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Are electron trajectories diffusion-like or smooth?

Jarek Duda

Published Oct 24, 2026• 0 citations
Publisher

Abstract

Same quantum density, different hidden electron paths - diffusion-like (Nelson/MERW), smooth with velocity memory, Bohm, psMERW. Diffusion-like paths give imaginary-time correlations and kinetic energy from increments, no smooth path can (TRK slope) - but smooth paths are valid samplers of response via Dalgarno-Lewis. Larmor clock on tunnelling Rb atoms: residence of stochastic paths agrees with weak value and measured decrease at lower energy if velocity memory << traversal, longer memory goes continuously to Bohm, 12-64% too long. psMERW splittings -0.956 ω/Δ from E_n-(D/2Δ)<V''>, at Compton gap it is minus Darwin term - excluded by Lamb shift. Code and data included.

DOI
10.5281/zenodo.23609213

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Same quantum density, different hidden electron paths - diffusion-like (Nelson/MERW), smooth with velocity memory, Bohm, psMERW. Diffusion-like paths give imaginary-time correlations and kinetic energy from increments, no smooth path can (TRK slope) - but smooth paths are valid samplers of response via Dalgarno-Lewis. Larmor clock on tunnelling Rb atoms: residence of stochastic paths agrees with weak value and measured decrease at lower energy if velocity memory << traversal, longer memory goes continuously to Bohm, 12-64% too long. psMERW splittings -0.956 ω/Δ from E_n-(D/2Δ)<V''>, at Compton gap it is minus Darwin term - excluded by Lamb shift. Code and data included.
As written by statistician George Box "All models are wrong, but some are useful", standard diffusion derivation or Feynman path ensembles use nonphysical infinite velocity/kinetic energy nowhere differentiable trajectories - what seems wrong, might be only our approximation to simplify mathematics. This article introduces some basic tools to investigate this issue. To consider ensembles of more physical finite velocity trajectories, we can work in (x,v)(x,v)(x,v) phase space like in Langevin equation with velocity controlling spatial steps, here also controlled with spatial potential V(x)V(x)V(x). There are derived and compared 4 approaches to predict stationary probability distributions: using Boltzmann ensemble of steps/points in space (GRW - generic random walk) or in phase space (psGRW), and analogously Boltzmann ensemble of paths in space (MERW - maximal entropy random walk) and in phase space (psMERW), also generalized to L{é}vy flights. Path ensembles generally have much stronger Anderson-like localization, MERW has stationary distribution exactly as quantum ground state. Proposed novel MERW in phase space has some slight differences, which might be distinguished experimentally. For example for 1D infinite potential well: ρ=1ρ=1ρ=1 stationary distribution for step ensemble, ρ∼sin⁡2ρ\sim \sin^2ρ∼sin2 for path ensemble (as in QM), and ρ∼sin⁡ρ\sim \sinρ∼sin for proposed smooth path ensembles - more frequently approaching the barriers due to randomly gained velocity.
The fact that the kinetic energy of a particle cannot exceed its overall energy implies that the velocity - i.e. the derivative of the trajectory - should be bounded. This means, in effect, that all the trajectories are differentiable (smooth). However, at first glance, there seems to be no direct requirement that the velocities continuously depend on time. In this paper, we show that the properties of electromagnetic field necessitate that the velocities are continuous functions of time - moreover, that they are at least as continuous as the Brownian motion.
A solution ψψψ to Schrödinger's equation needs some degree of regularity in order to allow the construction of a Bohmian mechanics from the integral curves of the velocity field ℏℑ(▽ψ/mψ).\hbar \Im \left( \bigtriangledown ψ/mψ\right) .ℏℑ(▽ψ/mψ). In the case of one specific non-differentiable weak solution ΨΨΨ we show how Bohmian trajectories can be obtained for ΨΨΨ from the trajectories of a sequence Ψn→Ψ.Ψ_{n}\rightarrow Ψ.Ψn​→Ψ. (For any real ttt the sequence $Ψ_{n}\left( t,\cdot \right) $ converges strongly.) The limiting trajectories no longer need to be differentiable. This suggests a way how Bohmian mechanics might work for arbitrary initial vectors ΨΨΨ in the Hilbert space on which the Schrödinger evolution % Ψ\mapsto e^{-iht}Ψ acts.
How much time does a tunneling particle spend in a barrier? A Larmor clock, one proposal to answer this question, measures the interaction between the particle and the barrier region using an auxiliary degree of freedom of the particle to clock the dwell time inside the barrier. We report on precise Larmor time measurements of ultracold ^{87}Rb atoms tunneling through an optical barrier, which confirm longstanding predictions of tunneling times. We observe that atoms generally spend less time tunneling through higher barriers and that this time decreases for lower energy particles. For the lowest measured incident energy, at least 90% of transmitted atoms tunneled through the barrier, spending an average of 0.59±0.02 ms inside. This is 0.11±0.03 ms faster than atoms traversing the same barrier with energy close to the barrier's peak and 0.21±0.03 ms faster than when the atoms traverse a barrier with 23% less energy.
How much time does a tunneling particle spend in a barrier? A Larmor clock, one proposal to answer this question, measures the interaction between the particle and the barrier region using an auxiliary degree of freedom of the particle to clock the dwell time inside the barrier. We report on precise Larmor time measurements of an ultra-cold gas of 87^{87}87Rb atoms tunneling through an optical barrier. The data capture distinctive features that confirm longstanding predictions of tunneling times. In particular, we demonstrate that atoms spend less time tunneling through higher barriers and that this time decreases for lower energy particles. For the lowest measured incident energy, at least 90%90\%90% of transmitted atoms tunneled through the barrier, spending an average of 0.59(2)0.59(2)0.59(2)ms inside. This is 0.11(3)0.11(3)0.11(3)ms faster than atoms traversing the same barrier with energy close to the barrier's peak and 0.21(3)0.21(3)0.21(3)ms faster than when the atoms traverse a barrier with 23%23\%23% less energy.