Generalisations of Heisenberg's inequality

1983 ◽  
pp. 443-449 ◽  
Author(s):  
Michael Cowling ◽  
John F. Price
Keyword(s):  
Heisenberg’S Inequality

HEISENBERG'S INEQUALITY IN SOBOLEV SPACES

2000 ◽  
Vol 20 (1) ◽  
pp. 113-121
Author(s):  
Minyou Qi ◽  
Guji Tian
Keyword(s):  
Sobolev Spaces ◽  
Heisenberg’S Inequality

On Heisenberg's Uncertainty Principle and the CCR

1993 ◽  
Vol 48 (3) ◽  
pp. 447-451 ◽  
Author(s):  
Reinhard Honegger
Keyword(s):  
Uncertainty Principle ◽  
Heisenberg Uncertainty Principle ◽  
Commutation Relations ◽  
Integrable Functions ◽  
Heisenberg Uncertainty ◽  
Heisenberg’S Inequality ◽  
Fock Representations ◽  
Heisenberg's Uncertainty Principle ◽  
Square Integrable Functions ◽  
Rigorous Mathematical Analysis

Abstract Realizing the canonical commutation relations (CCR) [N, Θ] = - i as N = - i d/dϑ and Θ to be the multiplication by ϑ on the Hilbert space of square integrable functions on [0, 2π], in the physical literature there seems to be some contradictions concerning the Heisenberg uncertainty principle ⟨ΔN⟨ ⟨ΔΘ⟨ ≥ 1/4. The difficulties may be overcome by a rigorous mathematical analysis of the domain of state vectors, for which Heisenberg's inequality is valid. It is shown that the exponentials exp {i t N} and exp{i sΘ} satisfy some commutation relations, which are not the Weyl relations. Finally, the present work aims at a better understanding of the phase and number operators in non-Fock representations.

Nonclassical properties of the Arik–Coon q−1-oscillator coherent states on the noncommutative complex plane ℂq

2017 ◽  
Vol 14 (11) ◽  
pp. 1750165 ◽  
Author(s):  
H. Fakhri ◽  
M. Sayyah-Fard
Keyword(s):  
Complex Plane ◽  
Coherent States ◽  
Uncertainty Relation ◽  
Photon Statistics ◽  
Squeezing Effect ◽  
Nonclassical Properties ◽  
Heisenberg’S Inequality ◽  
Photon Antibunching ◽  
Poissonian Statistics

This work has focused on the violation of uncertainty relation, squeezing effect, photon antibunching and sub-Poissonian statistics for the Arik–Coon [Formula: see text]-oscillator coherent states associated with the noncommutative complex plane [Formula: see text]. It is shown that one has to use a generalized definition for the covariance between the operators [Formula: see text] and [Formula: see text]. For [Formula: see text], Heisenberg's inequality violation with two different behaviors related to the role of the deformation parameter [Formula: see text] on the variances of the position and momentum operators is illustrated. We conclude that both weak and strong squeezing effects are exhibited by the [Formula: see text]-coherent states. In particular, strong squeezing effect is a direct consequent of the violation of Heisenberg's inequality. Moreover, the photon antibunching and sub-Poissonian photon statistics are two features of the [Formula: see text]-coherent states which are realized simultaneously with the squeezing effects. Clearly, the three later behaviors are different from their corresponding counterparts in the Arik–Coon [Formula: see text]-oscillator coherent states associated with a commutative complex plane.

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