Uncertainty relation for angle from a quantum-hydrodynamical perspective

Gazeau, Jean‐Pierre; Koide, T.

doi:10.1016/j.aop.2020.168159

Cited by 11 publications

(28 citation statements)

References 31 publications

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“…See Ref. [22,53,57,58] for details. It is worth mentioning that quantum hydrodynamics has an advantage to discuss quantum behaviors in generalized coordinates [22].…”

Section: Schrödinger Equationmentioning

confidence: 99%

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Uncertainty Relations in Hydrodynamics

Matos

Kodama

Koide

2020

Water

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The qualitative behaviors of uncertainty relations in hydrodynamics are numerically studied for fluids with low Reynolds numbers in 1+1 dimensional system. We first give a review for the formulation of the generalized uncertainty relations in the stochastic variational method (SVM), following the work by two of the present authors [Phys. Lett. A 382, 1472 (2018)]. In this approach, the origin of the finite minimum value of uncertainty is attributed to the non-differentiable (virtual) trajectory of a quantum particle and then both of the Kennard and Robertson-Schrödinger inequalities in quantum mechanics are reproduced. The same non-differentiable trajectory is applied to the motion of fluid elements in the Navier-Stokes-Fourier equation or the Navier-Stokes-Korteweg equation. By introducing the standard deviations of position and momentum for fluid elements, the uncertainty relations in hydrodynamics are derived. These are applicable even to the Gross-Pitaevskii equation and then the field-theoretical uncertainty relation is reproduced. We further investigate numerically the derived relations and find that the behaviors of the uncertainty relations for liquid and gas are qualitatively different. This suggests that the uncertainty relations in hydrodynamics are used as a criterion to classify liquid and gas in fluid.

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“…See Ref. [22,53,57,58] for details. It is worth mentioning that quantum hydrodynamics has an advantage to discuss quantum behaviors in generalized coordinates [22].…”

Section: Schrödinger Equationmentioning

confidence: 99%

“…This generalized formulation is studied in Refs. [21,22] and the Kennard-type and Robertson-Schrödinger-type inequalities are derived in hydrodynamics. Recently the Kennard inequality of quantum mechanics was investigated in a different stochastic approach [23].…”

Section: Introductionmentioning

confidence: 99%

Uncertainty Relations in Hydrodynamics

Matos

Kodama

Koide

2020

Water

Self Cite

View full text Add to dashboard Cite

show abstract

“…See Ref. [22,53,56,57] for details. It is worth mentioning that quantum hydrodynamics has an advantage to discuss quantum behaviors in generalized coordinates [22].…”

Section: Schrödinger Equationmentioning

confidence: 99%

“…This generalized formulation is studied in Refs. [21,22] and the Kennard-type and Robertson-Schrödinger-type inequalities are derived in hydrodynamics. Recently the Kennard inequality of quantum mechanics is investigated in a different stochastic approach [23].…”

Section: Introductionmentioning

confidence: 99%

Uncertainty Relations in Hydrodynamics

Matos¹,

Kodama²,

Koide³

2020

Preprint

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Uncertainty relations in hydrodynamics are numerically studied. We first give a review for the formulation of the generalized uncertainty relations in the stochastic variational method (SVM), following the work by two of the present authors [Phys.\ Lett.\ A\textbf{382}, 1472 (2018)]. In this approach, the origin of the finite minimum value of uncertainty is attributed to the non-differentiable (virtual) trajectory of a quantum particle and then both of the Kennard and Robertson-Schr\"{o}dinger inequalities in quantum mechanics are reproduced. The same non-differentiable trajectory is applied to the motion of fluid elements in the Navier-Stokes-Fourier equation or the Navier-Stokes-Korteweg equation. By introducing the standard deviations of position and momentum for fluid elements, the uncertainty relations in hydrodynamics are derived. These are applicable even to the Gross-Pitaevskii equation and then the field-theoretical uncertainty relation is reproduced. We further investigate numerically the derived relations and find that the behaviors of the uncertainty relations for liquid and gas are qualitatively different. This suggests that the uncertainty relations in hydrodynamics are used as a criterion to classify liquid and gas in fluid.

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“…(The basic theory of the stochastic calculus of variations had been developed by Yasue [13]. For very recent progress in stochastic variational principle, see [6,14,15].) Being beautiful, the Nelson stochastic quantization is not necessarily useful in applications to multi-dimensional systems.…”

Section: Introductionmentioning

confidence: 99%

Maupertuis-Hamilton least action principle in the space of variational parameters for Schrödinger dynamics; A dual time-dependent variational principle

Takatsuka

2020

J. Phys. Commun.

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Time-dependent variational principle (TDVP) provides powerful methods in solving the timedependent Schröinger equation. As such Kan developed a TDVP (Kan 1981 Phys. Rev. A 24, 2831) and found that there is no Legendre transformation in quantum variational principle, suggesting that there is no place for the Maupertuis reduced action to appear in quantum dynamics. This claim is puzzling for the study of quantum-classical correspondence, since the Maupertuis least action principle practically sets the very basic foundation of classical mechanics. Zambrini showed within the theory of stochastic calculus of variations that the Maupertuis least action principle can lead to the Nelson stochastic quantization theory (Zambrini 1984 J. Math. Phys. 25, 1314. We here revisit the basic aspect of TDVP and reveal the hidden roles of Maupertuis-Hamilton least action in the Schrödinger wavepacket dynamics. On this basis we propose a dual least (stationary) action principle, which is composed of two variational functionals; one responsible for 'energy related dynamics' and the other for 'dynamics of wave-flow'. The former is mainly a manifestation of particle nature in wave-particle duality, while the latter represents that of matter wave. It is also shown that by representing the TDVP in terms of these inseparably linked variational functionals the problem of singularity, which is inherent to the standard TDVPs, is resolved. The structure and properties of this TDVP are also discussed.By contrast, the TDVP gives far more practical methodologies. Indeed, there have been proposed various elaborated formalisms such as those of the so-called Dirac-Frenkel, MaLachlan [16], Kan [17], Kramer and Saraceno [18], and so on. Broeckhove, Lathouwers, Kesteloot and van Leuven [19] gave a unified account over these methods. These methods and the variants are widely used in many fields of physics and chemistry. (For more recent progress in other general theories of TDVP, we refer to [20][21][22][23][24][25][26] .) Yet, it is known that the TDVP generally faces a divergence problem in handling nonlinear variational parameters, which arises from the inversion of possibly singular matrices.An overwhelming advantage of TDVP is that it transforms the Schrödinger partial differential equation to a set of coupled ordinary differential equations over the space of variational parameters. In this context, Kan [17] determinedly states in his seminal paper that 'any time-dependent description of quantum systems derived from the variational principle is equivalent to Hamilton's description of a classical system'. On the other hand, Kan has explicitly shown that there does not exist Legendre transformation in his TDVP. Lack of the Legendre transformation suggests that there is no place for the Maupertuis reduced (or abbreviated) action [27, 28] to appear, and therefore the Maupertuis-Hamilton least action principle will not have direct significance in TDVP. It is not comfortable however to accept this assertion, since the least action principle of Maupertu...

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Uncertainty relation for angle from a quantum-hydrodynamical perspective

Cited by 11 publications

References 31 publications

Uncertainty Relations in Hydrodynamics

Uncertainty Relations in Hydrodynamics

Uncertainty Relations in Hydrodynamics

Maupertuis-Hamilton least action principle in the space of variational parameters for Schrödinger dynamics; A dual time-dependent variational principle

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