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Hamiltonian Structure of the Schrödinger Classical Dynamical System

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Abstract

The connection between quantum mechanics and classical statistical mechanics has motivated in the past the representation of the Schrödinger quantum-wave equation in terms of “projections” onto the quantum configuration space of suitable phase-space asymptotic kinetic models. This feature has suggested the search of a possible exact super-dimensional classical dynamical system (CDS), denoted as Schrödinger CDS, which uniquely determines the time-evolution of the underlying quantum state describing a set of N like and mutually interacting quantum particles. In this paper the realization of the same CDS in terms of a coupled set of Hamiltonian systems is established. These are respectively associated with a quantum-hydrodynamic CDS advancing in time the quantum fluid velocity and a further one the RD-CDS, describing the relative dynamics with respect to the quantum fluid.

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Notes

  1. In fact according ’t Hooft “Contrary to common belief, it is not difficult to construct deterministic (i.e., classical dynamical) models (for the Schroedinger equation) where stochastic behavior is correctly described... What is difficult, however, is to obtain a Hamiltonian that is bounded from below” (see Ref. [6]).

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Acknowledgments

Work developed within the research project of the Albert Einstein Center for Gravitation and Astrophysics, Czech Science Foundation No. 14-37086G (M.T.). One of the authors (M.T.) acknowledges the hospitality of the Department of Mechanical Engineering, Ben Gurion University of the Negev, Be’er Sheva, Israel, during the preparation of the manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematics and Geosciences, University of Trieste, Via Valerio 12, 34127, Trieste, Italy

    Massimo Tessarotto

  2. Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava, Bezručovo nám.13, 74601, Opava, Czech Republic

    Massimo Tessarotto

  3. Department of Mechanical Engineering, Ben-Gurion University of the Negev, P.O. Box 653, Beersheba, 84105, Israel

    Michael Mond

  4. Department of Mathematics, Universities of West Indies, Mona, Jamaica

    Davide Batic

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  1. Massimo Tessarotto
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Correspondence to Massimo Tessarotto.

Appendix: Derivation of Euler Equation

Appendix: Derivation of Euler Equation

To prove Eq. (52) let us apply term by term the operator \(\frac{1}{m }\nabla \) to the quantum phase-function equation (45). Then invoking the definition (42) there it follows

$$\begin{aligned} \frac{\partial }{\partial t}\mathbf {V+}\frac{1}{2m^{2}}\nabla \left( \nabla S-\frac{q}{c}\mathbf {A}\right) ^{2}=-\frac{q}{mc}\frac{ \partial }{\partial t}\mathbf {A}-\frac{1}{m}\nabla q\phi -\frac{1}{m}\nabla \widehat{U}_{QM}. \end{aligned}$$
(146)

Consider, as an illustration, the case of the 1-body Schrödinger equation. In this case the following identity applies to the 1-body vector

$$\begin{aligned} \frac{1}{2m^{2}}\nabla \left( \nabla S-\frac{q}{c}\mathbf {A}\right) ^{2}= & {} \frac{1}{m^{2}}\left( \nabla S-\frac{q}{c}\mathbf {A}\right) \cdot \nabla \left( \nabla S-\frac{q}{c}\mathbf {A}\right) \nonumber \\&+\,\frac{1}{m^{2}}\left( \nabla S-\frac{q}{c}\mathbf {A}\right) \times \left[ \nabla \times \left( \nabla S-\frac{q}{c}\mathbf {A}\right) \right] \nonumber \\= & {} \mathbf {V}\cdot \nabla \mathbf {V-}\frac{q}{mc}\mathbf {V}\times \mathbf {B.} \end{aligned}$$
(147)

As a consequence the Euler equation (52) immediately follows.

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Tessarotto, M., Mond, M. & Batic, D. Hamiltonian Structure of the Schrödinger Classical Dynamical System. Found Phys 46, 1127–1167 (2016). https://doi.org/10.1007/s10701-016-0012-0

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