Skip to main content

Mathematics for Physics

98 Commutators

An operator built from a product difference that is used to measure whether two operators can be applied in either order without change.

The definition of the commutator. The commutator of two operators is \(AB-BA\). This principle is used to test whether the product order matters.

The commutator is

\[ [A,B] = AB - BA \]

where

  • \(A\) and \(B\) are operators.
  • \([A,B]\) is their commutator.

The canonical relation \([X,P]=i\hbar\). Position and momentum do not commute. This principle is used to write the canonical commutation relation and to derive the uncertainty principle.

The position-momentum commutator is

\[ [X,P] = i\hbar \]

where

  • \(X\) is the position operator.
  • \(P\) is the momentum operator.
  • \(\hbar\) is the reduced Planck constant.

Conservation when \([A,H]=0\). If \([A,H]=0\), then \(A\) is conserved. This principle is used to identify constants of the motion.

Simultaneous eigenbases. If \([A,B]=0\), the operators share a common eigenbasis. This principle is used to measure \(A\) and \(B\) together.

Note: Also called a commutation relation when set equal to a specific value. Also denoted \([A,B]\).

98.1 References

  1. Hall, B. C. Quantum Theory for Mathematicians. Springer, 2013. — \([A,B]=AB-BA\); \([X,P]=i\hbar\).
  2. Sakurai, J. J., & Napolitano, J. Modern Quantum Mechanics. Cambridge University Press, 2021. — \([A,B]=AB-BA\); \([x,p_{x}]=i\hbar\).