Page 294 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

Appendix B

Fourier series and Fourier integral

Fourier series
An arbitrary function f (è) which satis®es the Dirichlet conditions can be expanded as
1
a 0 X
f (è) ˆ ‡ (a n cos nè ‡ b n sin nè) (B:1)
2
nˆ1
where è is a real variable, n is a positive integer, and the coef®cients a n and b n are
constants. The Dirichlet conditions specify that f (è) is single-valued, is continuous
except for a ®nite number of ®nite discontinuities, and has a ®nite number of maxima
and minima. The series expansion (B.1) of the function f (è) is known as a Fourier
series.
We note that
cos n(è ‡ 2ð) ˆ cos nè
sin n(è ‡ 2ð) ˆ sin nè
so that each term in equation (B.1) repeats itself in intervals of 2ð. Thus, the function
f (è) on the left-hand side of equation (B.1) has the property
f (è ‡ 2ð) ˆ f (è)
which is to say, f (è) is periodic with period 2ð. For convenience, we select the range
ÿð < è < ð for the period, although any other range of width 2ð is acceptable. If a
function F(j) has period p, then it may be converted into a function f (è) with period
2ð by introducing the new variable è de®ned by è 2ðj=p, so that
f (è) F(2ðj= p). If a non-periodic function F(è) is expanded in a Fourier series, the
function f (è) obtained from equation (B.1) is identical with F(è) over the range
ÿð < è < ð, but outside that range the two functions do not agree.
To ®nd the coef®cients a n and b n in the Fourier series, we ®rst multiply both sides
of equation (B.1) by cos mè and integrate from ÿð to ð. The resulting integrals are
evaluated in equations (A.12), (A.14), and (A.16). For n ˆ 0, all the integrals on the
right-hand side vanish except the ®rst, so that
ð

a 0
f (è)dè ˆ 3 2ð ˆ ða 0
ÿð 2
For m . 0, all the integrals on the right-hand side vanish except for the one in which
n ˆ m, giving

285