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

Asked by: Saed Moh'd Alazar

Answer

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

'In a way science is a key to the gates of heaven, and the same key opens the gates of hell, and we do not have any instructions as to which is which gate.
Shall we throw away the key and never have a way to enter the gates of heaven? Or shall we struggle with the problem of which is the best way to use the key?'