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Viscous control of minimum uncertainty state

Tomoi Koide C

Published Apr 23, 2021• 0 citations
OpenAlex

Abstract

A minimum uncertainty state for position and momentum is obtained in quantum viscous hydrodynamics which is defined through the Navier-Stokes-Korteweg (NSK) equation. This state is the generalization of the coherent state and its uncertainty is given by a function of the coefficient of viscosity. The uncertainty can be smaller than the standard minimum value in quantum mechanics, $\hbar/2$, when the coefficient of viscosity is smaller than a critical value which is similar in magnitude to the Kovtun-Son-Starinets (KSS) bound.

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A minimum uncertainty state for position and momentum is obtained in quantum viscous hydrodynamics which is defined through the Navier-Stokes-Korteweg (NSK) equation. This state is the generalization of the coherent state and its uncertainty is given by a function of the coefficient of viscosity. The uncertainty can be smaller than the standard minimum value in quantum mechanics, ℏ/2\hbar/2ℏ/2, when the coefficient of viscosity is smaller than a critical value which is similar in magnitude to the Kovtun-Son-Starinets (KSS) bound.
Abstract A minimum uncertainty state for position and momentum of a fluid element is obtained. We consider a general fluid described by the Navier–Stokes–Korteweg (NSK) equation, which reproduces the behaviors of a standard viscous fluid, a fluid with the capillary action and a quantum fluid, with the proper choice of parameters. When the parameters of the NSK equation is adjusted to reproduce Madelung’s hydrodynamic representation of the Schrödinger equation, the uncertainty relation of a fluid element reproduces the Kennard and the Robertson–Schrödinger inequalities in quantum mechanics. The derived minimum uncertainty state is the generalization of the coherent state and its uncertainty is given by a function of the shear viscosity. The viscous uncertainty can be smaller than the inviscid minimum value when the shear viscosity is smaller than a critical value which is similar in magnitude to the Kovtun–Son–Starinets bound. This uncertainty reflects the information of the fluctuating microscopic degrees of freedom in the fluid and will modify the standard hydrodynamic scenario, for example, in heavy-ion collisions.
Quantum mechanics has been argued to be a coarse-graining of some underlying deterministic theory. Here we support this view by establishing a map between certain solutions of the Schroedinger equation, and the corresponding solutions of the irrotational Navier–Stokes equation for viscous fluid flow. As a physical model for the fluid itself we propose the quantum probability fluid. It turns out that the (state-dependent) viscosity of this fluid is proportional to Planck’s constant, while the volume density of entropy is proportional to Boltzmann’s constant. Stationary states have zero viscosity and a vanishing time rate of entropy density. On the other hand, the nonzero viscosity of nonstationary states provides an information-loss mechanism whereby a deterministic theory (a classical fluid governed by the Navier–Stokes equation) gives rise to an emergent theory (a quantum particle governed by the Schroedinger equation).
The coherent state basis is employed to obtain, in the semiclassical limit, the Euler equation of hydrodynamics corresponding to the single-particle Schrodinger equation. This may also correspond to a collective vibrational mode of the system. By introducing unobserved modes, the concept of viscosity (and the Navier-Stokes equation) is extracted. The motivation for the study stems from the physics that underlies the increasing use of macro-concepts and of hydrodynamical ideas in nuclear physics, particularly in the description of heavy-ion reactions and of fission and fusion processes.
Uncertainty relations are key components in the understanding of the nature of quantum mechanics. In particular, entropic relations are preferred in the study of angular position and angular momentum states. We propose a new form of angle–angular momentum state that provides, for all practical purposes, a lower bound on the entropic uncertainty relation, , for any given angular uncertainty, thus improving upon previous bounds. We establish this by comparing this sum with the absolute minimum value determined by a global numerical search. These states are convenient to work with both analytically and experimentally, which suggests that they may be of use for quantum information purposes.
The energy, position, and momentum eigenstates of a para-Bose oscillator system were considered in paper I. Here we consider the Bargmann or the analytic function description of the para-Bose system. This brings in, in a natural way, the coherent states ‖z;α〉 defined as the eigenstates of the annihilation operator ?. The transformation functions relating this description to the energy, position, and momentum eigenstates are explicitly obtained. Possible resolution of the identity operator using coherent states is examined. A particular resolution contains two integrals, one containing the diagonal basis ‖z;α〉〈z;α‖ and the other containing the pseudodiagonal basis ‖z;α〉〈−z;α‖. We briefly consider the normal and antinormal ordering of the operators and their diagonal and discrete diagonal coherent state approximations. The problem of constructing states with a minimum value of the product of the position and momentum uncertainties and the possible α dependence of this minimum value is considered.