Motion of vortices implies chaos in Bohmian mechanics

Abstract

– Bohmian mechanics is a causal interpretation of quantum mechanics in which particles describe trajectories guided by the wave function. The dynamics in the vicinity of nodes of the wave function, usually called vortices, is regular if they are at rest. However, vortices generically move during time evolution of the system. We show that this movement is the origin of chaotic behavior of quantum trajectories. As an example, our general result is illustrated numerically in the two-dimensional isotropic harmonic oscillator. De Broglie-Bohm’s (BB) approach to quantum mechanics has experienced an increased popularity in recent years. This is due to the fact that it combines the accuracy of the standard quantum description with the intuitive insight derived from the causal trajectory formalism, thus providing a powerful theoretical tool to understand the physical mechanisms underlying microscopic phenomena [1, 2]. Although the behavior of quantum trajectories is very different from classical solutions it can be used to gain intuition in many physical phenomena. Numerous examples can be found in different areas of research. In particular, we can mention studies of barrier tunneling in smooth potentials [3], the quantum back-reaction problem [4] and ballistic transport of electrons in nanowires [5]. According to the BB theory of quantum motion, a particle moves in a deterministic orbit under the influence of the external potential and a quantum potential generated by the wave function. This quantum potential can be very intricate because it encodes information on wave interferences. Based on it, Bohm already predicted complex behavior of the quantum trajectories in his seminal work [6]. This was recently confirmed in several studies when presence of chaos in various systems has been shown numerically [7–9]. However, the mechanisms that cause such a complex behavior is still lacking. In this letter we show that movement of the zeros of the wave function, commonly known as vortices, implies chaos in the dynamics of quantum trajectories. Such a movement perturbs the velocity field producing transverse homoclinic orbits that generate the well known Smale horseshoes which is the origin of complex behavior. Our assertion is based on an analytical proof in a simplified model which resembled the velocity field near the vortices. In addition, we present a numerical study in a 2-D

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Cite this paper

@inproceedings{Wisniacki2005MotionOV, title={Motion of vortices implies chaos in Bohmian mechanics}, author={Diego Wisniacki and Enrique Pujals}, year={2005} }