Page 298 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

Fourier series and Fourier integral 289
ð

e i(nÿm)è dè ˆ 2ðä mn
ÿð
The ®nal result is
ð 1
1 2 X 2
jf (è)j dè ˆ jc n j (B:16)
2ð ÿð nˆÿ1
which is one form of Parseval's theorem. Other forms of Parseval's theorem are
obtained using the various alternative Fourier expansions.
Parseval's theorem is also known as the completeness relation and may be used to
verify that the set of functions e inè for ÿ1 < n < 1 are complete, as discussed in
Section 3.4. If some of the terms in the Fourier series are missing, so that the set of
basis functions in the expansion is incomplete, then the corresponding coef®cients on
the right-hand side of equation (B.16) will also be missing and the equality will not
hold.

Fourier integral
The Fourier series expansions of a function f (x) of the variable x over the range
ÿl < x < l may be generalized to the case where the range is in®nite, i.e., where
ÿ1 < x < 1. By a suitable limiting process in which l !1, equations (B.11) and
(B.12) may be extended to the form

1 1 ikx
f (x) ˆ p  g(k)e dk (B:17)
2ð ÿ1

1 1 ÿikx
g(k) ˆ p  f (x)e dx (B:18)
2ð ÿ1
Equation (B.17) is the Fourier integral representation of f (x). The function g(k) is the
Fourier transform of f (x), which in turn is the inverse Fourier transform of g(k).
For any function f (x) which satis®es the Dirichlet conditions over the range
ÿ1 < x < 1 and for which the integral

1
2
jf (x)j dx
ÿ1
converges, the Fourier integral in equation (B.17) converges to f (x) wherever f (x)is
continuous and to the mean value of f (x) at any point of discontinuity.
In some applications a function f (x, t), where x is a distance variable and t is the
time, is represented as a Fourier integral of the form

1 1 i[kxÿù(k)t]
f (x, t) ˆ p  G(k)e dk (B:19)
2ð ÿ1
where the frequency ù(k) depends on the variable k. In this case the Fourier transform
g(k) takes the form
g(k) ˆ G(k)e ÿiù(k)t
and equation (B.18) may be written as

1 1 ÿi[kxÿù(k)t]
G(k) ˆ p  f (x, t)e dx (B:20)
2ð ÿ1
The functions f (x, t) and G(k) are, then, a generalized form of Fourier transforms.