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  Volume 6, Issue 2, Article 45
On the Refined Heisenberg-Weyl Type Inequality

    Authors: John Michael Rassias,  
    Keywords: Heisenberg-Weyl Type Inequality, Uncertainty Principle, Gram determinant.  
    Date Received: 11/01/05  
    Date Accepted: 17/03/05  
    Subject Codes:

26, 33, 42, 60, 62.

    Editors: Saburou Saitoh,  

The well-known second moment Heisenberg-Weyl inequality (or uncertainty relation) states: Assume that $ f:mathbb{R}to mathbb{C}$ is a complex valued function of a random real variable $ x$ such that $ fin L^{2}(mathbb{R})$, where $ mathbb{R}=(-infty ,infty )$. 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 {widehat{f} } rightvert^2$ is at least $ {E_{mathbb{R},leftvert f rightvert^2} } mathord{left/ {vphantom {{E_{... ...ftvert f rightvert^2} } {4pi }}} right. kern-nulldelimiterspace} {4pi }$, where $ widehat{f} $ is the Fourier transform of $ f$, $ widehat{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 $, and $ E_{mathbb{R},leftvert f rightvert^2} =int_{mathbb{R}} {leftvert {fleft( x right)} rightvert^2dx} $. This uncertainty relation is well-known in classical quantum mechanics. In 2004, the author generalized the afore-mentioned result to the higher order moments for $ L^2(mathbb{R})$ functions $ f.$ In this paper, a refined form of the generalized Heisenberg-Weyl type inequality is established.

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