Chaos in Bohmian Quantum Mechanics: A Short Review
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Chaos in Bohmian Quantum Mechanics: A Short Review George Contopoulos1* and Athanasios C. Tzemos1** 1
Research Center for Astronomy and Applied Mathematics of the Academy of Athens, Soranou Efessiou 4, GR-11527 Athens, Greece
Received August 03, 2020; revised August 28, 2020; accepted September 08, 2020
Abstract—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. MSC2010 numbers: 37N20, 81Q50 DOI: 10.1134/S1560354720050056 Keywords: chaos, Bohmian mechanics, entanglement
1. INTRODUCTION In quantum mechanics (QM) the state of a quantum particle described by a wavevector |Ψ evolves according to the time-dependent Schr¨odinger equation which in the position representation reads: 2 ∂Ψ(r, t) ∇2 + V Ψ(r, t) = i , (1.1) − 2m ∂t where V = V (r, t) is the potential, is Planck’s constant, m is the mass and Ψ = r|Ψ is the wave function corresponding to the state vector |Ψ. In the standard approach of QM (the Copenhagen approach, see, e. g., [1]) trajectories are not considered, because the Heisenberg uncertainty does not allow the simultaneous determination of positions and velocities. Bohmian quantum mechanics (BQM) is an alternative interpretation of QM. It is a nonlocal pilot wave theory whose principles were first developed by Louis de Broglie [2, 3] and then by David Bohm [4, 5].1) According to BMQ, the quantum particles follow certain trajectories guided by the wave function according to a set of deterministic equations, the so-called Bohmian equations: ∇Ψ dr = . (1.2) m dt Ψ *
E-mail: [email protected] E-mail: [email protected] 1) BQM is also relate
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