Page 83 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

74 General principles of quantum theory

(á i ÿ á ) ø (x)ø i (x)w(x)dx ˆ 0 (3:21)
j j
When j ˆ i, the integral in equation (3.21) cannot vanish because the

product ø ø i and the function w(x) are always positive. Therefore, we have
i

á i ˆ á and the eigenvalues á i are real. For the situation where i 6ˆ j and
i

á i 6ˆ á , the integral in equation (3.21) must vanish,
j

ø (x)ø i (x)w(x)dx ˆ 0 (3:22)
j
Thus, the set of functions ø i (x) for non-degenerate eigenvalues are mutually
orthogonal when integrated with a weighting function w(x). Eigenfunctions
corresponding to degenerate eigenvalues can be made orthogonal as discussed
earlier.
The discussion above may be generalized to more than one variable. In the
general case, equation (3.18) is replaced by
^
Aø i (q 1 , q 2 , ...) ˆ á i w(q 1 , q 2 , ...)ø i (q 1 , q 2 , ...) (3:23)
and equation (3.22) by

ø (q 1 , q 2 , ...)ø i (q 1 , q 2 , ...)w(q 1 , q 2 , ...)dq 1 dq 2 ... ˆ 0 (3:24)
j
Equation (3.18) can also be transformed into the more usual form, equation
(3.5). We ®rst de®ne a set of functions ö i (x)as

ö i (x) ˆ [w(x)] 1=2 ø i (x) ˆ ø i (x)=u(x) (3:25)
where

u(x) ˆ [w(x)] ÿ1=2 (3:26)
The function u(x) is real because w(x) is always positive and u(x) is positive
because we take the positive square root. If w(x) approaches in®nity at any
^
point within the range of hermiticity of A (as x approaches in®nity, for
example), then ø i (x) must approach zero such that the ratio ö i (x) approaches
zero. Equation (3.18) is now multiplied by u(x) and ø i (x) is replaced by
u(x)ö i (x)
^
2
u(x)Au(x)ö i (x) ˆ á i w(x)[u(x)] ö i (x)
^
^
^
If we de®ne an operator B by the relation B ˆ u(x)Au(x) and apply equation
(3.26), we obtain
^
Bö i (x) ˆ á i ö i (x)
which has the form of equation (3.5). We observe that