On the Sharpened Heisenberg-weyl Inequality
نویسنده
چکیده
The well-known second order moment Heisenberg-Weyl inequality (or uncertainty relation) in Fourier Analysis 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, f (x) = ∫ R e f̂ (ξ) dξ, and E|f |2 = ∫ R |f (x)| 2 dx. This uncertainty relation is well-known in classical quantum mechanics. In 2004, the author generalized the afore-mentioned result to higher order moments and in 2005, he investigated a Heisenberg-Weyl type inequality without Fourier transforms. In this paper, a sharpened form of this generalized Heisenberg-Weyl inequality is established in Fourier analysis. Afterwards, an open problem is proposed on some pertinent extremum principle.These results are useful in investigation of quantum mechanics.
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