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Journal of Jinan University

Generalizations of Weyl's and Heisenberg's uncertainty inequality

Yang Enhao

Published Jan 1, 2000• 0 citations
OpenAlex

Abstract

New generalizations of the Weyl's and Heisenberg's uncertainty inequality, which describe the uncertain principle of quantum mechanics, are established by means of an auxiliary definite integral identity constructed herein. [

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Generalized uncertainty principles

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The phenomenon in the essence of classical uncertainty principles is well known since the thirties of the last century. We introduce a new phenomenon which is in the essence of a new notion that we introduce: "Generalized Uncertainty Principles". We show the relation between classical uncertainty principles and generalized uncertainty principles. We generalized "Landau-Pollak-Slepian" uncertainty principle. Our generalization relates the following two quantities and two scaling parameters: 1) The weighted time spreading $\int_{-\infty}^\infty |f(x)|^2w_1(x)dx$, ($w_1(x)$ is a non-negative function). 2) The weighted frequency spreading $\int_{-\infty}^\infty |\hat{f}(ω)|^2w_2(ω)dω$. 3) The time weight scale $a$, ${w_1}_a(x)=w_1(xa^{-1})$ and 4) The frequency weight scale $b$, ${w_2}_b(ω)=w_2(ωb^{-1})$. "Generalized Uncertainty Principle" is an inequality that summarizes the constraints on the relations between the two spreading quantities and two scaling parameters. For any two reasonable weights $w_1(x)$ and $w_2(ω)$, we introduced a three dimensional set in $R^3$ that is in the essence of many uncertainty principles. The set is called "possibility body". We showed that classical uncertainty principles (such as the Heiseneberg-Pauli-Weyl uncertainty principle) stem from lower bounds for different functions defined on the possibility body. We investigated qualitative properties of general uncertainty principles and possibility bodies. Using this approach we derived new (quantitative) uncertainty principles for Landau-Pollak-Slepian weights. We found the general uncertainty principles related to homogeneous weights, $w_1(x)=w_2(x)=x^k$, $k\in N$, up to a constant.

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Since the publication of Heisenberg’s paper1 on the “anschaulichen Inhalt” of quantum mechanics, discussions of the fundamental limitations on the accuracy of physical measurements have been much in the foreground. According to Heisenberg, the quantum mechanics implies that it is impossible to measure simultaneous values of a coordinate and its conjugate momentum with unlimited precision. Instead, if Δp be the estimated error or uncertainty in a momentum and Δq that in the associated coordinate one must have the inequality, ΔpΔq>h2π \Delta p\Delta q>\frac{h}{{2\pi}} ΔpΔq>2πh​ (1) This inequality has come to be known quite generally as Heisenberg’s uncertainty relation.
ABSTRACT. The well-known second moment Heisenberg-Weyl inequality (or uncertainty relation) states: Assume that f: R → C is a complex valued function of a random real variable x such that f ∈ L2 (R), where R = (−∞, ∞). Then the product of the second moment of the random real x for |f | 2 and the second moment of the random real ξ for ∣ f ∣ 2 is at least E 2 R,|f | 4π, where f is the Fourier transform of f, f (ξ) = ∫ R e−2iπξxf (x) dx and f (x) = ∫ R e2iπξx ˆ f (ξ) dξ, and E 2 R,|f | = ∫ R |f (x)|2 dx. This uncertainty relation is wellknown in classical quantum mechanics. In 2004, the author generalized the afore-mentioned result to the higher order moments for L2 (R) functions f. In this paper, a refined form of the generalized Heisenberg-Weyl type inequality is established.
The phenomenon in the essence of classical uncertainty principles is well known since the thirties of the last century. We introduce a new phenomenon which is in the essence of a new notion that we introduce: "Generalized Uncertainty Principles". We show the relation between classical uncertainty principles and generalized uncertainty principles. We generalized "Landau-Pollak-Slepian" uncertainty principle. Our generalization relates the following two quantities and two scaling parameters: 1) The weighted time spreading ∫−∞∞∣f(x)∣2w1(x)dx\int_{-\infty}^\infty |f(x)|^2w_1(x)dx∫−∞∞​∣f(x)∣2w1​(x)dx, (w1(x)w_1(x)w1​(x) is a non-negative function). 2) The weighted frequency spreading ∫−∞∞∣f^(ω)∣2w2(ω)dω\int_{-\infty}^\infty |\hat{f}(ω)|^2w_2(ω)dω∫−∞∞​∣f^​(ω)∣2w2​(ω)dω. 3) The time weight scale aaa, w1a(x)=w1(xa−1){w_1}_a(x)=w_1(xa^{-1})w1​a​(x)=w1​(xa−1) and 4) The frequency weight scale bbb, w2b(ω)=w2(ωb−1){w_2}_b(ω)=w_2(ωb^{-1})w2​b​(ω)=w2​(ωb−1). "Generalized Uncertainty Principle" is an inequality that summarizes the constraints on the relations between the two spreading quantities and two scaling parameters. For any two reasonable weights w1(x)w_1(x)w1​(x) and w2(ω)w_2(ω)w2​(ω), we introduced a three dimensional set in R3R^3R3 that is in the essence of many uncertainty principles. The set is called "possibility body". We showed that classical uncertainty principles (such as the Heiseneberg-Pauli-Weyl uncertainty principle) stem from lower bounds for different functions defined on the possibility body. We investigated qualitative properties of general uncertainty principles and possibility bodies. Using this approach we derived new (quantitative) uncertainty principles for Landau-Pollak-Slepian weights. We found the general uncertainty principles related to homogeneous weights, w1(x)=w2(x)=xkw_1(x)=w_2(x)=x^kw1​(x)=w2​(x)=xk, k∈Nk\in Nk∈N, up to a constant.