and 
,
If 
, then 
, and we say ``
and
commute;'' otherwise, 
gives us a measure of the
error we make if we assume 
and 
do commute. As an example,
consider our operators 
and 
. Then,
Thus 
and 
do not commute and the extent of
their non-commutivity is measured by 
. The extent of this
non-commutivity is
directly related to the Heisenberg uncertainty principle 
. Note that in the classical limit (
), 
, 
and 
do commute. This
corresponds directly to the fact that in classical physics
there is no limit to how precisely x and p may be defined so that
.
As a final exercise, we verify that we get this same result for the commutator when carried out in the momentum representation,
