Page 323 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

314 Appendix F
á
1
1
d i i‡ j ÿr d i X (ÿ1) á ᇠj‡i X (ÿ1) (á ‡ j ‡ i)! ᇠj
(r e ) ˆ r ˆ r
dr i dr i á! á! (á ‡ j)!
áˆ0 áˆ0
1
X (á ‡ j ‡ i)!
ˆ r j (ÿr) á (F:14)
á!(á ‡ j)!
áˆ0
Comparison of equations (F.13) and (F.14) yields the result that
j
j ÿ j r
Ë (r) ˆ (ÿ1) r e d i (r i‡ j ÿr
e )
i
dr i
From equation (F.12) we obtain
(i ‡ j)! d i
j j ÿ j r i‡ j ÿr
L (r) ˆ (ÿ1) r e (r e )
i‡ j i
i! dr
Finally, replacing i by the original index k (ˆ i ‡ j), we have
k! ÿ j r d kÿ j k ÿr
j
j
L (r) ˆ (ÿ1) r e (r e ) (F:15)
k (k ÿ j)! dr kÿ j
Equation (F.15) for the associated Laguerre polynomials is the analog of (F.5) for the
Laguerre polynomials and, in fact, when j ˆ 0, equation (F.15) reduces to (F.5).
Differential equation
The differential equation satis®ed by the associated Laguerre polynomials may be
obtained by repeatedly differentiating equations (F.8) j times
3 2
r d L k ‡ (2 ÿ r) d L k ‡ (k ÿ 1) dL k ˆ 0
dr 3 dr 2 dr
4 3 2
r d L k ‡ (3 ÿ r) d L k ‡ (k ÿ 2) d L k ˆ 0
dr 4 dr 3 dr 2
.
. .
j
d j‡2 L k d j‡1 L k d L k
r ‡ (j ‡ 1 ÿ r) ‡ (k ÿ j) ˆ 0
dr j‡2 dr j‡1 dr j
j
When the polynomials L (r) are introduced with equation (F.9), the differential
k
equation is
2
d L j dL j j
r k ‡ (j ‡ 1 ÿ r) k ‡ (k ÿ j)L (r) ˆ 0 (F:16)
dr 2 dr k
Integral relations
In order to obtain the orthogonality and normalization relations of the associate
Laguerre polynomials, we make use of the generating function (F.10). We multiply
together g(r, s; j), g(r, t; j), and the factor r j‡í ÿr and then integrate over r to give
e
an integral that we abbreviate with the symbol I

1
1
s t
1 X X á â 1 j
j
e
e
I r j‡í ÿr g(r, s; j)g(r, t; j)dr ˆ r j‡í ÿr L (r)L (r)dr
â
á
0 ሠj ∠j á!â! 0
(F:17)