JIPAM logo: Home Link
 
Home Editors Submissions Reviews Volumes RGMIA About Us
 

   
  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,  
 
    Abstract:

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.

         
       
  Download Screen PDF
  Download Print PDF
  Send this article to a friend
  Print this page
  

      search [advanced search] copyright 2003 terms and conditions login