Generic Bohmian Trajectories of an Isolated Particle

Mark P. Davidson

George Contopoulos, Athanasios C. Tzemos

This is a short review of the theory of chaos in Bohmian quantum mechanics based on our series of works in this field. Our first result is the development of a generic theoretical mechanism responsible for the generation of chaos in an arbitrary Bohmian system (in 2 and 3 dimensions). This mechanism allows us to explore the effect of chaos on Bohmian trajectories and study in detail (both analytically and numerically) the different kinds of Bohmian trajectories where, in general, chaos and order coexist. Finally, we explore the effect of quantum entanglement on the evolution of the Bohmian trajectories and study chaos and ergodicity in qubit systems which are of great theoretical and practical interest. We find that the chaotic trajectories are also ergodic, i. e., they give the same final distribution of their points after a long time regardless of their initial conditions. In the case of strong entanglement most trajectories are chaotic and ergodic and an arbitrary initial distribution of particles will tend to Born’s rule over the course of time. On the other hand, in the case of weak entanglement the distribution of Born’s rule is dominated by ordered trajectories and consequently an arbitrary initial configuration of particles will not tend, in general, to Born’s rule unless it is initially satisfied. Our results shed light on a fundamental problem in Bohmian mechanics, namely, whether there is a dynamical approximation of Born’s rule by an arbitrary initial distribution of Bohmian particles.

Athanasios C. Tzemos

G. Contopoulos, Christos Efthymiopoulos

Johannes Fankhauser

Abstract Pilot wave theory endows particles with de nite positions at all times governed by deterministic dynamics. However, individual particle trajectories are generically undetectable by experiment. This idea might seem to be contested in light of two proposals: (1) So-called ‘weak velocity measurements’, allegedly detecting Bohmian trajectories by weakly probing a quantum system without essentially disturbing it, and (2) the so-called ‘surrealistic’ trajectories experiment which supposedly establishes a conflict between the ‘actual’ position of a particle and its position derived from pilot wave theory. Although both attempts shed light on the nature of Bohmian particles, neither constitute empirical or theoretical evidence in favour or against pilot wave theory. Both instances admit a straightforward standard quantum mechanical interpretation compatible with the predictions of Bohmian theories. It is concluded that the puzzles arise from the absence of a coherent account of what quantum mechanical measurements signify.

Dandan Li, Jinqiao Duan, Lin Li +1 more

Bohmian mechanics is a non-relativistic quantum theory based on a particle approach. In this paper, we study the Schrödinger equation with a rapidly oscillating potential and the associated Bohmian trajectory. We prove that the corresponding Bohmian trajectory converges locally in a measure, and the limit coincides with the Bohmian trajectory for the effective Schrödinger equation on a finite time interval. This is beneficial for efficient simulation of the Bohmian trajectories in oscillating potential fields.