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

Abstract

We develop a wavepacket approach to the diffraction of charged particles by a thin material target and we use the de Broglie-Bohm quantum trajectories to study various phenomena in this context. We construct a particle wave function model given as the sum of two terms ψ = ψ ingoing +ψ outgoing , each having a wavepacket form with longitudinal and transverse quantum coherence lengths both finite. We find the form of the separator, i.e.the limit between the domains of prevalence of the ingoing and outgoing quantum flow. The structure of the quantum-mechanical currents in the neighborhood of the separator implies the formation of an array of quantum vortices (nodal point -X point complexes). The X point gives rise to stable and unstable manifolds, whose directions determine the scattering of the de Broglie -Bohm trajectories. We show how the deformation of the separatior near Bragg angles explains the emergence of a diffraction pattern by the de Broglie -Bohm trajectories. We calculate the arrival time distributions for particles scattered at different angles. A main prediction is that the arrival time distributions have a dispersion proportional to v −1 0 × the largest of the longitudinal and transverse coherence lengths, where v 0 is the mean velocity of incident particles. We also calculate time-of-flight differences ∆T for particles scattered in different angles. The predictions of the de Broglie -Bohm theory for ∆T turn to be different from estimates of the same quantity using other theories on time observables like the sum-over-histories or the Kijowski approach. We propose an experimental setup aiming to test such predictions. Finally, we explore the semiclassical limit of short wavelength and short quantum coherence lengths, and demonstrate how, in this case, results with the de Broglie -Bohm trajectories are similar to the classical results of Rutherford scattering.

Applied Bohmian mechanics

“…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.…”

Atom-Diffraction from Surfaces with Defects: A Fermatian, Newtonian and Bohmian Joint View

“…This type of motion is a confirmation of the former results found by Yinnon et al [ 27 ]. Actually, recently, Efthymiopoulos et al [ 60 ] formally determined the conditions under which such temporary nodal regions in scattering problems will appear, which happens to be in the regions where the amplitudes (modulus) of the incoming and outgoing waves become equal. This boundary, where both values are the same and that have important dynamical consequences, is what they called “separator”.…”

Μελέτη Της Στοχαστικότητας Τροχιών Μη Γραμμικών Δυναμικών Συστημάτων

“…Σκέδαση φορτισμένων σωματιδίων σχέση: 42]). Ο υπολογισμός του S ef f (k 0 ; r, t) βρίσκεται στο παράρτημα Γʹ.…”

“…Στην περίπτωση που η εξερχόμενη κυματοσυνάρτηση δίνεται από τη σχέση (5.25), η συνθήκη (5.29) παίρνει τη μορφή: exp [42]). Η χρονική εξέλιξη της μεταβατικής ζώνης στο επίπεδο (R, z) εξαρτάται από τη χρονική εξέλιξη των πλατών του εισερχόμενου και του εξερχόμενου κύματος σε κάθε σημείο του θεσεογραφικού χώρου.…”