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  Volume 5, Issue 1, Article 4
On the Heisenberg-Pauli-Weyl Inequality

    Authors: John Michael Rassias,  
    Keywords: Pascal Identity, Plancherel-Parseval-Rayleigh Identity, Lagrange Identity, Gaussian function, Fourier transform, Moment, Bessel equation, Hermite polynomials, Heisenberg-Pauli-Weyl Inequality.  
    Date Received: 02/01/03  
    Date Accepted: 04/03/03  
    Subject Codes:

Primary: 26Xxx; Secondary: 42Xxx, 60Xxx,

    Editors: Alexander G. Babenko,  

In 1927, W. Heisenberg demonstrated the impossibility of specifying simultaneously the position and the momentum of an electron within an atom.The following result named, Heisenberg inequality, is not actually due to Heisenberg. In 1928, according to H. Weyl this result is due to W. Pauli.The said inequality states, as follows: Assume that $ f:\mathbb{R}% \rightarrow \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 $ \left\vert f\right\vert ^{2}$ and the second moment of the random real $ \xi $ for $ \left\vert {\hat{f}% }\right\vert ^{2}$is at least $ {E_{\left\vert f\right\vert ^{2}}}\left/ {% 4\pi }\right. $, where $ {\hat{f}}$ is the Fourier transform of $ f$, such that $ {\hat{f}}\left( \xi \right) =\int_{R}{e^{-2i\pi \xi x}}f\left( x\right) dx$ and $ f\left( x\right) =\int_{R}{e^{2i\pi \xi x}}\hat{f}\left( \xi \right) d\xi $, $ i=\sqrt{-1}$ and $ E_{\left\vert f\right\vert ^{2}}=\int_{R}{\left\vert {f\left( x\right) }\right\vert ^{2}dx}$. In this paper we generalize the afore-mentioned result to the higher moments for $ L^{2}$ functions $ f$ and establish the Heisenberg-Pauli-Weyl inequality.

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