Mathematical Methods for Physicists: A concise introduction - Site Map

SPECIAL FUNCTIONS OF MATHEMATICAL PHYSICS
Now H n …x† satis®es Hermite's equation
Hn 00 …x†2xHn…x†‡2nH 0
n …x† ˆ0:
Eliminating Hn 00 …x† from the last two equations, we obtain
which reduces to
2xH 0
n…x†2nH n …x† ˆ2H n …x†‡2xH 0
n…x†H 0
n‡1…x†
H 0
n‡1…x† ˆ2…n ‡ 1†H n …x†:
Replacing n by n ‡ 1 in Eq. (7.44), we have
H 0
n‡1…x† ˆ2xH n‡1 …x†H n‡2 …x†:
Combining this with Eq. (7.45) we obtain
H n‡2 …x† ˆ2xH n‡1 …x†2…n ‡ 1†H n …x†:
This will quickly give the higher polynomials.
…7:45†
…7:46†
Generating function for the H n …x†
By using Rodrigues' formula we can also ®nd a generating formula for the H n …x†.
This is
…x; t† ˆe 2txt2 ˆ e fx2 …tx† 2g ˆ X1
nˆ0
Di€erentiating Eq. (7.47) n times with respect to t we get
e x2
@n
@t n ˆ e e…tx†2 x2 …1† n @ n
@x n ˆ X1
e…tx†2
H n …x†
t n :
n!
kˆ0
H n‡k …x† tk
k! :
Put t ˆ 0 in the last equation and we obtain Rodrigues' formula
H n …x† ˆ…1† n e x2
dn
dx n …ex2 †:
…7:47†
These are de®ned by
The orthogonal Hermite functions
from which we have
F n …x† ˆe x2 =2 H n …x†;
DF n …x† ˆxF n …x†‡e x2 =2 H 0
n…x†;
…7:48†
D 2 F n …x† ˆe x2 =2 H 00
n …x†2xe x2 =2 H 0
n…x†‡x 2 e x2 =2 H n …x†F n …x†
ˆ e x2 =2 ‰H 00
n …x†2xH 0
n…x†Š ‡ x 2 F n …x†F n …x†;
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