Physics:Quantum Commutator - Revision history

WikiHarold: Expand short Quantum intro

2026-05-23T22:58:45Z

Expand short Quantum intro

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	<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">		<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">
	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>		'''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>		<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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WikiHarold: Move Quantum references inline

2026-05-23T22:15:12Z

Move Quantum references inline

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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}}

WikiHarold: Replace Quantum references with Wikipedia-sourced references

2026-05-23T22:03:01Z

Replace Quantum references with Wikipedia-sourced references

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	== 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=Griffiths \|first=David J. \|title=Introduction to Quantum Mechanics \|edition=2nd \|publisher=Prentice Hall \|year=2004 \|isbn=0-13-805326-X}}
	* {{Cite book \|last=~~Dirac~~ \|first=~~P. A. M.~~ \|title=~~The Principles of~~ Quantum ~~Mechanics~~ \|publisher=~~Oxford University Press~~ \|year=~~1930~~}}		* {{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}}

WikiHarold: Use Quantum See also index module

2026-05-23T21:57:43Z

Use Quantum See also index module

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	== See also ==		== See also ==
	* [[Physics:~~Quantum mechanics]]~~		{{#invoke:PhysicsQC\|tocHeadingAndList\|Physics:Quantum basics/See also}}
	* [[Physics:Quantum Matrix mechanics]]
	* [[Physics:Quantum ~~Uncertainty principle]]~~
	* quantum operators
	* quantum observables

	== References ==		== References ==

WikiHarold: Remove leading BOM from Quantum page

2026-05-23T21:55:36Z

Remove leading BOM from Quantum page

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

WikiHarold: Repair Quantum page See also and references

2026-05-23T21:53:38Z

Repair Quantum page See also and references

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	{{Short description\|Operator expression measuring non-commutativity}}		{{Short description\|Operator expression measuring non-commutativity}}
	{{Quantum book backlink\|Foundations}}		{{Quantum book backlink\|Foundations}}
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	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]].

	== ~~Related concepts~~ ==		== See also ==
	* [[Physics:Quantum mechanics]]		* [[Physics:Quantum mechanics]]
	* [[Physics:Quantum Matrix mechanics]]		* [[Physics:Quantum Matrix mechanics]]

WikiHarold: Restore Quantum article header layout

2026-05-23T21:41:42Z

Restore Quantum article header layout

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	{{Quantum article nav\|previous=Physics:Quantum Matrix mechanics\|previous label=Matrix mechanics\|next=Physics:Quantum Measurement problem\|next label=Measurement problem}}		{{Quantum article nav\|previous=Physics:Quantum Matrix mechanics\|previous label=Matrix mechanics\|next=Physics:Quantum Measurement problem\|next label=Measurement problem}}

	~~[[File~~:~~Quantum_Commutator_educational_yellow.png\|thumb\|right\|A commutator compares doing two quantum operations in different orders.]]~~		<div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">

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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

	<math>[A,B] = AB - BA.</math>		<math>[A,B] = AB - BA.</math>
			</div>

			<div style="width:300px;">
			[[File:Quantum_Commutator_educational_yellow.png\|thumb\|280px\|A commutator compares doing two quantum operations in different orders.]]
			</div>

			</div>

	== Role in quantum mechanics ==		== Role in quantum mechanics ==

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2026-05-23T09:59:24Z

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	* [[Physics:Quantum Matrix mechanics]]		* [[Physics:Quantum Matrix mechanics]]
	* [[Physics:Quantum Uncertainty principle]]		* [[Physics:Quantum Uncertainty principle]]
	* ~~[[Physics:Quantum operator]]~~		* quantum operators
	* ~~[[Physics:Quantum observable]]~~		* quantum observables

	== References ==		== References ==

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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}}

			[[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]].

Maintenance script: Create compact quantum concept page

2026-05-23T09:52:41Z

Create compact quantum concept page

New page

{{Short description|Operator expression measuring non-commutativity}}
{{Quantum book/methods backlink|Foundations}}

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

<math>[A,B] = AB - BA.</math>

== 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

<math>[x,p] = i\hbar.</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]].

== Related concepts ==
* [[Physics:Quantum mechanics]]
* [[Physics:Quantum Matrix mechanics]]
* [[Physics:Quantum Uncertainty principle]]
* [[Physics:Quantum operator]]
* [[Physics:Quantum observable]]

== References ==
{{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}}