Physics:Quantum Commutator: Difference between revisions
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{{Short description|Operator expression measuring non-commutativity}} | {{Short description|Operator expression measuring non-commutativity}} | ||
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[[File:Quantum_Commutator_educational_yellow.png|thumb|right|A commutator compares doing two quantum operations in different orders.]] | |||
In quantum mechanics, a '''commutator''' measures how much two operators fail to commute. For two operators <math>A</math> and <math>B</math>, the commutator is | In quantum mechanics, a '''commutator''' measures how much two operators fail to commute. For two operators <math>A</math> and <math>B</math>, the commutator is | ||
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The canonical position and momentum commutator is | The canonical position and momentum commutator is | ||
<math>[x,p] = i\hbar.</math> | <math>[x,p] = i\hbar I.</math> | ||
This relation underlies the [[Physics:Quantum Uncertainty principle|uncertainty principle]] and is one of the basic structures of [[Physics:Quantum Matrix mechanics|matrix mechanics]]. | This relation underlies the [[Physics:Quantum Uncertainty principle|uncertainty principle]] and is one of the basic structures of [[Physics:Quantum Matrix mechanics|matrix mechanics]]. | ||
Revision as of 09:57, 23 May 2026
In quantum mechanics, a commutator measures how much two operators fail to commute. For two operators and , the commutator is
Role in quantum mechanics
Commutators are central because quantum observables are represented by operators. If two observables have a nonzero commutator, the corresponding quantities generally cannot both have sharply defined values in the same state.
The canonical position and momentum commutator is
This relation underlies the uncertainty principle and is one of the basic structures of matrix mechanics.
Related concepts
- Physics:Quantum mechanics
- Physics:Quantum Matrix mechanics
- Physics:Quantum Uncertainty principle
- Physics:Quantum operator
- Physics:Quantum observable
References
- "Commutator". https://mathworld.wolfram.com/Commutator.html.
- Dirac, P. A. M. (1930). The Principles of Quantum Mechanics. Oxford University Press.
Author: Harold Foppele