Page 297 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

288 Appendix B

a n ÿ ib n
c n , n . 0
2
a n ‡ ib n
c ÿn , n . 0 (B:9)
2
a 0
c 0
2
The coef®cients c n in equation (B.8) may be obtained from (B.9) with a n and b n given
by (B.2). The result is
ð
1 ÿinè
c n ˆ f (è)e dè (B:10)
2ð ÿð
which applies to all values of n, positive and negative, including n ˆ 0. We note in

passing that c ÿn is the complex conjugate c of c n .
n
In terms of the distance variable x, equations (B.8) and (B.10) become
1
X
f (x) ˆ c n e inðx=l (B:11)
nˆÿ1
1 l ÿinðx=l
c n ˆ f (x)e dx (B:12)
2l ÿl
while in terms of the time t, they take the form
1
X inùt
f (t) ˆ c n e (B:13)
nˆÿ1
ù ð=ù ÿinùt
c n ˆ f (t)e dt (B:14)
2ð ÿð=ù

Parseval's theorem
We now investigate the relation between the average of the square of f (è) and the
coef®cients in the Fourier series for f (è). For this purpose we select the Fourier series
(B.8), although any of the other expansions would serve as well. In this case the
2
average of jf (è)j over the interval ÿð < è < ð is
ð
1 jf (è)j dè
2
2ð ÿð
The square of the absolute value of f (è) in equation (B.8) is
2
1 1 1
X X X
2 i(nÿm)è
jf (è)j ˆ c n e inè ˆ c c n e (B:15)
m

nˆÿ1 mˆÿ1 nˆÿ1
where the two independent summations have been assigned different dummy indices.
Integration of both sides of equation (B.15) over è from ÿð to ð gives
ð X X ð

1
1
2 i(nÿm)è
jf (è)j dè ˆ c c n e dè
m
ÿð mˆÿ1 nˆÿ1 ÿð
Since m and n are integers, the integral on the right-hand side vanishes except when
m ˆ n, so that we have