Nodal points and the transition from ordered to chaotic Bohmian trajectories

TL;DR

This paper investigates how nodal points influence the transition from ordered to chaotic trajectories in a 2D quantum system using Bohmian mechanics, combining analytical and numerical methods.

Contribution

It provides a formal proof of nodal point-free domains, constructs regular orbit series, and elucidates the role of X-points and bifurcations in chaos generation.

Findings

Existence of bounded nodal point-free regions

Identification of saddle (X) points near nodal points

Mechanism of chaos generation via nodal-X point complexes

Abstract

We explore the transition from order to chaos for the Bohmian trajectories of a simple quantum system corresponding to the superposition of three stationary states in a 2D harmonic well with incommensurable frequencies. We study in particular the role of nodal points in the transition to chaos. Our main findings are: a) A proof of the existence of bounded domains in configuration space which are devoid of nodal points, b) An analytical construction of formal series representing regular orbits in the central domain as well as a numerical investigation of its limits of applicability. c) A detailed exploration of the phase-space structure near the nodal point. In this exploration we use an adiabatic approximation and we draw the flow chart in a moving frame of reference centered at the nodal point. We demonstrate the existence of a saddle point (called X-point) in the vicinity of the nodal point which plays a key role in the manifestation of exponential sensitivity of the orbits. One of the invariant manifolds of the X-point continues as a spiral terminating at the nodal point. We find cases of Hopf bifurcation at the nodal point and explore the associated phase space structure of the nodal point - X-point complex. We finally demonstrate the mechanism by which this complex generates chaos. Numerical examples of this mechanism are given for particular chaotic orbits, and a comparison is made with previous related works in the literature.