Page 308 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

Hermite polynomials 299

For convenience, we have abbreviated the integral with the symbol I. To evaluate the
left integral, we substitute the analytical forms for the generating functions from
equation (D.1) to give

1 2 2 2 1 2
e
I ˆ e ÿî e 2îsÿs e 2îtÿt dî ˆ e 2st e ÿ(îÿsÿt) d(î ÿ s ÿ t) ˆ ð 1=2 2st
ÿ1 ÿ1
where equation (A.5) has been used. We next expand e 2st in the power series (A.1) to
obtain
1 n n n
X 2 s t
I ˆ ð 1=2
n!
nˆ0
Substitution of this expression for I into equation (D.11) gives
1 n n 1 1 n m 1
X 2 (st) X X s t 2
ð 1=2 ˆ e ÿî H n (î)H m (î)dî (D:12)
n! n!m!
nˆ0 nˆ0 mˆ0 ÿ1
On the left-hand side, we see that there are no terms for which the power of s is not
equal to the power of t. Therefore, terms on the right-hand side with n 6ˆ m must
vanish, giving

1 2
e ÿî H n (î)H m (î)dî ˆ 0, n 6ˆ m (D:13)
ÿ1
The Hermite polynomials H n (î) form an orthogonal set over the range ÿ1 < î < 1
2
n
with a weighting factor e ÿî . If we equate coef®cients of (st) on each side of equation
(D.12), we obtain

1 2
n
2
e ÿî [H n (î)] dî ˆ 2 n!ð 1=2
ÿ1
which may be combined with equation (D.13) to give

1 2
n
e ÿî H n (î)H m (î)dî ˆ 2 n!ð 1=2 ä nm (D:14)
ÿ1
Completeness
If we de®ne the set of functions ö n (î)as
2
n
ð
e
ö n (î) (2 n!) ÿ1=2 ÿ1=4 ÿî =2 H n (î) (D:15)
then equation (D.14) shows that the members of this set are orthonormal with
1
weighting factor unity. We can also demonstrate that this set is complete.
We begin with the integral formula (A.8) which, with suitable de®nitions for the
parameters, may be written as

1 2 2
e ÿ(s =4)‡iîs ds ˆ 2ð 1=2 ÿî (D:16)
e
ÿ1
If we replace e ÿî 2 in equation (D.3) by the integral in (D.16), we obtain for H n (î)
(ÿ1) n î 2 @ n 1 ÿ(s =4)‡iîs (ÿ1) n î 2 1 ÿs =4 @ n iîs
2
2
H n (î) ˆ e e ds ˆ e e e ds
2ð 1=2 @î n 2ð 1=2 @î n
ÿ1 ÿ1
(ÿi) n î 2 1 ÿ(s =4)‡iîs n
2
ˆ e e s ds
2ð 1=2 ÿ1
1 See D. Park (1992) Introduction to the Quantum Theory, 3rd edition (McGraw-Hill, New York), p. 565.