Page 313 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

304 Appendix E
2
d v dv
2
(1 ÿ ì ) ‡ 2(l ÿ 1)ì ‡ 2lv ˆ 0
dì 2 dì
We now differentiate r times more and obtain
r
d r‡2 v d r‡1 v d v
2
(1 ÿ ì ) ‡ 2(l ÿ r ÿ 1)ì ‡ (r ‡ 1)(2l ÿ r) ˆ 0 (E:8)
dì r‡2 dì r‡1 dì r
If we let r ˆ l and de®ne w as
l
d v d l 2 l
w ˆ (ì ÿ 1)
dì l dì l
then equation (E.8) reduces to
2
d w dw
2
(1 ÿ ì ) ÿ 2ì ‡ l(l ‡ 1)w ˆ 0
dì 2 dì
which is just Legendre's differential equation (E.7). Since the polynomials P l (ì)
represent all of the solutions of equation (E.7), these polynomials must be multiples of
w, so that
d l 2 l
P l (ì) ˆ c l (ì ÿ 1)
dì l
l
The proportionality constants c l may be evaluated by setting the term in ì , namely
d l (2l)!
2l l
c l l ì ˆ c l ì
dì l!
l
equal to the term in ì in equation (E.2), i.e.,
(2l)! ì l
l
2 (l!) 2
Thus, we have
1
c l ˆ l
2 l!
and
1 d l 2 l
P l (ì) ˆ (ì ÿ 1) (E:9)
2 l! dì l
l
This expression (equation (E.9)) is Rodrigues' formula.

Associated Legendre polynomials
m
The associated Legendre polynomials P (ì) are de®ned in terms of the Legendre
l
polynomials P l (ì)by
m
d P l (ì)
2 m=2
m
P (ì) (1 ÿ ì ) (E:10)
l dì m
where m is a positive integer, m ˆ 0, 1, 2, .. . , l.If m ˆ 0, then the corresponding
associated Legendre polynomial is just the Legendre polynomial of degree l.If m . l,
then the corresponding associated Legendre polynomial vanishes.
The generating functions g (m) (ì, s) for the associated Legendre polynomials may
be found from equation (E.1) by letting