Mathematical Methods for Physicists: A concise introduction - Site Map

SPECIAL FUNCTIONS OF MATHEMATICAL PHYSICS
Substituting these into Eq. (7.38) we obtain
X 1
jˆ0
… j ‡ 1†… j ‡ 2†a j‡2 ‡ 2… j†a j x j ˆ 0:
For a power series to vanish the coecient of each power of x must be zero; this
gives
… j ‡ 1†… j ‡ 2†a j‡2 ‡ 2… j†a j ˆ 0;
from which we obtain the recurrence relations
a j‡2 ˆ
2… j †
… j ‡ 1†… j ‡ 2† a j:
…7:40†
We obtain polynomial solutions of Eq. (7.38) when ˆ n, a positive integer. Then
Eq. (7.40) gives
For even n, Eq. (7.40) gives
a n‡2 ˆ a n‡4 ˆˆ0:
a 2 ˆ…1† 2n
2! a 0; a 4 ˆ…1† 2 2 2 …n 2†n
a
4! 0 ; a 6 ˆ…1† 3 2 3 …n 4†…n 2†n
a
6!
0
and generally
a n ˆ…1† n=2 2 n=2 n…n 2†4 2
a
n!
0 :
This solution is called a Hermite polynomial of degree n and is written H n …x†. If
we choose
we can write
H n …x† ˆ…2x† n
a 0 ˆ …1†n=2 2 n=2 n!
n…n 2†4 2 ˆ …1†n=2 n!
…n=2†!
n…n 1†
…2x† n2 ‡
1!
n…n 1†…n 2†…n 3†
…2x† n4 ‡: …7:41†
2!
When n is odd the polynomial solution of Eq. (7.38) can still be written as Eq.
(7.41) if we write
In particular,
a 1 ˆ …1†…n1†=2 2n!
…n=2 1=2†! :
H 0 …x† ˆ1; H 1 …x† ˆ2x; H 3 …x† ˆ4x 2 2; H 3 …x† ˆ8x 2 12x;
H 4 …x† ˆ16x 4 48x 2 ‡ 12; H 5 …x† ˆ32x 5 160x 3 ‡ 120x; ... :
312