Generalization of Uncertainty Relation for Quantum and Stochastic Systems

Abstract

The generalized uncertainty relation applicable to quantum and stochastic systems is derived within the stochastic variational method. This relation not only reproduces the well-known inequality in quantum mechanics but also is applicable to the Gross-Pitaevskii equation and the Navier-Stokes-Fourier equation, showing that the finite minimum uncertainty between the position and the momentum is not an inherent property of quantum mechanics but a common feature of stochastic systems. We further discuss the possible implication of the present study in discussing the application of the hydrodynamic picture to microscopic systems, like relativistic heavy-ion collisions.

11 pages, 1 figure, the Robertson-Schroedinger uncertainty relation is discussed. Accepted for publication in Phys, Lett. A

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• Uncertainty Relations in Hydrodynamics
• Uncertainty relation for angle from a quantum-hydrodynamical perspective
• Reinterpreting Shock Wave Structure Predictions using the Navier-Stokes Equations
• Quantum Mechanics from Stochastic Processes
• Variational formulation of compressible hydrodynamics in curved spacetime and symmetry of stress tensor
• Analytic Continuation of Stochastic Mechanics
• Viscous control of minimum uncertainty state in hydrodynamics
• On the dual structure of the Schrödinger dynamics
• Torsion-Driven Nonlinearity in Spinless Quantum Mechanics
• Recasting Navier-Stokes Equations
• Development of a Stochastic Interpretation of Quantum Mechanics by E. Nelson. Derivation of the Schrodinger-Euler-Poisson Equations