Page 81 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

72 General principles of quantum theory

hermitian operator involve several variables and that the weighting function
must be used. The functions are, therefore, orthogonal with respect to the
weighting function w(q 1 , q 2 , ...).
If the weighting function is real and positive, then we can de®ne ö 1 and ö 2
as
p  p 
ö 1 ˆ wø 1 , ö 2 ˆ wø 2

The functions ö 1 and ö 2 are then mutually orthogonal with respect to a
^
weighting function of unity. Moreover, if the operator A is hermitian with
^
respect to ø 1 and ø 2 with a weighting function w, then A is hermitian with
respect to ö 1 and ö 2 with a weighting function equal to unity.
If two or more linearly independent eigenfunctions have the same eigen-
value, so that the eigenvalue is degenerate, the orthogonality theorem does not
apply. However, it is possible to construct eigenfunctions that are mutually
orthogonal. Suppose there are two independent eigenfunctions ø 1 and ø 2 of
^
the operator A with the same eigenvalue á. Any linear combination
c 1 ø 1 ‡ c 2 ø 2 , where c 1 and c 2 are any pair of complex numbers, is also an
^
eigenfunction of A with the same eigenvalue, so that
^
^
^
A(c 1 ø 1 ‡ c 2 ø 2 ) ˆ c 1 Aø 1 ‡ c 2 Aø 2 ˆ á(c 1 ø 1 ‡ c 2 ø 2 )
From any pair ø 1 , ø 2 which initially are not orthogonal, we can construct by
selecting appropriate values for c 1 and c 2 a new pair which are orthogonal. By
selecting different sets of values for c 1 , c 2 , we may obtain in®nitely many new
pairs of eigenfunctions which are mutually orthogonal.
As an illustration, suppose the members of a set of functions ø 1 , ø 2 , ... , ø n
are not orthogonal. We de®ne a new set of functions ö 1 , ö 2 , ... , ö n by the
relations

ö 1 ˆ ø 1
ö 2 ˆ aö 1 ‡ ø 2

ö 3 ˆ b 1 ö 1 ‡ b 2 ö 2 ‡ ø 3
.
. .

If we require that ö 2 be orthogonal to ö 1 by setting hö 1 j ö 2 iˆ 0, then the
constant a is given by

a ˆÿhø 1 j ø 2 i=hø 1 j ø 1 iˆÿhö 1 j ø 2 i=hö 1 j ö 1 i
and ö 2 is determined. We next require ö 3 to be orthogonal to ö 1 and to ö 2 ,
which gives