Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Hunan University of Science and Engineering

Generalized Uncertainty Relation and Minimum Uncertainty of Position

Fang Wan

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

From heisenberg’s uncertainty relation, generalized uncertainty relation of quantum system under Planck scale is given and thus minimum uncertainty of position is obtained. The significance of generalized uncertainty relation is elaborated further.

Related papers

On Majorization Uncertainty Relations in the Presence of a Minimal Length

Alexey E. Rastegin

The emergence of a minimal length at the Planck scale is consistent with modern developments in quantum gravity. This is taken into account by transforming the Heisenberg uncertainty principle into the generalized uncertainty principle. Here, the position-momentum commutator is modified accordingly. In this paper, majorization uncertainty relations within the generalized uncertainty principle are considered. Dealing with observables with continuous spectra, each of the axes of interest is divided into a set of non-intersecting bins. Such formulation is consistent with real experiments with a necessarily limited precision. On the other hand, the majorization approach is mainly indicative for high-resolution measurements with sufficiently small bins. Indeed, the effects of the uncertainty principle are brightly manifested just in this case. The current study aims to reveal how the generalized uncertainty principle affects the leading terms of the majorization bound for position and momentum measurements. Interrelations with entropic formulations of this principle are briefly discussed.

Physics•Dec 14, 2022•1 citation

The generalized uncertainty principle within the ordinary framework of quantum mechanics

Y.V. Przhiyalkovskiy

A proper deformation of the underlying coordinate and momentum commutation relations in quantum mechanics provides a phenomenological approach to account for the influence of gravity on small scales. Introducing the squared momentum term results in a generalized uncertainty principle, which limits the minimum uncertainty in particle position to the Planck length. However, such a deformation of the commutator significantly changes the formalism, making it separate from the canonical formalism of quantum mechanics. In this study, it is shown that the deformed algebra of position and momentum operators can be incorporated into the framework of ordinary quantum mechanics.

arXiv:2407.09123•Jul 12, 2024•1 citation

Generalized uncertainty relations

Robert Warren Finkel

The mathematical content of the uncertainty principle is contained in special states akin to, and including, states of minimum uncertainty. Specific uncertainty relations are produced by averaging observables in these states. The approach is shown to give more flexible and complete expressions of indeterminacy than the familiar generalized Heisenberg inequality: statistical measures other than variances can be employed, the method is not limited to describing uncertainties in terms of products of statistical dispersions, and the resulting inequalities have the most stringent bounds attainable independent of the system state. Applications are then made to find uncertainty relations for position-momentum average displacements from the means, time-energy indeterminacy, and uncertainties in angular momentum components.

Physical Review A•Feb 1, 1987•37 citations

On Uncertainty Relations at Planck's Scale

Branko G Dragovich

A brief review of the previous research on the Heisenberg uncertainty relations at the Planck scale is given. In this work, investigation of the uncertainty principle extends to p-adic and adelic quantum mechanics. In particular, p-adic analogs of the Heisenberg algebra and uncertainty relation are introduced. Unlike ordinary quantum theory, adelic quantum approach provides a promising framework to probe space below the Planck length.

arXiv:hep-th/0405067•To appear in the proceedings of•May 8, 2004•1 citation

Uncertainty relations for generalized quantum measurements and completely positive maps

Masanao Ozawa

The Heisenberg uncertainty relation for measurement noise and disturbance states that any position measurement with noise /spl epsiv/ brings the momentum disturbance not less than h/2/spl epsiv/. However, this relation holds only for restricted class of measurements. Here, a generalized uncertainty relation for measurement noise and disturbance is formalized and proven, which holds for all the possible quantum measurements. For this purpose, all the possible quantum measurements are characterized by naturally acceptable axioms. The measurement noise and disturbance are defined rigorously for any such general quantum measurements.

IEEE International Workshop on Workload Characterization (WWC)•Apr 23, 2004•0 citations

Heisenberg's uncertainty relation: Violation and reformulation

Masanao Ozawa

The uncertainty relation formulated by Heisenberg in 1927 describes a trade-off between the error of a measurement of one observable and the disturbance caused on another complementary observable so that their product should be no less than a limit set by Planck's constant. In 1980, Braginsky, Vorontsov, and Thorne claimed that this relation leads to a sensitivity limit for gravitational wave detectors. However, in 1988 a model of position measurement was constructed that breaks both this limit and Heisenberg's relation. Here, we discuss the problems as to how we reformulate Heisenberg's relation to be universally valid and how we experimentally quantify the error and the disturbance to refute the old relation and to confirm the new relation.

arXiv:1402.5601•Journal of Physics Conference Series•Apr 14, 2014•12 citations
The emergence of a minimal length at the Planck scale is consistent with modern developments in quantum gravity. This is taken into account by transforming the Heisenberg uncertainty principle into the generalized uncertainty principle. Here, the position-momentum commutator is modified accordingly. In this paper, majorization uncertainty relations within the generalized uncertainty principle are considered. Dealing with observables with continuous spectra, each of the axes of interest is divided into a set of non-intersecting bins. Such formulation is consistent with real experiments with a necessarily limited precision. On the other hand, the majorization approach is mainly indicative for high-resolution measurements with sufficiently small bins. Indeed, the effects of the uncertainty principle are brightly manifested just in this case. The current study aims to reveal how the generalized uncertainty principle affects the leading terms of the majorization bound for position and momentum measurements. Interrelations with entropic formulations of this principle are briefly discussed.
A proper deformation of the underlying coordinate and momentum commutation relations in quantum mechanics provides a phenomenological approach to account for the influence of gravity on small scales. Introducing the squared momentum term results in a generalized uncertainty principle, which limits the minimum uncertainty in particle position to the Planck length. However, such a deformation of the commutator significantly changes the formalism, making it separate from the canonical formalism of quantum mechanics. In this study, it is shown that the deformed algebra of position and momentum operators can be incorporated into the framework of ordinary quantum mechanics.
The mathematical content of the uncertainty principle is contained in special states akin to, and including, states of minimum uncertainty. Specific uncertainty relations are produced by averaging observables in these states. The approach is shown to give more flexible and complete expressions of indeterminacy than the familiar generalized Heisenberg inequality: statistical measures other than variances can be employed, the method is not limited to describing uncertainties in terms of products of statistical dispersions, and the resulting inequalities have the most stringent bounds attainable independent of the system state. Applications are then made to find uncertainty relations for position-momentum average displacements from the means, time-energy indeterminacy, and uncertainties in angular momentum components.
The Heisenberg uncertainty relation for measurement noise and disturbance states that any position measurement with noise /spl epsiv/ brings the momentum disturbance not less than h/2/spl epsiv/. However, this relation holds only for restricted class of measurements. Here, a generalized uncertainty relation for measurement noise and disturbance is formalized and proven, which holds for all the possible quantum measurements. For this purpose, all the possible quantum measurements are characterized by naturally acceptable axioms. The measurement noise and disturbance are defined rigorously for any such general quantum measurements.
The uncertainty relation formulated by Heisenberg in 1927 describes a trade-off between the error of a measurement of one observable and the disturbance caused on another complementary observable so that their product should be no less than a limit set by Planck's constant. In 1980, Braginsky, Vorontsov, and Thorne claimed that this relation leads to a sensitivity limit for gravitational wave detectors. However, in 1988 a model of position measurement was constructed that breaks both this limit and Heisenberg's relation. Here, we discuss the problems as to how we reformulate Heisenberg's relation to be universally valid and how we experimentally quantify the error and the disturbance to refute the old relation and to confirm the new relation.