Page 307 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

298 Appendix D

Another recurrence relation may be obtained by differentiating equation (D.1) with
respect to î to obtain
1 dH n s n
X
2
2se 2îsÿs ˆ
dî n!
nˆ0
Replacing the exponential on the left-hand side using equation (D.1) gives
1 n 1 n
X s X dH n s
2s H n (î) ˆ
nˆ0 n! nˆ0 dî n!
n
If we then equate the coef®cients of s , we obtain the desired result
dH n
ˆ 2nH nÿ1 (î) (D:6)
dî
The relations (D.5) and (D.6) may be combined to give a third recurrence relation.
Addition of the two equations gives

d
H n‡1 (î) ˆ 2î ÿ H n (î) (D:7)
dî
With this recurrence relation, a Hermite polynomial may be obtained from the
preceding polynomial. By applying the relation (D.7) to H n (î) k times, we have
k
d
H n‡k (î) ˆ 2î ÿ H n (î) (D:8)
dî

Differential equation
To ®nd the differential equation that is satis®ed by the Hermite polynomials, we ®rst
differentiate the second recurrence relation (D.6) and then substitute (D.6) with n
replaced by n ÿ 1 to eliminate the ®rst derivative of H nÿ1 (î)
2
d H n dH nÿ1
ˆ 2n ˆ 4n(n ÿ 1)H nÿ2 (î) (D:9)
dî 2 dî
Replacing n by n ÿ 1 in the ®rst recurrence relation (D.5), we have
H n (î) ÿ 2îH nÿ1 (î) ‡ 2(n ÿ 1)H nÿ2 (î) ˆ 0
which may be used to eliminate H nÿ2 (î) in equation (D.9), giving
2
d H n
‡ 2nH n (î) ÿ 4nîH nÿ1 (î) ˆ 0
dî 2
Application of equation (D.6) again to eliminate H nÿ1 (î) yields
2
d H n ÿ 2î dH n ‡ 2nH n (î) ˆ 0 (D:10)
dî 2 dî
which is the Hermite differential equation.

Integral relations
To obtain the orthogonality and normalization relations for the Hermite polynomials,
we multiply together the generating functions g(î, s) and g(î, t), both obtained from
equation (D.1), and the factor e ÿî 2 and then integrate over î

n m
1
1
1 2 X X s t 1 2
I e ÿî g(î, s)g(î, t)dî ˆ e ÿî H n (î)H m (î)dî (D:11)
n!m!
ÿ1 nˆ0 mˆ0 ÿ1