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  Volume 6, Issue 1, Article 11
On the Heisenberg-Weyl Inequality

    Authors: John Michael Rassias,  
    Keywords: Heisenberg-Weyl Inequality, Uncertainty Principle, Absolute Moment, Gaussian, Extremum Principle.  
    Date Received: 20/09/04  
    Date Accepted: 25/11/04  
    Subject Codes:

26Dxx, 30Xxx, 33Xxx, 42Xxx, 43Xxx, 60Xxx

    Editors: George Anastassiou,  

In 1927, W. Heisenberg demonstrated the impossibility of specifying simultaneously the position and the momentum of an electron within an atom.The well-known second moment Heisenberg-Weyl inequality states: Assume that $ f: mathbb{R} to mathbb{C}$ is a complex valued function of a random real variable $ x$ such that $ f in L^{2}(mathbb{R})$. Then the product of the second moment of the random real $ x$ for $ leftvert f rightvert^2$ and the second moment of the random real $ xi $ for $ leftvert {hat{f} } rightvert^2$ is at least $ {E_{leftvert f rightvert^2} } mathord{left/ {vphantom {{E_{leftvert f rightvert^2} } {4pi }}} right. kern-nulldelimiterspace} {4pi }$, where $ hat{f}$ is the Fourier transform of $ f$, such that $ hat{f} left( xi right)=int_mathbb{R} {e^{-2ipi xi x}} fleft( x right)dx$ and $ fleft( x right)=int_mathbb{R} {e^{2ipi xi x}} hat {f}left( xi right)dxi $, $ i=sqrt {-1} $ and $ E_{leftvert f rightvert^2} =int_mathbb{R} {leftvert {fleft( x right)} rightvert^2dx} $. In 2004, the author generalized the afore-mentioned result to the higher order absolute moments for $ L^{2 }$ functions $ f$ with orders of moments in the set of natural numbers . In this paper, a new generalization proof is established with orders of absolute moments in the set of non-negative real numbers. Afterwards, an application is provided by means of the well-known Euler gamma function and the Gaussian function and an open problem is proposed on some pertinent extremum principle. This inequality can be applied in harmonic analysis and quantum mechanics.

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