Physics:Quantum Commutator

In quantum mechanics, a commutator measures how much two operators fail to commute. For two operators $A$ and $B$, the commutator is

$[A,B]=AB-BA.$

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

$[x,p]=i\hslash .$

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