Commutator

The commutator of two operators is

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

It measures order-dependence: if $[A,B]\ne0$, applying $A$ and then $B$ differs from applying $B$ and then $A$ on at least some states in the common domain. In quantum mechanics, commutators organize compatibility, uncertainty relations, symmetry generators, and Heisenberg-picture dynamics.

Formula Hook

The canonical position-momentum commutator is

$$[\hat x,\hat p]=i\hbar.$$

For observables $A$ and $B$, the Robertson uncertainty relation includes the expectation value of the commutator:

$$\Delta A\,\Delta B \ge \frac{1}{2}\, \lvert\langle[A,B]\rangle\rvert.$$

Canonical Home

See Commutators for the Core discussion, Commutators and Anticommutators for algebraic identities, and Canonical Commutation Relations for the reference formula sheet.

Common Confusions

• $[A,B]=0$ is an operator statement; for unbounded operators, domains matter.
• A vanishing commutator is not the same thing as statistical independence.
• $[A,B]$ and $\{A,B\}=AB+BA$ encode different algebraic information.
• The classical Poisson bracket is related to, but not identical with, the quantum commutator.

Related Entries

• Observable
• Hamiltonian
• Eigenstate

References

• L. E. Ballentine, Quantum Mechanics: A Modern Development, 2nd ed., World Scientific, 2014.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.