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 precision with which certain pairs of physical properties of a particle, known as complementary variables, such as position x and momentum p, can be known. (ħ is the reduced Planck constant, h / 2π). , their standard deviations in an ensemble of individual measurements on similarly prepared systems, then "There are, in principle, no restrictions on the precisions of individual measurements and ".)

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• 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 precision with which certain pairs of physical properties of a particle, known as complementary variables, such as position x and momentum p, can be known. Introduced first in 1927, by the German physicist Werner Heisenberg, it states that the more precisely the position of some particle is determined, the less precisely its momentum can be known, and vice versa. The formal inequality relating the standard deviation of position σx and the standard deviation of momentum σp was derived by Earle Hesse Kennard later that year and by Hermann Weyl in 1928: (ħ is the reduced Planck constant, h / 2π). Historically, the uncertainty principle has been confused with a somewhat similar effect in physics, called the observer effect, which notes that measurements of certain systems cannot be made without affecting the systems, that is, without changing something in a system. Heisenberg offered such an observer effect at the quantum level (see below) as a physical "explanation" of quantum uncertainty. It has since become clear, however, that the uncertainty principle is inherent in the properties of all wave-like systems, and that it arises in quantum mechanics simply due to the matter wave nature of all quantum objects. Thus, the uncertainty principle actually states a fundamental property of quantum systems, and is not a statement about the observational success of current technology. It must be emphasized that measurement does not mean only a process in which a physicist-observer takes part, but rather any interaction between classical and quantum objects regardless of any observer.(N.B. on precision: If δx and δp are the precisions of position and momentum obtained in an individual measurement and , their standard deviations in an ensemble of individual measurements on similarly prepared systems, then "There are, in principle, no restrictions on the precisions of individual measurements and , but the standard deviations will always satisfy ".) Since the uncertainty principle is such a basic result in quantum mechanics, typical experiments in quantum mechanics routinely observe aspects of it. Certain experiments, however, may deliberately test a particular form of the uncertainty principle as part of their main research program. These include, for example, tests of number–phase uncertainty relations in superconducting or quantum optics systems. Applications dependent on the uncertainty principle for their operation include extremely low-noise technology such as that required in gravitational wave interferometers. (en)
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• Position and momentum probability densities for an initially Gaussian distribution. From top to bottom, the animations show the cases Ω=ω, Ω=2ω, and Ω=ω/2. Note the tradeoff between the widths of the distributions.
• Propagation of de Broglie waves in 1d—real part of the complex amplitude is blue, imaginary part is green. The probability of finding the particle at a given point x is spread out like a waveform, there is no definite position of the particle. As the amplitude increases above zero the curvature reverses sign, so the amplitude begins to decrease again, and vice versa—the result is an alternating amplitude: a wave.
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• Propagation of a de broglie plane wave.svg
• Propagation of a de broglie wavepacket.svg
• Position_and_momentum_of_a_Gaussian_initial_state_for_a_QHO,_narrow.gif
• Position_and_momentum_of_a_Gaussian_initial_state_for_a_QHO,_wide.gif
• Position_and_momentum_of_a_Gaussian_initial_state_for_a_QHO,_balanced.gif
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• Uncertainty principle
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• 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 precision with which certain pairs of physical properties of a particle, known as complementary variables, such as position x and momentum p, can be known. (ħ is the reduced Planck constant, h / 2π). , their standard deviations in an ensemble of individual measurements on similarly prepared systems, then "There are, in principle, no restrictions on the precisions of individual measurements and ".) (en)
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• Uncertainty principle (en)
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