Order in de Broglie - Bohm quantum mechanics

TL;DR

This paper investigates conditions under which quantum trajectories in de Broglie-Bohm mechanics are ordered rather than chaotic, analyzing their properties through perturbative and numerical methods, and exploring implications for quantum relaxation.

Contribution

It provides a theoretical and numerical analysis of ordered quantum trajectories, including conditions for their existence and their impact on quantum relaxation in de Broglie-Bohm theory.

Findings

Ordered trajectories can be established avoiding nodal encounters.

Series expansions effectively describe trajectories and compare well with numerical results.

Order in the system can suppress quantum relaxation even with chaotic initial ensembles.

Abstract

A usual assumption in the so-called {\it de Broglie - Bohm} approach to quantum dynamics is that the quantum trajectories subject to typical `guiding' wavefunctions turn to be quite irregular, i.e. {\it chaotic} (in the dynamical systems' sense). In the present paper, we consider mainly cases in which the quantum trajectories are {\it 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 H\'{e}non - 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 {\it 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.