Page 86 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics
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3.5 Simultaneous eigenfunctions 77
X
f (q 1 , q 2 , ...) ø (q9 1 , q9 2 , ...)f (q9 1 , q9 2 , ...)w(q9 1 , q9 2 , ...)dq9 1 ,dq9 2 , ...
i
i
3 ø i (q 1 , q 2 , ...)
Interchanging the order of summation and integration gives
" #
X
f (q 1 , q 2 , ...) ø (q9 1 , q9 2 , ...)ø i (q 1 , q 2 , ...)
i
i
3 f (q9 1 , q9 2 , ...)w(q9 1 , q9 2 , ...)dq 1 dq 2 ...
so that the completeness relation takes the form
X
w(q9 1 , q9 2 , ...) ø (q9 1 , q9 2 , ...)ø i (q 1 , q 2 , ...) ä(q 1 ÿ q9 1 )ä(q 2 ÿ q9 2 ) ...
i
i
(3:32)
3.5 Simultaneous eigenfunctions
Suppose the members of a complete set of functions ø i are simultaneously
^
^
eigenfunctions of two hermitian operators A and B with eigenvalues á i and â i ,
respectively
^
Aø i á i ø i
^
Bø i â i ø i
^
^
If we operate on the ®rst eigenvalue equation with B and on the second with A,
we obtain
^ ^ ^
BAø i á i Bø i á i â i ø i
^^ ^
ABø i â i Aø i á i â i ø i
from which it follows that
^^
^ ^
(AB ÿ BA)ø i [A, B]ø i 0
^ ^
Thus, the functions ø i are eigenfunctions of the commutator [A, B] with
eigenvalues equal to zero. An operator that gives zero when applied to any
^
^
member of a complete set of functions is itself zero, so that A and B commute.
^
^
We have just shown that if the operators A and B have a complete set of
^
^
simultaneous eigenfunctions, then A and B commute.
We now prove the converse, namely, that eigenfunctions of commuting
operators can always be constructed to be simultaneous eigenfunctions.
^
^
^ ^
^
Suppose that Aø i á i ø i and that [A, B] 0. Since A and B commute, we
have