Mathematical Methods for Physicists: A concise introduction - Site Map

HERMITE'S EQUATION
but H 00
n …x†2xH 0
n…x† ˆ2nH n …x†, so we can rewrite the last equation as
which gives
D 2 F n …x† ˆe x2 =2 ‰2nH 0
n…x†Š ‡ x 2 F n …x†F n …x†
ˆ2nF n …x†‡x 2 F n …x†F n …x†;
D 2 F n …x†x 2 F n …x†‡…2n ‡ 1†F n …x† ˆ0:
…7:49†
We can now show that the set fF n …x†g is orthogonal in the in®nite range
1 < x < 1. Multiplying Eq. (7.49) by F m …x† we have
F m …x†D 2 F n …x†x 2 F n …x†F m …x†‡…2n ‡ 1†F n …x†F m …x† ˆ0:
Interchanging m and n gives
F n …x†D 2 F m …x†x 2 F m …x†F n …x†‡…2m ‡ 1†F m …x†F n …x† ˆ0:
Subtracting the last two equations from the previous one and then integrating
from 1 to ‡1, wehave
Z 1
Z
1 1
I n;m ˆ F n …x†F m …x†dx ˆ
…Fn 00 F
1
2…n m†
m FmF 00
n †dx:
1
The integration by parts gives
Z
2…n m†I n;m ˆ FnF 0
m FmF 0 1
1
n 1 …FnF 0
m 0 FmF 0 n†dx:
0
1
Since the right hand side vanishes at both limits and if m 6ˆ m, we have
I n;m ˆ
Z 1
1
F n …x†F m …x†dx ˆ 0:
…7:50†
When n ˆ m we can proceed as follows
I n;n ˆ
Z 1
1
e x2 H n …x†H n …x†dx ˆ
Z 1
1
e x2 D n …e x2 †D m …e x2 †dx:
R R
Integration by parts, that is, udv ˆ uv vdu with u ˆ e
x 2 D n …e x2 †
and v ˆ D n1 …e x2 †, gives
I n;n ˆ
Z 1
1
‰2xe x2 D n …e x2 †‡e x2 D n‡1 …e x2 †ŠD n1 …e x2 †dx:
By using Eq. (7.43) which is true for y ˆ…1† n D n q ˆ…1† n D n …e x2 † we obtain
I n;n ˆ
Z 1
2ne x2 D n1 …e x2 †D n1 …e x2 †dx ˆ 2nI n1;n1 :
1
315