Page 98 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

3.7 Postulates of quantum mechanics 89

the degeneracy of the eigenvalue ë i and let jiái, á ˆ 1, 2, ... , g i , be the
^
orthonormal eigenkets of A. We assume that the subset of kets corresponding
to each eigenvalue ë i has been made orthogonal by the Schmidt procedure
outlined in Section 3.3.
If the eigenkets jiái constitute a discrete set, we may expand the state vector
jØi as
g i
X X
jØiˆ c iá jiái (3:48)
i áˆ1
where the expansion coef®cients c iá are
c iá ˆhiájØi (3:49)
The expansion of the bra vector hØj is, therefore, given by
g j
X X

hØjˆ c hjâj (3:50)
jâ
j âˆ1
where the dummy indices i and á have been replaced by j and â.
^
The expectation value of A is obtained by substituting equations (3.48) and
(3.50) into (3.46)
g j g i g j g i
X X X X X X X X
^
hAiˆ c c iá hjâjAjiáiˆ c c iá ë i hjâjiái
jâ
jâ
j âˆ1 i áˆ1 j âˆ1 i áˆ1
g i
X X
2
ˆ jc iá j ë i (3:51)
i áˆ1
where we have noted that the kets jiái are orthonormal, so that
hjâjiáiˆ ä ij ä áâ
A comparison of equations (3.47) and (3.51) relates the probability P i to the
expansion coef®cients c iá
g i g i
X X
2 2
P i ˆ jc iá j ˆ jhiájØij (3:52)
áˆ1 áˆ1
where equation (3.49) has also been introduced. For the case where ë i is non-
degenerate, the index á is not needed and equation (3.52) reduces to
2 2
P i ˆjc i j ˆjhijØij
For a continuous spectrum of eigenkets with non-degenerate eigenvalues, it
is more convenient to write the eigenvalue equation (3.45) in the form
^
Ajëiˆ ëjëi
where ë is now a continuous variable and jëi is the eigenfunction whose
eigenvalue is ë. The expansion of the state vector Ø becomes