What is the physical meaning of a commutator in quantum mechanics?

Asked by: Saed Moh'd Alazar


A commutator in quantum mechanics tells us if we can measure two 'observables' at the same time. If the commutator of two 'observables' is zero, then they CAN be measured at the same time, otherwise there exists an uncertainty relation between the two. For example, the famous Heisenberg Uncertainty principle is a direct consequence of the fact that position and momentum do not commute, therefore we can not precisely determine position and momentum at the same time.

Mathematically, a commutator is written:

[A, B] = A*B - B*A

In 'normal' everyday mathematics, this would be zero. However, in quantum mechanics, the objects A and B correspond to 'observables'. So the commutator tells us that observing the system in different orders effects the outcome. Going back to the Heisenberg example, taking a measurement of the position of a particle will disturb its momentum.
Answered by: Tom Davis, B.S., Physics Grad Student, UBC, Vancouver