidental Agreement: The apparent agreement between the experimental data and the theoretical De Broglie speed was an artifact of the time-averaging of the moving pulse, not a validation of the measurement technique for stationary states.

5. Significance

Defense of Bohmian Mechanics: The paper demonstrates that the recent experiment does not constitute a valid challenge to Bohmian mechanics. The claim of nonzero velocity in stationary states is based on a misinterpretation of dynamic pulse data as static eigenstate data.
Clarification of Quantum Velocities: It rigorously distinguishes between the Bohm velocity (real, drives density flow) and the symmetric/osmotic velocity (imaginary component in derivation, potentially physical but directionless in net flow).
Cautionary Tale: The work highlights how a confluence of experimental error (misidentifying the state) and mathematical coincidence (time-averaging mimicking the target value) can create a convincing but false physical interpretation.
Future Directions: While the specific experiment is inconclusive, the paper validates the theoretical interest in the symmetric velocity. It suggests that generalized models (like the local many-worlds model or complex action theories) where $\vec{v}_s$ has ontological status remain viable and worthy of further investigation, independent of this specific experimental result.