Physics:Quantum Commutator: Difference between revisions

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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<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>
<math>[A,B] = AB - BA.</math>
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<math>[x,p] = i\hbar I.</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]].


== See also ==
== See also ==
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== References ==
== References ==
{{reflist|3}}
{{reflist|3}}
* {{Cite book |last=Griffiths |first=David J. |title=Introduction to Quantum Mechanics |edition=2nd |publisher=Prentice Hall |year=2004 |isbn=0-13-805326-X}}
* {{Cite book |last=Liboff |first=Richard L. |title=Introductory Quantum Mechanics |edition=4th |publisher=Addison-Wesley |year=2003 |isbn=0-8053-8714-5}}
* {{Cite book |last=McMahon |first=David |title=Quantum Field Theory |publisher=McGraw Hill |year=2008 |isbn=978-0-07-154382-8}}


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

Revision as of 22:15, 23 May 2026

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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.

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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