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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Hydrodynamic Model of Quantum Mechanics

Horst E. Wilhelm

Published Apr 15, 1970• 94 citations
Publisher

Abstract

An analysis of typical microphysical systems is presented in the hydrodynamic formulation of quantum mechanics. The emphasis is on the physical peculiarities appearing in the hydrodynamic picture, and on the mathematical treatment of the nonlinear quantum-hydrodynamic field equations. Further, quantum-hydro-dynamic uncertainty relations are derived, which relate the minimum uncertainty products to the interior quantum stresses.

DOI
10.1103/physrevd.1.2278

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Systems .........................................................................................337 21.5 The Iterative Finite Difference Method ......................................................339 21.6 Concluding Remarks and Future Research.................................................342 Acknowledgments..................................................................................................343 Bibliography ..........................................................................................................343 The quantum hydrodynamic formulation of quantum mechanics is intuitively appealing [1-7]. The quantum potential (Q) and its associated force fq = −∇Q appear on an equal footing with the classical potential and force in the equations of motion. Thus, this approach provides a unified computational method for including both classical and quantum mechanical effects in a molecular dynamics (trajectory based) calculation. However, despite their deceptively simple form, the quantum hydrodynamic equations of motion are very challenging to solve numerically. The quantum potential is non-local (i.e., the trajectories are coupled) and it is a non-linear function of the density (i.e., it depends upon the solution itself). Thus, unlike a typical classical potential, the quantum potential is manifestly time dependent and evolves in “lock-step” fashion with the dynamics. Furthermore, the quantum potential can

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An analysis of typical microphysical systems is presented in the hydrodynamic formulation of quantum mechanics. The emphasis is on the physical peculiarities appearing in the hydrodynamic picture, and on the mathematical treatment of the nonlinear quantum-hydrodynamic field equations. Further, quantum-hydro-dynamic uncertainty relations are derived, which relate the minimum uncertainty products to the interior quantum stresses.
The formulation of quantum mechanics in terms of the picture of trajectory ensemble or of the hydrodynamical picture, developed in a previous article, is further supplemented for some points. Individual trajectories for some typical non-stationary state are explicitly obtained, illustrating that they generally show complicated fluctuations due to the action of quantum potential. Next, emphasis is placed on the fact that the Schrödinger field admits a mechanical model. Also the Klein-Gordon field is represented with a quasi-irrotational flow under some additional scalar field. This representation of Klein-Gordon field suggests its possible generalization to include the vortical motion. Such treatments of interacting Klein-Gordon and electromagnetic fields involve the formalism of Dirac's recent classical electrodynamics as a limiting case. Some remarks are given on this Dirac's theory. Finally, the picture of trajectory ensemble is generalized so as to represent wave equations with sources. In an appendix criticism is given on the attempt to interpret quantum mechanics in terms of Markoff process.
The hydrodynamical form of Wigner–Dunkl quantum mechanics has been formulated. The main step to get such formulation was to derive a pair of Madelung-like equations (called Madelung–Wigner–Dunkl equations) from the Schrödinger equation appearing in Wigner–Dunkl quantum mechanics. Some aspects of both equations contained in the pair, i.e. the inhomogeneous continuity equation and the quantum Hamilton–Jacobi equation, have been investigated and discussed. With certain assumption, the relation between the expectation value of the quantum potential appearing in the quantum Hamilton–Jacobi equation and the so-called Fisher information has been derived. Then an uncertainty relation between momentum and position which may be tighter than the standard uncertainty relation has been derived by using Cramér–Rao inequality and Fisher information.
Quantum hydrodynamics is a formulation of quantum mechanics based on the probability density and flux (current) density of a quantum system. It can be used to define trajectories which allow for a particle-based interpretation of quantum mechanics, commonly known as Bohmian mechanics. However, quantum hydrodynamics rests on the usual time-dependent formulation of quantum mechanics where time appears as a parameter. This parameter describes the correlation of the state of the quantum system with an external system -- a clock -- which behaves according to classical mechanics. With the Exact Factorization of a quantum system into a marginal and a conditional system, quantum mechanics and hence quantum hydrodynamics can be generalized for quantum clocks. In this article, the theory is developed and it is shown that trajectories for the quantum system can still be defined, and that these trajectories depend conditionally on the trajectory of the clock. Such trajectories are not only interesting from a fundamental point of view, but they can also find practical applications whenever a dynamics relative to an external time parameter is composed of "fast" and "slow" degrees of freedom and the interest is in the fast ones, while quantum effects of the slow ones (like a branching of the wavepacket) cannot be neglected. As an illustration, time- and clock-dependent trajectories are calculated for a model system of a non-adiabatic dynamics, where an electron is the quantum system, a nucleus is the quantum clock, and an external time parameter is provided, e.g. via an interaction with a laser field that is not treated explicitly.
Systems .........................................................................................337 21.5 The Iterative Finite Difference Method ......................................................339 21.6 Concluding Remarks and Future Research.................................................342 Acknowledgments..................................................................................................343 Bibliography ..........................................................................................................343 The quantum hydrodynamic formulation of quantum mechanics is intuitively appealing [1-7]. The quantum potential (Q) and its associated force fq = −∇Q appear on an equal footing with the classical potential and force in the equations of motion. Thus, this approach provides a unified computational method for including both classical and quantum mechanical effects in a molecular dynamics (trajectory based) calculation. However, despite their deceptively simple form, the quantum hydrodynamic equations of motion are very challenging to solve numerically. The quantum potential is non-local (i.e., the trajectories are coupled) and it is a non-linear function of the density (i.e., it depends upon the solution itself). Thus, unlike a typical classical potential, the quantum potential is manifestly time dependent and evolves in “lock-step” fashion with the dynamics. Furthermore, the quantum potential can
In a recent paper, we have shown how in Madelung's hydrodynamic formulation of quantum mechanics, the uncertainties are related to the phase and amplitude of the complex wave function. Now we include a dissipative environment via a nonlinear modified Schrödinger equation. The effect of the environment is described by a complex logarithmic nonlinearity that vanishes on average. Nevertheless, there are various changes in the dynamics of the uncertainties originating from the nonlinear term. Again, this is illustrated explicitly using generalized coherent states as examples. With particular focus on the quantum mechanical contribution to the energy and the uncertainty product, connections can be made with the thermodynamic properties of the environment.