Elements of Modern Physics
128
(n, l = n 0 + 1, j = n 0 + 3/2) → (n 0 + 1, l = n 0 , j = n 0 + 1/2)
(n, l = n 0 + 1, j = n 0 + 1/2) → (n 0 + 1, l = n 0 , j = n 0 ± 1/2)
(4.120)
(n, l = n 0 – 1, j = n 0 – 1/2) → (n 0 + 1, l = n 0 , j = n 0 ± 1/2)
(n, l = n 0 – 1, j = n 0 – 3/2) → (n 0 + 1, l = n 0 , j = n 0 ± 1/2)
(4.121)
where n 0 ≥ 2, and n > n 0 + 1. Hence, the number of fine-structure lines for
transitions from n → n 0 + 1 states is
N = 7 + 6 (n 0 – 2), n → n 0 + 1, n > n 0 + 1 ≥ 3
(4.122)
Thus, each line in the Paschen series consists of 13 lines, each line in the
Brackett series consists of 19 lines, etc.
4. For a transition n → n′, n > n′, the highest frequency corresponds to the
transition
(n, j = 3/2) → (n′, j = 1/2)
(4.123)
and is given by
v = v 0 +
2 2
2
2
'
2
2
| |
|
|
(3 2 )
(3 4 ')
4
4'
α
α
−
−
−
n
n
Z
E
Z
E
n
n
n h
n h
(4.124)
where v 0 is the frequency in the absence of fine structure correction given
by
v 0 =
'
n
n
E E
h
−
(4.125)
Example 5
The spin 1/2 space contains only two linearly independent states α and β as
defined in Eq. (4.78). It is convenient to regard these states with S z = ±
1
2
as
two-component column vectors
α =
1
0
 
 
 
(4.126)
β =
0
1
 
 
 