Page 309 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

300 Appendix D

The function ö n (î) as de®ned by equation (D.15) then becomes
(ÿi) n î =2 1 ÿ(s =4)‡iîs n
2
2
ö n (î) ˆ e e s ds (D:17)
ð
2(2 ðn!) 1=2 1=4 ÿ1
n
We now evaluate the summation
1
X
ö n (î)ö n (î9)
nˆ0
by substituting equation (D.17) twice, once with the dummy variable of integration s
and once with s replaced by t. Since the functions ö n (î) are real, they equal their
complex conjugates. These substitutions give
1 1 1 1 n
X 1 (î ‡î9 )=2 ÿ[(s ‡t )=4]‡i(îs‡î9t) X (ÿ1) n
2
2
2
2
ö n (î)ö n (î9) ˆ e e (st) ds dt
4ð 3=2 2 n!
n
nˆ0 ÿ1 ÿ1 nˆ0
n
since (ÿi) 2n ˆ (ÿ1) . The summation on the right-hand side is easily evaluated using
equation (A.1)
1 n n
X (ÿ1) st ÿst=2
ˆ e
n! 2
nˆ0
Noting that
2
s ‡ t 2 st (s ‡ t) 2
‡ ˆ
4 2 4
we have
1 1 1
X 1 (î ‡î9 )=2 ÿ[(s‡t) =4]‡i(îs‡î9t)
2
2
2
ö n (î)ö n (î9) ˆ e e ds dt (D:18)
4ð 3=2
nˆ0 ÿ1 ÿ1
The double integral may be evaluated by introducing the new variables u and v
s ‡ t s ÿ t
u ˆ , v ˆ or s ˆ u ‡ v, t ˆ u ÿ v
2 2
ds dt ˆ 2du dv
The double integral is thereby factored into

1 2 1 2
e
2 e ÿu ‡i(î‡î9)u du e i(îÿî9)v dv ˆ 2 3 ð 1=2 ÿ(î‡î9) =4 3 2ðä(î ÿ î9)
ÿ1 ÿ1
where the ®rst integral is evaluated by equation (A.8) and the second by (C.6).
Equation (D.18) now becomes
1
X [(î ‡î9 )=2]ÿ[(î‡î9) =4] (îÿî9) =4
2
2
2
2
ö n (î)ö n (î9) ˆ e ä(î ÿ î9) ˆ e ä(î ÿ î9)
nˆ0
Applying equation (C.5e), we obtain the completeness relation for the functions ö n (î)
1
X
ö n (î)ö n (î9) ˆ ä(î ÿ î9) (D:19)
nˆ0
demonstrating, according to equation (3.31), that the set ö n (î) is a complete set.