The conventional symbol used for f( ) is
 (), is Greek symbol for capital theta, and we will stick to this convention as it will help you when you refer to other books. Similarly  denotes a function of  , ( ). |
Seperating the variables,  ( ,  ) = ( )  (). |
( L 2 / 2I )  = E  |
| Substituting |
^ 2  = -2IE/  2 |
(3.15) |
| = [(1/sin 2 )  2 /   2 + (1/sin )  /  sin  /  ] 
|
| |
= (1/sin 2 ) (  2 /  2) +  (1/sin  )  /   sin  /  |
(3.16) |
Dividing by Q  , multiplying by sin2 and rearranging, |
| |
1/   2 /  2 = -1/ sin   /  sin  /   - 2IE/  2 sin 2 |
(3.17) |
we see that the left hand side depends only on  and the right hand side depends only on  . Since  and  are independent, each side must be equal to the same constant. Calling this constant – ml2 we get |
| |
 2 /   2 = –ml2 |
(3.18) |
setting x = cos, sinbecomes  = (1-x 2 ) 1/2 |
Setting 2IE /  2 = l ( l +1 ) |
| And by using (3.18), (3.17) becomes |
| |
(1-x 2 ) d 2 /d 2– 2xd /d +{ l ( l +1) – ml 2 / (1- x 2 ) } = 0 |
(3.19) |
This is called the associated Legendre equation. The solutions for this equation are “well behaved” (ie, single valued, differentiable and finite) for the following conditions |
| a) l = 0,1,2,…… |
| |
b) 1  m l  l |
(3.20) |
This implies that only when l is a positive integer and the absolute value of ml is less than or equal to the value of l , acceptable or well behaved solutions exist. The normalized solutions for equation (3.15) for l = 0, 1 and 2 are given in the Table 3.1. Substitute the first three solutions in eq (3.15) and verify that these satisfy the equation. |
| |
Table 3.1 A few normalized solutions to equation 3.15. The literature symbol for these solutions is Y l m (  ,  ) , and for simplicity, we take m = ml . |
| |
l |
m = m l |
Y lm (  ,  ) |
0 |
0 |
(4  ) -1/2 |
1 |
0 |
(3/4  ) 1/2 cos  |
1 |
 1
|
 ( 3/8  ) 1/2 sin  e  i 
|
2 |
0 |
(5/16  ) 1/2 (3 cos 2  - 1) |
2 |
 1
|
(15/16  ) 1/2 cos  sin  e i |
2 |
 2
|
(15 / 32  ) 1/2 sin 2 e 2 i |
|
| |
