Origin of chaos near critical points of quantum flow

TL;DR

This paper investigates how chaos arises near moving quantum nodal points in the de Broglie-Bohm framework, revealing the role of nodal point-X-point complexes and their scattering effects on quantum trajectories.

Contribution

It provides a theoretical and numerical analysis of chaos near quantum nodal points, introducing formulas for the nodal point-X-point complex and conditions for chaos emergence.

Findings

Chaos results from scattering with nodal point-X-point complexes.

Lyapunov exponents scale inversely with nodal point speed.

Numerical experiments confirm the theoretical predictions.

Abstract

The general theory of motion in the vicinity of a moving quantum nodal point (vortex) is studied in the framework of the de Broglie - Bohm trajectory method of quantum mechanics. Using an adiabatic approximation, we find that near any nodal point of an arbitrary wavefunction $\psi$ there is an unstable point (called X-point) in a frame of reference moving with the nodal point. We find general formulae for the nodal point - X-point complex as well as necessary and sufficient conditions of validity of the adiabatic approximation. Chaos emerges from the consecutive scattering events of the orbits with nodal point - X-point complexes. A theoretical model is constructed yielding the local value of the Lyapunov characteristic number in a scattering event, which scales as an inverse power of the speed of the nodal point in the rest frame, or proportionally to the size of the nodal point X- point complex. The results of detailed numerical experiments with different wavefunctions possessing multiple moving nodal points are reported. The statistics of the Lyapunov characteristic numbers of the orbits are found and compared to the number of encounter events of each orbit with the nodal point X-point complexes. Various phenomena appearing at first as counter-intuitive find a straightforward explanation.