Page 82 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

3.3 Hermitian operators 73

b 1 ˆÿhö 1 j ø 3 i=hö 1 j ö 1 i
b 2 ˆÿhö 2 j ø 3 i=hö 2 j ö 2 i
In general, we have
sÿ1
X
ö s ˆ ø s ‡ k si ö i
iˆ1

k si ˆÿhö i j ø s i=hö i j ö i i
This construction is known as the Schmidt orthogonalization procedure. Since
the initial selection for ö 1 can be any of the original functions ø i or any linear
combination of them, an in®nite number of orthogonal sets ö i can be obtained
by the Schmidt procedure.
We conclude that all eigenfunctions of a hermitian operator are either
mutually orthogonal or, if belonging to a degenerate eigenvalue, can be chosen
to be mutually orthogonal. Throughout the remainder of this book, we treat all
the eigenfunctions of a hermitian operator as an orthogonal set.

Extended orthogonality theorem
The orthogonality theorem can also be extended to cover a somewhat more
general form of the eigenvalue equation. For the sake of convenience, we
present in detail the case of a single variable, although the treatment can be
generalized to any number of variables. Suppose that instead of the eigenvalue
^
equation (3.5), we have for a hermitian operator A of one variable
^
Aø i (x) ˆ á i w(x)ø i (x) (3:18)
where the function w(x) is real, positive, and the same for all values of i.
Therefore, equation (3.18) can also be written as
^

A ø (x) ˆ á w(x)ø (x) (3:19)
j
j
j

Multiplication of equation (3.18) by ø (x) and integration over x give
j

^

ø (x)Aø i (x)dx ˆ á i ø (x)ø i (x)w(x)dx (3:20)
j
j
^
Now, the operator A is hermitian with respect to the functions ø i with a
weighting function equaling unity, so that the integral on the left-hand side of
equation (3.20) becomes

^

^
ø (x)Aø i (x)dx ˆ ø i (x)A ø (x)dx ˆ á ø (x)ø i (x)w(x)dx
j j j j
where equation (3.19) has been used as well. Accordingly, equation (3.20)
becomes