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The commutator, defined in section 3.1.2, is
very important in quantum mechanics. Since a definite value of
observable A can be assigned to a system only if the system is in an
eigenstate of , then we can simultaneously assign definite
values to two observables A and B only if the system is in an
eigenstate of both and . Suppose the system has a
value of for observable A and for observable B. The we
require



(64) 




If we multiply the first equation by and the second by
then we obtain



(65) 




and, using the fact that
is an eigenfunction of
and , this becomes



(66) 




so that if we subtract the first equation from the second, we obtain

(67) 
For this to hold for general eigenfunctions, we must have
, or
. That is, for two
physical quantities to be simultaneously observable, their operator
representations must commute.
Section 8.8 of Merzbacher [2] contains some useful
rules for evaluating commutators. They are summarized below.

(68) 

(69) 

(70) 

(71) 

(72) 

(73) 

(74) 
If and are two operators which commute with their
commutator, then

(75) 

(76) 
We also have the identity (useful for coupledcluster theory)

(77) 
Finally, if
then the uncertainties
in A and B, defined as
, obey the
relation^{1}

(78) 
This is the famous Heisenberg uncertainty principle. It is easy
to derive the wellknown relation

(79) 
from this generalized rule.
Next: Linear Vector Spaces in
Up: Mathematical Background
Previous: Unitary Operators
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David Sherrill
20060815