arXiv:1210.1138

The Schrödinger equation and its generalized analogs derived by the quantum hydrodynamic representation

Piero Chiarelli

Published Oct 3, 2012• 1 citation

Abstract

The derivation of the Schrödinger-like equations from the system of equations of the quantum hydrodynamic analogy (QHA) is analyzed in presence of fluctuations. If in absence of fluctuation each QHA solution can be tracked back to the equivalent one of the Schrödinger problem, in presence of fluctuations the work shows that the ensemble of solution of the QHA equations is wider than the Schrödinger one and the tracking-back procedure from the QHA problem to the original Schrödinger equation is not generally possible. The break-down of the mathematical correspondence is inspected and correlated to the characteristics of the states. Finally, the paper shows that if the form of Schrödinger equation is generalized, the QHA stochastic case as well as its classical limit can be translated into a Schrödinger-like representation.

DOI: 10.48550/arxiv.1210.1138

Related papers

Is the quantum hydrodynamic analogy more general than the Schroedinger approach?

Piero Chiarelli

The quantum hydrodynamic analogy (QHA), equivalent to the Schrödinger equation, is investigated and extended to the stochastic case. The investigation shows that in addition to reproducing the standard quantum mechanics the QHA model is able to generally describe the quantum stochastic dynamics leading to the dissipative Schrödinger equation given by Kostin [55] as a particular case. The inspection shows that the QHA is well suited for the treatment of problems where local noise (spatially distributed one) is introduced. In this case the analysis shows that the bi-univocal correspondence between the QHA and the Schrödinger approach breaks down and that the states described by the QHA do not have their corresponding ones into the Schrödinger description.

arXiv:1205.6131May 28, 20121 citation

Can fluctuating quantum states acquire the classical behavior on large scale?

Piero Chiarelli

The quantum hydrodynamic analogy (QHA) equivalent to the Schrodinger equation is generalized to its stochastic version by a systematic technique. On large scale, the quantum stochastic hydrodynamic analogy (QSHA) shows dynamics that under some circumstances may acquire the classical evolution. The QSHA puts in evidence that in presence of spatially distributed noise the quantum pseudo-potential restores the quantum behavior on a distance shorter than the correlation length of fluctuations (named here lc) of the quantum wave function modulus. The quantum mechanics is achieved in the deterministic limit when lc tends to infinity with respect to the scale of the problem. When the physical length of the problem is of order or larger than lc, the quantum potential may have a finite range of efficacy maintaining the non-local behavior on a distance lL (named here "quantum non-locality length") depending both by the noise amplitude and by the inter-particle strength of interaction. In the deterministic limit (quantum mechanics) the model shows that the "quantum non-locality length" lLalso becomes infinite. The QSHA unveils that in linear systems fluctuations are not sufficient to break the quantum non-locality showing that lL is infinite even if lc is finite.

arXiv:1107.4198Jul 21, 201131 citations

The Classical Mechanics from the Quantum Equation

Piero Chiarelli, E. Piaggio

This work shows that the stochastic generalization of the quantum hydrodynamic analogy (QHA) has its corresponding stochastic Schrödinger equation (SSE) as similarly happens for the deterministic limit. The SSE owns an imaginary random noise that has a finite correlation distance, so that when the physical length of the problem is much smaller than it, the SSE converges to the standard Schrödinger equation comprehending it. The model shows that in non-linear (weakly bounded) systems, the term responsible of the non-local interaction in the SSE may have a finite range of efficacy maintaining its non-local effect on a finite distance. A non-linear SSE that describes the related large-scale classical dynamics is derived. The work also shows that at the edge between the quantum and the classical regime the SSE can lead to the semi-empirical Gross-Pitaevskii equation.

Physical Science International JournalJan 1, 201316 citations

Large-Scale classical behavior of fluctuating quantum states in non-linear systems

Piero Chiarelli

The hydrodynamic analogy (QHA) equivalent to the Schrodinger equation is generalized to its stochastic version by a systematic technique. On large scale, the stochastic hydrodynamic analogy (QSHA) shows dynamics that under some circumstances may acquire the classical evolution. The QSHA puts in evidence that in presence of spatially distributed noise the pseudo-potential restores the behavior on a distance shorter than the correlation of fluctuations (named here lc) of the wave function modulus. The mechanics is achieved in the deterministic limit when lc tends to infinity with respect to the scale of the problem. When the physical of the problem is of order or larger than lc, the potential may have a finite range of efficacy maintaining the non-local behavior on a distance lL (named here quantum non-locality length) depending both by the noise amplitude and by the inter-particle strength of interaction. In the deterministic limit (quantum mechanics) the model shows that the quantum non-locality length lLalso becomes infinite. The QSHA unveils that in linear systems fluctuations are not sufficient to break the non-locality showing that lL is infinite even if lc is finite.

Jul 21, 20111 citation

Eulerian and Newtonian dynamics of quantum particles

Sergey A. Rashkovskiy

We derive the classical equations of hydrodynamic type (Euler equation and the continuity equation) from which the Schrodinger equation follows as a limit case. It is shown that the statistical ensemble corresponding to quantum system and described by the Schrodinger equation, can be considered as an inviscid gas obeying the ideal gas law with quickly oscillating sign-alternating temperature. This statistical ensemble performs the complex movements consisting of smooth average movement and fast oscillations. It is shown that average movements of statistical ensemble are described by Schrodinger equation. A model of quantum motion within the limits of classical mechanics which corresponds to the considered hydrodynamic system is suggested.

arXiv:1302.2502Progress of Theoretical and Experimental PhysicsJun 1, 20134 citations

Generalized Equations in Quantum Mechanics and Brownian Theory

Pierre-Henri Chavanis

We discuss formal analogies between a nonlinear Schrödinger equation derived by the author from the theory of scale relativity and the equations of Brownian theory. By using the Madelung transformation, the nonlinear Schrödinger equation takes the form of hydrodynamic equations involving a friction force, an effective thermal pressure, a pressure due to the self-interaction, and a quantum potential. These hydrodynamic equations have a form similar to the damped Euler equations obtained for self-interacting Brownian particles in the theory of simple liquids. In that case, the temperature is due to thermal motion and the pressure arises from spatial correlations between the particles. More generally, the correlations can be accounted for by using the dynamical density functional theory. We determine the excess free energy of Brownian particles that reproduces the standard quantum potential. We then consider a more general form of excess free energy functionals and propose a new class of generalized Schrödinger equations. For a certain form of excess free energy, we recover the generalized Schrödinger equation associated with the Tsallis entropy considered in a previous paper.

SymmetryDec 13, 20236 citations