On the Heisenberg-pauli-weyl Inequality
نویسندگان
چکیده
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 thatf : R → C is a complex valued function of a random real variable x such that f ∈ L(R). Then the product of the second moment of the random real x for |f | and the second moment of the random real ξ for ∣∣∣f̂ ∣∣∣2is at least E|f |2 /4π , where f̂ is the Fourier transform of f , such that f̂ (ξ) = ∫ R e−2iπξxf (x) dx and f (x) = ∫ R ef̂ (ξ) dξ, i = √ −1 and E|f |2 = ∫ R |f (x)| dx. In this paper we generalize the afore-mentioned result to the higher moments for L functions f and establish the Heisenberg-Pauli-Weyl inequality.
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