Physics:Quantum Commutator: Difference between revisions

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{{Short description|Operator expression measuring non-commutativity}}
{{Short description|Operator expression measuring non-commutativity}}
{{Quantum book/methods backlink|Foundations}}
{{Quantum book backlink|Foundations}}
{{Quantum article nav|previous=Physics:Quantum Matrix mechanics|previous label=Matrix mechanics|next=Physics:Quantum Measurement problem|next label=Measurement problem}}


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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<math>[A,B] = AB - BA.</math>
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'''Commutator''' is a Book I topic in the Quantum Collection. 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<ref>{{Cite book |last=Griffiths |first=David J. |title=Introduction to Quantum Mechanics |edition=2nd |publisher=Prentice Hall |year=2004 |isbn=0-13-805326-X}}</ref>
 
<math>[A,B] = AB - BA.</math> Commutators are important because they encode the non-classical algebra of observables. When two operators do not commute, the order of operations matters and the corresponding measurements generally cannot both have sharp values. This structure underlies uncertainty relations, angular momentum algebra, symmetry generators, time evolution, and the operator language used throughout quantum mechanics.
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[[File:Quantum_Commutator_educational_yellow.png|thumb|280px|A commutator compares doing two quantum operations in different orders.]]
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== Role in quantum mechanics ==
== Role in quantum mechanics ==
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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]]<ref>{{Cite book |last=Liboff |first=Richard L. |title=Introductory Quantum Mechanics |edition=4th |publisher=Addison-Wesley |year=2003 |isbn=0-8053-8714-5}}</ref> and is one of the basic structures of [[Physics:Quantum Matrix mechanics|matrix mechanics]].


== Related concepts ==
== See also ==
* [[Physics:Quantum mechanics]]
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}
* [[Physics:Quantum Matrix mechanics]]
* [[Physics:Quantum Uncertainty principle]]
* [[Physics:Quantum operator]]
* [[Physics:Quantum observable]]


== References ==
== References ==
{{reflist|3}}
{{reflist|3}}
* {{Cite web |title=Commutator |url=https://mathworld.wolfram.com/Commutator.html |website=Wolfram MathWorld |access-date=2026-05-23}}
* {{Cite book |last=Dirac |first=P. A. M. |title=The Principles of Quantum Mechanics |publisher=Oxford University Press |year=1930}}


{{Author|Harold Foppele}}
{{Author|Harold Foppele}}

Latest revision as of 22:58, 23 May 2026

← Back to Foundations Book I
← Previous : Matrix mechanics
Next : Measurement problem →

Commutator is a Book I topic in the Quantum Collection. In quantum mechanics, a commutator measures how much two operators fail to commute. For two operators A and B, the commutator is[1]

[A,B]=AB−BA. Commutators are important because they encode the non-classical algebra of observables. When two operators do not commute, the order of operations matters and the corresponding measurements generally cannot both have sharp values. This structure underlies uncertainty relations, angular momentum algebra, symmetry generators, time evolution, and the operator language used throughout quantum mechanics.

A commutator compares doing two quantum operations in different orders.

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

[x,p]=iℏI.

This relation underlies the uncertainty principle[2] and is one of the basic structures of matrix mechanics.

See also

Table of contents (217 articles)

Index

Full contents

References

  1. ↑ Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 0-13-805326-X. 
  2. ↑ Liboff, Richard L. (2003). Introductory Quantum Mechanics (4th ed.). Addison-Wesley. ISBN 0-8053-8714-5. 


Author: Harold Foppele


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