Page 84 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

3.4 Eigenfunction expansions 75

^ ^ ^
ø Aø i dx ˆ ö uAuö i dx ˆ ö Bö i dx
j j j

^
^
^

(Aø j ) ø i dx ˆ (Auö j ) uö i dx ˆ (Bö j ) ö i dx
^
Since A is hermitian with respect to the ø i s, the two integrals on the left of
each equation equal each other, from which it follows that

^
^

ö Bö i dx ˆ (Bö j ) ö i dx
j
^
and B is therefore hermitian with respect to the ö i s.
3.4 Eigenfunction expansions
Consider a set of orthonormal eigenfunctions ø i of a hermitian operator. Any
arbitrary function f of the same variables as ø i de®ned over the same range of
these variables may be expanded in terms of the members of set ø i
X
f ˆ a i ø i (3:27)
i
where the a i s are constants. The summation in equation (3.27) converges to the
function f if the set of eigenfunctions is complete.By complete we mean that
no other function g exists with the property that hg j ø i iˆ 0 for any value of i,
where g and ø i are functions of the same variables and are de®ned over the
same variable range. As a general rule, the eigenfunctions of a hermitian
operator are not only orthogonal, but are also complete. A mathematical
criterion for completeness is presented at the end of this section.
The coef®cients a i are evaluated by multiplying (3.27) by the complex

conjugate ø of one of the eigenfunctions, integrating over the range of the
j
variables, and noting that the ø i s are orthonormal
* +
X X
hø j j f iˆ ø j a i ø i ˆ a i hø j j ø i iˆ a j

i i
Replacing the dummy index j by i,wehave

a i ˆhø i j f i (3:28)
Substitution of equation (3.28) back into (3.27) gives
X
f ˆ hø i j f iø i (3:29)
i