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物理

物理学咬文嚼字之四十四 Uncertaintyof the Uncertainty Principle（下）

曹则贤

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

经典不确定性所谓的＂不能够无限精确地确定一个物理量＂给欧洲的物理学家们带来了惶恐,但是经典世界里存在exact positioning的说法似乎也是毫无根据的.确定公路上一辆汽车的位置,精确到毫米量级都是没必要的、原则上不可能的（用汽车的哪个部位标定它的位置呢,部位又如何定位呢？）,遑论无限精

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经典不确定性所谓的＂不能够无限精确地确定一个物理量＂给欧洲的物理学家们带来了惶恐,但是经典世界里存在exact positioning的说法似乎也是毫无根据的.确定公路上一辆汽车的位置,精确到毫米量级都是没必要的、原则上不可能的（用汽车的哪个部位标定它的位置呢,部位又如何定位呢？）,遑论无限精
许多人以为uncertainty principle是量子理论的一个基本原理，但它不足是．它来自对任何正定空间都成立的Schwarz不等式，不过反映物理学中变量通过乘法或者某个方程耦合到一起的一个推论，同各种境遇中的两难选择产生了联想．量子力学中的uncertainty relation基于算符的非对易性，没有任何测量可以精确到能强化其正确性的地步．从经典扩散方程也能得出uncertainty relation关系．关于uncertainty principle有大量的错误诠释和滥用，中文语境下是翻译成“测不准原理”还是“不确定性原理”也值得商榷．
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.
The uncertainty principle plays an important role in both physics and mathematics, and it also plays an important role in information science. This paper reviewed the Heisenberg’s uncertainty principles, windowed uncertainty principles, logarithmic uncertainty principles, entropic uncertainty principles and so on in the fractional Fourier transform domains and linear canonical transform domains in great details. The development and applications of generalized uncertainty principles were analyzed, and the trend and direction were given as well.
The most famous contribution of Heisenberg is uncertainty principle. But the original uncertainty principle is improper. Considering all the possible situations (including the case that people can create laws) and applying Neutrosophy and Quad-stage Method, this paper presents "certainty-uncertainty principles" with general form and variable dimension fractal form. According to the classification of Neutrosophy, "certainty-uncertainty principles" can be divided into three principles in different conditions: "certainty principle", namely a particle’s position and momentum can be known simultaneously; "uncertainty principle", namely a particle’s position and momentum cannot be known simultaneously; and neutral (fuzzy) "indeterminacy principle", namely whether or not a particle’s position and momentum can be known simultaneously is undetermined.