15.4 Fourier series 425
The potential at point P is then
When Ris large enough (R 1 >> 1 r), the expansion of the potential can be truncated after
the leading term,
This is the potential that dominates the long-range interactions of a neutral polar
molecule.
0 Exercises 5, 6
15.4 Fourier series
The Fourier-series representation of a functionf(x)is the expansion of the function
in terms of the set of trigonometric functions
cos 1 nx, n 1 = 1 0, 1, 2, =
sin 1 nx, n 1 = 1 1, 2, 3, =
(15.31)
in the interval−π 1 ≤ 1 x 1 ≤ 1 π(we consider intervals of arbitrary width later in this section).
These functions are orthogonal in the interval (and in every interval of width 2 π),
(15.32)
(15.33)
(15.34)
and the corresponding normalization integrals are
(15.35)
(15.36)
Z
−+=>
ππsin sinnx nx dx π if 0n
Z
−+=
=
πππ
π
cos cosnx nx dx
n
n
20
0
if
if
Z
−+=, ,
ππcos sinmx nx dx 0 all m n
Z
−+=, ≠
ππsin sinmx nx dx 0 m n
Z
−+=, ≠
ππcos cosmx nx dx 0 m n
V
R
≈
μ
ε
cosθ
4
02π
V
R
Q
R
Q
R
=+++
μ
εεε
cosθ
444
02304506πππ