Quantum Vortices and Trajectories in Particle Diffraction

Abstract

We investigate the phenomenon of the diffraction of charged particles by thin material targets using the method of the de Broglie-Bohm quantum trajectories. The particle wave function can be modeled as a sum of two terms ψ = ψ ingoing + ψ outgoing . A thin separator exists between the domains of prevalence of the ingoing and outgoing wavefunction terms. The structure of the quantum-mechanical currents in the neighborhood of the separator implies the formation of an array of quantum vortices. The flow structure around each vortex displays a characteristic pattern called 'nodal point -X point complex'. The X point gives rise to stable and unstable manifolds. We find the scaling laws characterizing a nodal point-X point complex by a local perturbation theory around the nodal point. We then analyze the dynamical role of vortices in the emergence of the diffraction pattern. In particular, we demonstrate the abrupt deflections, along the direction of the unstable manifold, of the quantum trajectories approaching an X-point along its stable manifold. Theoretical results are compared to numerical simulations of quantum trajectories. We finally calculate the times of flight of particles following quantum trajectories from the source to detectors placed at various scattering angles θ, and thereby propose an experimental test of the de Broglie -Bohm formalism.

Applied Bohmian mechanics

“…In a series of works published later on by Wisniacki and coworkers [282,283] it was shown that the origin of Bohmian chaos is in the evolution in time of those vortices. An analogous conclusion was also found by Efthymiopoulos and coworkers [197,[284][285][286][287], who showed that chaos is due to the presence of moving quantum vortices forming nodal point-X-point complexes. In particular, in reference [197] a theoretical analysis of the dependence of Lyapunov exponents of Bohmian trajectories on the size and speed of the quantum vortices is presented, which explains their earlier numerical findings [284,285].…”

“…More recently and independently, Efthymiopoulos and coworkers have also tackled a similar issue by studying the diffraction of charged particles by thin material targets [197,198]. The exhaustive analysis presented by these authors is in agreement with the earlier findings by Hirschfelder and coworkers.…”

“…Because part of the wave is still undergoing a motion towards the surface while the other part is already being deflected, a web of nodes emerges from their interference (this is precisely the idea behind the analytical treatment developed by Efthymiopoulos et al in [197,198]). As can be shown numerically [205,206], if the circulation integral is performed along the part of a Bohmian trajectory that encloses a node of the wave function, the result is a multiple of the quantum flux 2π /m, in agreement with the result discussed in Sect.…”

Wavepacket approach to particle diffraction by thin targets: Quantum trajectories and arrival times

“…of ours [51], in which we implemented the de Broglie -Bohm approach in the case of electron diffraction through a thin crystal. In that study, however, we assumed a planar wave model for the propagation of the electron wavefunction in the longitudinal direction.…”

“…In contrast, in the present paper we assume instead a finite longitudinal quantum coherence length. This assumption leads to a number of crucial new elements with respect to [51]. In fact, in order to achieve our goal we derive a wavefunction model by a refined implementation of basic scattering theory, so as to account for a fully-localized in space description of scattering.…”