Page 321 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

312 Appendix F
k á á
X (ÿ1) k! d
k
k
k
[(d=dr) ÿ 1] ˆ (ÿ1) [1 ÿ (d=dr)] ˆ (ÿ1) k
á!(k ÿ á)! dr á
áˆ0
so that
L k (r) ˆ [(d=dr) ÿ 1] r k (F:7)
k
á
where we have noted that (ÿ1) ˆ (ÿ1) ÿá .
From equation (F.2), (F.5), (F.6), or (F.7), we observe that the polynomial L k (r)isof
degree k and we may readily obtain the ®rst few polynomials of the set
L 0 (r) ˆ 1
L 1 (r) ˆ 1 ÿ r
L 2 (r) ˆ 2 ÿ 4r ‡ r 2

2
L 3 (r) ˆ 6 ÿ 18r ‡ 9r ÿ r 3
We also note that L k (0) ˆ k!.

Differential equation
Equation (F.5) can be used to ®nd the differential equation satis®ed by the polynomials
L k (r). We note that the function f (r) de®ned as
k ÿr
f (r) r e
satis®es the relation
df
r ‡ (r ÿ k)f ˆ 0
dr
If we differentiate this expression k ‡ 1 times, we obtain
2
d f (k) df (k)
r ‡ (1 ‡ r) ‡ (k ‡ 1) f (k) ˆ 0
dr 2 dr
where f (k) is the kth derivative of f (r). Since from equation (F.5) we have
f (k) ˆ e ÿr L k (r)
the Laguerre polynomials L k (r) satisfy the differential equation
2
d L k dL k
r ‡ (1 ÿ r) ‡ kL k (r) ˆ 0 (F:8)
dr 2 dr

Associated Laguerre polynomials
j
The associated Laguerre polynomials L (r) are de®ned in terms of the Laguerre
k
polynomials by
d j
j
L (r) L k (r) (F:9)
k dr j
j
k
Since L k (r) is a polynomial of degree k, L (r) is a constant and L (r) ˆ 0 for j . k.
k
k
The generating function g(r, s; j) for the associated Laguerre polynomials with ®xed
j is readily obtained by differentiation of equation (F.1)