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Heisenberg uncertainty principle

In quantum physics, the outcome of even an ideal measurement of a system is not deterministic, but instead is characterized by a probability distribution, and the larger the associated standard deviation is, the more "uncertain" we might say that that characteristic is for the system. The Heisenberg uncertainty principle, or HUP, gives a lower bound on the product of the standard deviations of position and momentum for a system, implying that it is impossible to have a particle that has an arbitrarily well-defined position and momentum simultaneously. More precisely, the products of the standard deviations in each of the three spatial dimensions are bounded by

where is the reduced Planck constant; Δx, Δy, and Δz are the standard deviations of the three coordinates of position; and Δpx, Δpy, and Δpz are the standard deviations of the three components of momentum. The principle generalizes to many other pairs of quantities besides position and momentum (for example, angular momentum about two different axes), and can be derived directly from the axioms of quantum mechanics (particularly the de Broglie relations).

Note that the uncertainties in question are characteristic of the mathematical quantities themselves. In any real-world measurement, there will be additional uncertainties created by the non-ideal and imperfect measurement process. The uncertainty principle holds true regardless of whether the measurements are ideal (sometimes called von Neumann measurements) or non-ideal (Landau measurements). Note also that the product of the uncertainties, of order 10−35 joule-seconds, is so small that the uncertainty principle has negligible effect on objects of macroscopic scale, despite its importance for atoms and subatomic particles.

The uncertainty principle was an important step in the development of quantum mechanics when it was discovered by Werner Heisenberg in 1927. It is often confused with the observer effect.