A usual assumption in the so-called de Broglie -Bohm approach to quantum dynamics is that the quantum trajectories subject to typical 'guiding' wavefunctions turn to be quite irregular, i.e. chaotic (in the dynamical systems' sense). In the present paper, we consider mainly cases in which the quantum trajectories are ordered, i.e. they have zero Lyapunov characteristic numbers. We use perturbative methods to establish the existence of such trajectories from a theoretical point of view, while we analyze their properties via numerical experiments. Using a 2D harmonic oscillator system, we first establish conditions under which a trajectory can be shown to avoid close encounters with a moving nodal point, thus avoiding the source of chaos in this system. We then consider series expansions for trajectories both in the interior and the exterior of the domain covered by nodal lines, probing the domain of convergence as well as how successful the series are in comparison with numerical computations or regular trajectories. We then examine a Hénon -Heiles system possessing regular trajectories, thus generalizing previous results. Finally, we explore a key issue of physical interest in the context of the de Broglie -Bohm formalism, namely the influence of order in the so-called quantum relaxation effect. We show that the existence of regular trajectories poses restrictions to the quantum relaxation process, and we give examples in which the relaxation is suppressed even when we consider initial ensembles of only chaotic trajectories, provided, however, that the system as a whole is characterized by a certain degree of order.
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.
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.
We study analytically the orbits along the asymptotic manifolds from a complex unstable periodic orbit in a symplectic 4-D Froeschlé map. The orbits are given as convergent series. We compare the analytic results by truncating the series at various orders with the corresponding numerical results and we find agreement along a more extended length, as the order of truncation increases. The agreement is improved when the parameters approach those of the stability domain. Along the manifolds no terms with small divisors appear in the series. The same result is found if we use a parametrization method along the asymptotic curves. In the case of orbits starting close to the manifolds small divisors appear, but the orbits remain close to the manifolds for an extended period of time. If the parameters of the map are close to the stable domain the orbits recede and approach the origin several times and remain confined in a certain volume around the origin for a long time before escaping to large distances. For special sets of parameters we see resonance phenomena and the orbits take particular forms near every resonance.
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