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

Piero Chiarelli

Published Jul 21, 2011• 1 citation
OpenAlex

Abstract

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.

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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.
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.
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.
The paper investigates the non-local property of quantum mechanics in the quantum hydrodynamic analogy (QHA) given by Madelung. The role of the quantum potential in generating the non-local dynamics of quantum mechanics is analyzed. The work shows how in presence of noise the non-local properties as well as the quantization of the action are perturbed. The resulting stochastic QHA dynamics much depend by the strength of the interaction: Strongly bounded systems (such as linear ones) lead to quantum entangled stochastic behavior, while weakly bounded ones may be not able to maintain the quantum superposition of states on large distances and may loose their macro-scale quantum coherence acquiring the classical stochastic evolution . The work shows that in the frame of the stochastic approach it is possible to have freedom between quantum weakly bounded systems. The stochastic QHA model shows that the wave-function collapse to an eigenstates (deriving by interaction of a quantum microscopic system with a classical (macroscopic) one) can be described by the model itself as a kinetic quantum (relaxation) process to a stationary state. Since the time of the wave function decay to the eigenstate represents the minimum duration time of a measurement, the minimum uncertainty principle is shown to be compatible with the relativistic postulate about the light speed as the maximum velocity of transmission of interaction. About this topic, the paper shows that the Lorenz invariance of the relativistic quantum potential (coming from the Dirac equation) enforces the hypothesis that the superluminal transmission of information are not present in measurements on quantum entangled state.
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.
In the present paper the gas, liquid and solid phases made of structureless particles, are visited to the light of the quantum stochastic hydrodynamic analogy (SQHA). The SQHA shows that the open quantum mechanical behavior is maintained on a distance shorter than the theory-defined quantum correlation length (lc). When, the physical length of the problem is larger than lc, the model shows that the quantum (potential) interactions may have a finite range of interaction maintaining the non-local behavior on a finite distance quantum non-locality length lq. The present work shows that when the mean molecular distance is larger than the quantum non-locality length we have a classical phases (gas and van der Waals liquids) while when the mean molecular distance becomes smaller than lq or than lc we have phases such as the solid crystal or the superfluid one, respectively, that show quantum characteristics. The model agrees with Lindemann empirical law (and explains it), for the mean square deviation of atom from the equilibrium position at melting point of crystal, and shows a connection between the maximum density at the He lambda point and that one near the water-ice solidification point.
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.