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

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 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 $\psi = Re^{iS/\hbar}$, the velocity of interest, $\vec{v}_s = -\frac{\hbar}{m}\frac{\vec{\nabla}R}{R}$, is different from the Bohm velocity $\vec{v}_B=\frac{1}{m}\vec{\nabla}S$, 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 $\vec{v}_s$ 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.