Since the Fourier components were not defined at the classical frequencies, they could not be used to construct an exact trajectory, so that the formalism could not answer certain overly precise questions about where the electron was or how fast it was going. In March 1926, working in Bohr's institute, Heisenberg realized that the non-commutativity implies the uncertainty principle.

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 Heisenberg, the uncertainty principle states that the more precisely the position of some particle is determined, the less precisely its momentum can be predicted from initial conditions, and vice versa.

In his Chicago lecture he refined his principle: Kennard in 1927 first proved the modern inequality: where , and , are the standard deviations of position and momentum.

Heisenberg only proved relation () for the special case of Gaussian states. ===Terminology and translation=== Throughout the main body of his original 1927 paper, written in German, Heisenberg used the word "Ungenauigkeit" ("indeterminacy"), to describe the basic theoretical principle.

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: where is the reduced Planck constant, ). Historically, the uncertainty principle has been confused with a related effect in physics, called the observer effect, which notes that measurements of certain systems cannot be made without affecting the system, that is, without changing something in a system.

When the English-language version of Heisenberg's textbook, The Physical Principles of the Quantum Theory, was published in 1930, however, the translation "uncertainty" was used, and it became the more commonly used term in the English language thereafter. ===Heisenberg's microscope=== The principle is quite counter-intuitive, so the early students of quantum theory had to be reassured that naive measurements to violate it were bound always to be unworkable.

This directly contrasts with the Copenhagen interpretation of quantum mechanics, which is non-deterministic but lacks local hidden variables. In 1934, Popper published Zur Kritik der Ungenauigkeitsrelationen (Critique of the Uncertainty Relations) in Naturwissenschaften, and in the same year Logik der Forschung (translated and updated by the author as The Logic of Scientific Discovery in 1959), outlining his arguments for the statistical interpretation.

In 1935, Einstein, Podolsky and Rosen (see EPR paradox) published an analysis of widely separated entangled particles.

Nevertheless, in 1945, L.

While formulating the many-worlds interpretation of quantum mechanics in 1957, Hugh Everett III conjectured a stronger extension of the uncertainty principle based on entropic certainty.

This directly contrasts with the Copenhagen interpretation of quantum mechanics, which is non-deterministic but lacks local hidden variables. In 1934, Popper published Zur Kritik der Ungenauigkeitsrelationen (Critique of the Uncertainty Relations) in Naturwissenschaften, and in the same year Logik der Forschung (translated and updated by the author as The Logic of Scientific Discovery in 1959), outlining his arguments for the statistical interpretation.

He believed the "natural basic assumption" that a complete description of reality would have to predict the results of experiments from "locally changing deterministic quantities" and therefore would have to include more information than the maximum possible allowed by the uncertainty principle. In 1964, John Bell showed that this assumption can be falsified, since it would imply a certain inequality between the probabilities of different experiments.

Efforts to improve this bound are an active area of research. ===The Efimov inequality by Pauli matrices=== In 1976, Sergei P.

In 1982, he further developed his theory in Quantum theory and the schism in Physics, writing: [Heisenberg's] formulae are, beyond all doubt, derivable statistical formulae of the quantum theory.

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