Chaos in Bohmian Quantum Mechanics: A Short Review

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.

APPENDIX

The analytical formula of the wave function (7.6) in the position representation reads

$$\displaystyle\Psi=c_{1}\Psi_{R}\Psi_{L}+c_{2}\Psi_{L}\Psi_{R}$$

$$\displaystyle=\frac{(\omega_{x}\omega_{y})^{1/4}}{\sqrt{\pi}}\exp\left\{\frac{i}{2}\left[{a_{0}}^{2}\big{(}\sin(2{\omega_{x}}t)+\sin(2{\omega_{y}}t)\big{)}-({\omega_{x}}+{\omega_{y}})t\right]\right\}$$

$$\displaystyle\quad\times\exp\Big{\{}-\frac{1}{2}(\omega_{x}x^{2}+\omega_{y}y^{2})-a_{0}^{2}[\cos^{2}(\omega_{x}t)+\cos^{2}(\omega_{y}t)]\Big{\}}$$

$$\displaystyle\quad\times\Big{\{}\exp\big{(}f_{x}-f_{y}-i(g_{x}-g_{y})\big{)}c_{1}+\exp(-f_{x}+f_{y}+i(g_{x}-g_{y})c_{2}\Big{\}},$$

(A.1)

where

$$\displaystyle f_{x}=\sqrt{2\omega_{x}}{a_{0}}x\cos(\omega_{x}t),\quad f_{y}=\sqrt{2\omega_{y}}{a_{0}}y\cos(\omega_{y}t)$$

(A.2)

$$\displaystyle g_{x}=\sqrt{2\omega_{x}}{a_{0}}x\sin(\omega_{x}t),\quad g_{y}=\sqrt{2\omega_{y}}{a_{0}}y\sin(\omega_{y}t).$$

(A.3)

The Bohmian equations of motion produced by the wave function (A.1) are

$$\frac{dx}{dt}=-\frac{\sqrt{2\omega_{x}}{a_{0}}\left[A\cos(\omega_{x}t)+B\sin(\omega_{x}t)\right]}{G},$$

(A.4)

$$\frac{dy}{dt}=\frac{\sqrt{2\omega_{y}}{a_{0}}\left[A\cos(\omega_{y}t)+B\sin(\omega_{y}t)\right]}{G},$$

(A.5)

where

$$A=2{c_{1}}{c_{2}}{e^{2f_{x}+2f_{y}}}\sin\big{(}2(g_{x}-g_{y})\big{)},\quad B={c_{1}}^{2}{e^{4f_{x}}}-{c_{2}}^{2}{e^{4f_{y}}}$$

(A.6)

and

$$G={c_{1}}^{2}{e^{4f_{x}+4f_{y}}}+2{c_{1}}{c_{2}}{e^{2f_{x}+2f_{y}}}\cos\big{(}2(g_{x}+g_{y})\big{)}+{c_{2}}^{2}.$$

(A.7)