scholarly journals Uncertainty Relations in Hydrodynamics

2020 ◽  
Author(s):  
Gyell Gonçalves de Matos ◽  
Takeshi Kodama ◽  
Tomoi Koide
Keyword(s):  
Quantum Mechanics ◽  
Uncertainty Relation ◽  
Quantum Particle ◽  
Navier Stokes ◽  
Pitaevskii Equation ◽  
Uncertainty Relations ◽  
Derived Relations ◽  
Stochastic Variational Method ◽  
Virtual Trajectory ◽  
Fourier Equation

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.

scholarly journals Uncertainty Relations in Hydrodynamics

Water ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 3263
Author(s):  
Gyell Gonçalves de Matos ◽  
Takeshi Kodama ◽  
Tomoi Koide
Keyword(s):  
Reynolds Numbers ◽  
Navier Stokes ◽  
Pitaevskii Equation ◽  
Uncertainty Relations ◽  
Dimensional System ◽  
Derived Relations ◽  
Stochastic Variational Method ◽  
Low Reynolds Numbers ◽  
Virtual Trajectory ◽  
Fourier Equation

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.

scholarly journals Uncertainty Relations: Curiosities and Inconsistencies

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1640
Author(s):  
Krzysztof Urbanowski
Keyword(s):  
Quantum Theory ◽  
Lower Bound ◽  
Uncertainty Relation ◽  
Quantum Particle ◽  
Uncertainty Relations ◽  
Rigorous Analysis ◽  
Special Cases ◽  
The Status ◽  
Symmetric Quantum ◽  
Standard Deviations

Analyzing general uncertainty relations one can find that there can exist such pairs of non-commuting observables A and B and such vectors that the lower bound for the product of standard deviations ΔA and ΔB calculated for these vectors is zero: ΔA·ΔB≥0. Here we discuss examples of such cases and some other inconsistencies which can be found performing a rigorous analysis of the uncertainty relations in some special cases. As an illustration of such cases matrices (2×2) and (3×3) and the position–momentum uncertainty relation for a quantum particle in the box are considered. The status of the uncertainty relation in PT–symmetric quantum theory and the problems associated with it are also studied.

scholarly journals Relaxed uncertainty relations and information processing

2009 ◽  
Vol 9 (9&10) ◽  
pp. 801-832 ◽  
Author(s):  
G. Ver Steeg ◽  
S. Wehner
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Uncertainty Relation ◽  
Random Access ◽  
The Other ◽  
Uncertainty Relations ◽  
Expectation Values ◽  
Chsh Inequality ◽  
Local Correlations ◽  
Non Local

We consider a range of "theories'' that violate the uncertainty relation for anti-commuting observables derived. We first show that Tsirelson's bound for the CHSH inequality can be derived from this uncertainty relation, and that relaxing this relation allows for non-local correlations that are stronger than what can be obtained in quantum mechanics. We continue to construct a hierarchy of related non-signaling theories, and show that on one hand they admit superstrong random access encodings and exponential savings for a particular communication problem, while on the other hand it becomes much harder in these theories to learn a state. We show that the existence of these effects stems from the absence of certain constraints on the expectation values of commuting measurements from our non-signaling theories that are present in quantum theory.

NONLINEAR QUANTUM MECHANICS WITH NONCLASSICAL GRAVITATIONAL SELF-INTERACTION

1989 ◽  
Vol 04 (13) ◽  
pp. 3229-3267 ◽  
Author(s):  
A.D. POPOVA
Keyword(s):  
Quantum Mechanics ◽  
Quantum Particle ◽  
Superposition Principle ◽  
Uncertainty Relations ◽  
Nonlinear Quantum ◽  
Nonlinear Quantum Mechanics ◽  
Stationary Gravitational Field ◽  
Quantum Operators ◽  
Original Approach ◽  
Ordinary Quantum

The original approach for the self-consistent inclusion of gravity into quantum mechanics of a particle is developed. (There are no connections with second quantization.) The nonstandard action principle is constructed for the stationary situation: the quantum particle in a stationary state creating some nonclassical stationary gravitational field and interacting with it, The accompanying problem of covariantization of quantum operators is considered. The general theory is illustrated by the Newtonian-Schrödingerian and quasi classical limiting cases. The levels of applicability of ordinary quantum mechanics and the problems of measurements and interpretation of nonclassical gravity are discussed. The “uncertainty relations” connecting uncertainties of some “local” parts of curvature and those of the particle’s position and momentum are derived. The superposition principle is generalized on the base of some approximate action.

Time in Quantum Mechanics

2011 ◽  
Author(s):  
Jan Hilgevoord ◽  
David Atkinson
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Uncertainty Relation ◽  
Classical Mechanics ◽  
Special Problem ◽  
Uncertainty Relations ◽  
Operator Formalism ◽  
Heisenberg Uncertainty ◽  
Time In Quantum Mechanics ◽  
Time Operators

Unlike classical mechanics, quantum mechanics assumes the famous Heisenberg uncertainty relations. One of these concerns time: the energy–time uncertainty relation. Unlike the canonical position–momentum uncertainty relation, the energy–time relation is not reflected in the operator formalism of quantum theory. Indeed, it is often said and taken as problematic that there is not a so-called “time operator” in quantum theory. This chapter sheds light on these questions and others, including the absorbing matter of whether quantum mechanics allows for the existence of ideal clocks. The second section notes that quantum mechanics does not involve a special problem for time, and that there is no fundamental asymmetry between space and time in quantum mechanics over and above the asymmetry which already exists in classical physics. The third section studies time operators in detail. The fourth section discusses various uncertainty relations involving time.

