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Bohmian Mechanics, Open Quantum Systems and Continuous Measurements pp 125–180Cite as

Bohmian Stochastic Trajectories

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Abstract

In the last decades, many nonlinear extensions of the Schrödinger equation have been proposed in literature either to explore the fundamental aspects of quantum mechanics, with the usual linear theory representing only a limiting case, or to describe open quantum systems. For the description of nonconservative quantum systems, Kostin formulated in an heuristic way the so-called Schrödinger–Langevin (SL) equation or Kostin equation, for the Brownian motion. This equation has been subsequently rederived, improved and extended for its use in numerous applications, mainly without including the noise term. Numerous features of this SL equation can be better revealed within the framework of de Broglie–Bohm (quantum hydrodynamical trajectory formulation) of quantum mechanics. Within this formalism, several dissipative problems are presented and discussed: the Ramsauer–Townsend effect, the tunneling dynamics through a barrier, the plasma fluid formulation and the Lorentz–Abraham (extended electron) equation for a point-charge electron. These two last examples are also discussed in order to see the correspondence between classical and quantum dynamics. Very few applications of this SL equation are devoted to stochastic problems in the literature where the noise term needs to be included. The so-called Bohmian–Brownian motion is introduced in the context of surface diffusion with single adsorbates. An extension to interacting adsorbates is discussed within a simple, phenomenological model. Interestingly enough, this study leads us to quantum anomalous diffusion. The harmonic motion is also briefly considered in order to compare with the same open dynamics under the presence of a continuous quantum measurement (Chap. 4). Finally, a generalization of the SL equation is proposed for nonlinear dissipation.

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Notes

  1. 1.

    The \(\delta \)–function counts only one half when the integration is carried out from zero to infinity.

  2. 2.

    At finite coverages, one usually distinguishes between two diffusion coefficients: the tracer and the collective diffusion constants [117]. The first one refers to the self–diffusion process and focuses on the motion of a single adsorbate. On the contrary, the second is related to the collective motion of all adsorbates which is governed by Fick’s law. In any case, a Kubo–Green formula relates both diffusion coefficients with the velocity autocorrelation function of a single adsorbate or with the corresponding of the velocity of the center of mass, respectively.

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Authors and Affiliations

  1. Department of Sciences, Harvard-Westlake School and UCLA Extension, Los Angeles, CA, USA

    Antonio B. Nassar

  2. Superior Council for Scientific Research, Institute of Fundamental Physics , Madrid, Spain

    Salvador Miret-Artés

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Nassar, A.B., Miret-Artés, S. (2017). Bohmian Stochastic Trajectories. In: Bohmian Mechanics, Open Quantum Systems and Continuous Measurements. Springer, Cham. https://doi.org/10.1007/978-3-319-53653-8_3

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