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Entropic Uncertainty
In quantum mechanics, information theory, and Fourier analysis, the entropic uncertainty or Hirschman uncertainty is defined as the sum of the temporal and spectral Shannon entropies. It turns out that Heisenberg's uncertainty principle can be expressed as a lower bound on the sum of these entropies. This is ''stronger'' than the usual statement of the uncertainty principle in terms of the product of standard deviations. In 1957, Hirschman considered a function ''f'' and its Fourier transform ''g'' such that :g(y) \approx \int_^\infty \exp (-2\pi ixy) f(x)\, dx,\qquad f(x) \approx \int_^\infty \exp (2\pi ixy) g(y)\, dy ~, where the "≈" indicates convergence in 2, and normalized so that (by Plancherel's theorem), : \int_^\infty , f(x), ^2\, dx = \int_^\infty , g(y), ^2 \,dy = 1~. He showed that for any such functions the sum of the Shannon entropies is non-negative, : H(, f, ^2) + H(, g, ^2) \equiv - \int_^\infty , f(x), ^2 \log , f(x), ^2\, dx - \int_^\infty , g(y), ^2 \lo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uncertainty Principle
In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physical quantities of a particle, such as position, ''x'', and momentum, ''p'', can be predicted from initial conditions. Such variable pairs are known as complementary variables or canonically conjugate variables; and, depending on interpretation, the uncertainty principle limits to what extent such conjugate properties maintain their approximate meaning, as the mathematical framework of quantum physics does not support the notion of simultaneously well-defined conjugate properties expressed by a single value. The uncertainty principle implies that it is in general not possible to predict the value of a quantity with arbitrary certainty, even if all initial conditions are specified. Introduced first in 1927 by the German physicist Werner ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
