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 separator 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 0 − 1 × , 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 Δ T for particles scattered in different angles. The predictions of the de Broglie–Bohm theory for Δ T turn out 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.