On the Heisenberg-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 well-known second moment Heisenberg-Weyl inequality states: Assume that f : 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 ∣∣∣2 is 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 e f̂ (ξ) dξ, i = √ −1 and E|f |2 = ∫ R |f (x)| 2 dx. In 2004, the author generalized the afore-mentioned result to the higher order absolute moments for L 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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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 ...
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