Page 93 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

84 General principles of quantum theory
^
for n . 2 also equals P i . This property is consistent with the interpretation of
^
P i as a projection operator since the result of projecting jöi onto jii should be
the same whether the projection is carried out once, twice, or multiple times.
^
The operator P i is hermitian, so that the projection of jöi on jø i i is equal to
^
the projection of jø i i on jöi. To show that P i is hermitian, we let j÷iˆ
^
^
y
jöiˆjii in equation (3.41) and obtain P i ˆ P i .
The expansion of a function f (x) in terms of the orthonormal set ø i (x), as
shown in equation (3.27), may be expressed in terms of kets as
X X
jf iˆ a i jø i iˆ a i jii
i i
where jf i is regarded as a vector in ket space. The constants a i are the
projections of jf i on the `unit ket vectors' jii and are given by equation (3.28)

a i ˆhijf i
Combining these two equations gives equation (3.29), which when expressed
in Dirac notation is

X
jf iˆ jiihijf i
i
P
Since f (x) is an arbitrary function of x, the operator i jiihij must equal the
identity operator, so that
X
jiihijˆ 1 (3:43)
i
^
From the de®nition of P i in equation (3.42), we see that
X
^
P i ˆ 1
i
P
Since the operator i jiihij equals unity, it may be inserted at any point in an
equation. Accordingly, we insert it between the bra and the ket in the scalar
product of jf i with itself
* ! +
X
hf jf iˆ f jiihij f ˆ 1

i
where we have assumed jf i is normalized. This expression may be written as
X X X
2 2
hf jf iˆ hf jiihijf iˆ jhij f ij ˆ ja i j ˆ 1
i i i
Thus, the expression (3.43) is related to the completeness criterion (3.30) and
is called, therefore, the completeness relation.