Editing Canonical commutation relation (section)

{{Short description|Relation satisfied by conjugate variables in quantum mechanics}}

In [[quantum mechanics]], the '''canonical commutation relation''' is the fundamental relation between [[canonical conjugate]] quantities (quantities which are related by definition such that one is the [[Fourier transform]] of another). For example,
<math display="block">[\hat x,\hat p_x] = i\hbar \mathbb{I}</math>

between the position operator {{mvar|x}} and momentum operator {{mvar|p<sub>x</sub>}} in the {{mvar|x}} direction of a point particle in one dimension, where {{math|1= [''x'' , ''p''<sub>''x''</sub>] = ''x'' ''p''<sub>''x''</sub> − ''p''<sub>''x''</sub> ''x''}} is the [[Commutator#Ring theory|commutator]] of {{mvar|x}} and {{mvar|p<sub>x</sub> }}, {{mvar|i}} is the [[imaginary unit]], and {{math|ℏ}} is the [[reduced Planck constant]] {{math|''h''/2&pi;}}, and <math> \mathbb{I}</math> is the unit operator. In general, position and momentum are vectors of operators and their commutation relation between different components of position and momentum can be expressed as
<math display="block">[\hat x_i,\hat p_j] = i\hbar \delta_{ij},</math>
where <math>\delta_{ij}</math> is the [[Kronecker delta]].

This relation is attributed to [[Werner Heisenberg]], [[Max Born]] and [[Pascual Jordan]] (1925),<ref>{{cite web |title=The Development of Quantum Mechanics|url=https://www.heisenberg-gesellschaft.de/3-the-development-of-quantum-mechanics-1925-ndash-1927.html}}</ref><ref>{{Cite journal | last1 = Born | first1 = M. | last2 = Jordan | first2 = P. | doi = 10.1007/BF01328531 | title = Zur Quantenmechanik | journal = Zeitschrift für Physik | volume = 34 | pages = 858–888 | year = 1925 | issue = 1 |bibcode = 1925ZPhy...34..858B | s2cid = 186114542 }}</ref> who called it a "quantum condition" serving as a postulate of the theory; it was noted by [[Earle Hesse Kennard|E. Kennard]] (1927)<ref>{{Cite journal | last1 = Kennard | first1 = E. H. | title = Zur Quantenmechanik einfacher Bewegungstypen | doi = 10.1007/BF01391200 | journal = Zeitschrift für Physik | volume = 44 | issue = 4–5 | pages = 326–352 | year = 1927 |bibcode = 1927ZPhy...44..326K | s2cid = 121626384 }}</ref> to imply the [[Werner Heisenberg|Heisenberg]] [[uncertainty principle]]. The [[Stone–von Neumann theorem]] gives a uniqueness result for operators satisfying (an exponentiated form of) the canonical commutation relation.