scholarly journals COMMUTATOR ANOMALY IN NONCOMMUTATIVE QUANTUM MECHANICS

2006 ◽  
Vol 21 (39) ◽  
pp. 2971-2976 ◽  
Author(s):  
SAYIPJAMAL DULAT ◽  
KANG LI
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Uncertainty Relation ◽  
Uncertainty Relations ◽  
Noncommutative Phase Space ◽  
Noncommutative Quantum Mechanics ◽  
Physical Observable ◽  
Noncommutative Quantum

In this paper, the Schrödinger equation on noncommutative phase space is given by using a generalized Bopp's shift. Then the anomaly term of commutator of arbitrary physical observable operators on noncommutative phase space is obtained. Finally, the basic uncertainty relations for space–space and space–momentum as well as momentum–momentum operators in noncommutative quantum mechanics (NCQM), and uncertainty relation for arbitrary physical observable operators in NCQM are discussed.

scholarly journals Entropic uncertainty relation under correlated dephasing channels

2018 ◽  
Vol 96 (7) ◽  
pp. 700-704 ◽  
Author(s):  
Göktuğ Karpat
Keyword(s):  
Quantum Mechanics ◽  
Standard Deviation ◽  
Lower Bound ◽  
Quantum System ◽  
Quantum Channel ◽  
Uncertainty Relation ◽  
Open Quantum System ◽  
Uncertainty Relations ◽  
Entropic Uncertainty Relation ◽  
Incompatible Observables

Uncertainty relations are a characteristic trait of quantum mechanics. Even though the traditional uncertainty relations are expressed in terms of the standard deviation of two observables, there exists another class of such relations based on entropic measures. Here we investigate the memory-assisted entropic uncertainty relation in an open quantum system scenario. We study the dynamics of the entropic uncertainty and its lower bound, related to two incompatible observables, when the system is affected by noise, which can be described by a correlated Pauli channel. In particular, we demonstrate how the entropic uncertainty for these two incompatible observables can be reduced as the correlations in the quantum channel grow stronger.

scholarly journals Entropic Steering Criteria: Applications to Bipartite and Tripartite Systems

Entropy ◽  
2018 ◽  
Vol 20 (10) ◽  
pp. 763 ◽  
Author(s):  
Ana Costa ◽  
Roope Uola ◽  
Otfried Gühne
Keyword(s):  
Relative Entropy ◽  
Uncertainty Relation ◽  
Uncertainty Relations ◽  
W States ◽  
Entropic Uncertainty Relations ◽  
Entropic Uncertainty Relation ◽  
Local Measurements ◽  
Action At A Distance ◽  
Quantum Steering ◽  
Joint Convexity

The effect of quantum steering describes a possible action at a distance via local measurements. Whereas many attempts on characterizing steerability have been pursued, answering the question as to whether a given state is steerable or not remains a difficult task. Here, we investigate the applicability of a recently proposed method for building steering criteria from generalized entropic uncertainty relations. This method works for any entropy which satisfy the properties of (i) (pseudo-) additivity for independent distributions; (ii) state independent entropic uncertainty relation (EUR); and (iii) joint convexity of a corresponding relative entropy. Our study extends the former analysis to Tsallis and Rényi entropies on bipartite and tripartite systems. As examples, we investigate the steerability of the three-qubit GHZ and W states.

scholarly journals Vortex reconnections in classical and quantum fluids

SeMA Journal ◽  
2021 ◽  
Author(s):  
Alberto Enciso ◽  
Daniel Peralta-Salas
Keyword(s):  
Stokes Equations ◽  
Quantum Fluids ◽  
Navier Stokes ◽  
Pitaevskii Equation ◽  
Global Approximation ◽  
Rigorous Results ◽  
Navier Stokes Equations ◽  
Localization Principle ◽  
Global Smooth Solutions ◽  
Vortex Reconnection

AbstractWe review recent rigorous results on the phenomenon of vortex reconnection in classical and quantum fluids. In the context of the Navier–Stokes equations in $$\mathbb {T}^3$$ T 3 we show the existence of global smooth solutions that exhibit creation and destruction of vortex lines of arbitrarily complicated topologies. Concerning quantum fluids, we prove that for any initial and final configurations of quantum vortices, and any way of transforming one into the other, there is an initial condition whose associated solution to the Gross–Pitaevskii equation realizes this specific vortex reconnection scenario. Key to prove these results is an inverse localization principle for Beltrami fields and a global approximation theorem for the linear Schrödinger equation.

Quantum Mechanics

2017 ◽  
Author(s):  
Nicholas Manton ◽  
Nicholas Mee
Keyword(s):  
Quantum Mechanics ◽  
Ground State ◽  
Variational Principles ◽  
Free Particle ◽  
Uncertainty Relations ◽  
Probabilistic Interpretation ◽  
Historical Account ◽  
Potential Wells ◽  
Harmonic Oscillator Potential ◽  
Ground State Energies

In this chapter, the main features of quantum theory are presented. The chapter begins with a historical account of the invention of quantum mechanics. The meaning of position and momentum in quantum mechanics is discussed and non-commuting operators are introduced. The Schrödinger equation is presented and solved for a free particle and for a harmonic oscillator potential in one dimension. The meaning of the wavefunction is considered and the probabilistic interpretation is presented. The mathematical machinery and language of quantum mechanics are developed, including Hermitian operators, observables and expectation values. The uncertainty principle is discussed and the uncertainty relations are presented. Scattering and tunnelling by potential wells and barriers is considered. The use of variational principles to estimate ground state energies is explained and illustrated with a simple example.

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