Elements of Modern Physics
168
j 1 = l 1 + 1/2, j 2 = l 2 + 1/2 : J = l 1 + l 2 , l 1 + l 2 – 2, ..., 0
1
2
2
1
1
2 1
2
1
1
2
2
1/ 2,
1/ 2 :
,
1, ...,1
1/ 2,
1/ 2
= +
= −
= +
+ −
= −
= +
l
j l
j
J l l l l
j l
j l
j 1 = l 1 – 1/2, j 2 = l 2 – 1/2 : J = l 1 + l 2 – 2, l 1 + l 2 – 4, ..., 0
(5.76)
It is again observed that each J is repeated the same number of times in the
LS coupling scheme and n the j-j coupling scheme.
Example 3
The spin-orbit interaction splits the levels of the LS couplng scheme into multiplets.
The multiplet structure of the first few observed lines in mercury is as follows:
The triplet levels are split into (6s) (np)
3
P 2,1,0, (6s) (nd)
3
D 3 ,3,2,1, etc.
wereas (6s) (ns)
3
S 1 has only one level. The allowed transitions are:
(6s) (6p)
1
P 1 → (6s) (6s)
1
S 0 , λ = 1849.6 Å
(6s) (6p)
3
P 1 → (6s) (6s)
1
S 0 , λ = 2536.5 Å
(6s) (7s)
1
S 0 → (6s) (6p)
1
P 1 , λ = 10,139.7 Å
(5.77)
(6s) (7s)
3
S 1 → (6s) (6p)
3
P 0 , λ = 4046.6 Å
(6s) (7s)
3
S 1 → (6s) (6p)
3
P 1 , λ = 4358.4 Å
(6s) (7s)
3
S 1 → (6s) (6p)
3
P 2 , λ = 5460.7 Å
the other transitions between these multiplets being forbidden by the selection
rules, e.g. (6s) (6p)
3
P 0 → (6s) (6s)
1
S 0 is not allowed.
Example 4
The spin-orbit interaction breaks the degeneracy of a given LS level into levels
with different J values. It may be observed that the average of the L.S interaction,
summed over all the states of a given LS level, is zero, i.e.
Σ Σ
⋅
S
L
M M
L S = 0
(5.78)
(this follows from the fact that with a given orientation of S, for every term with
a given L, there is another term with –L). Now, the summation over the states
can equally well be carried over M J and J, which implies that
J
J M
Σ Σ L.S = 0
(5.79)
For a given J, the expectation value is the same for all M J values, so that this
relation is equivalent to
168
j 1 = l 1 + 1/2, j 2 = l 2 + 1/2 : J = l 1 + l 2 , l 1 + l 2 – 2, ..., 0
1
2
2
1
1
2 1
2
1
1
2
2
1/ 2,
1/ 2 :
,
1, ...,1
1/ 2,
1/ 2
= +
= −
= +
+ −
= −
= +
l
j l
j
J l l l l
j l
j l
j 1 = l 1 – 1/2, j 2 = l 2 – 1/2 : J = l 1 + l 2 – 2, l 1 + l 2 – 4, ..., 0
(5.76)
It is again observed that each J is repeated the same number of times in the
LS coupling scheme and n the j-j coupling scheme.
Example 3
The spin-orbit interaction splits the levels of the LS couplng scheme into multiplets.
The multiplet structure of the first few observed lines in mercury is as follows:
The triplet levels are split into (6s) (np)
3
P 2,1,0, (6s) (nd)
3
D 3 ,3,2,1, etc.
wereas (6s) (ns)
3
S 1 has only one level. The allowed transitions are:
(6s) (6p)
1
P 1 → (6s) (6s)
1
S 0 , λ = 1849.6 Å
(6s) (6p)
3
P 1 → (6s) (6s)
1
S 0 , λ = 2536.5 Å
(6s) (7s)
1
S 0 → (6s) (6p)
1
P 1 , λ = 10,139.7 Å
(5.77)
(6s) (7s)
3
S 1 → (6s) (6p)
3
P 0 , λ = 4046.6 Å
(6s) (7s)
3
S 1 → (6s) (6p)
3
P 1 , λ = 4358.4 Å
(6s) (7s)
3
S 1 → (6s) (6p)
3
P 2 , λ = 5460.7 Å
the other transitions between these multiplets being forbidden by the selection
rules, e.g. (6s) (6p)
3
P 0 → (6s) (6s)
1
S 0 is not allowed.
Example 4
The spin-orbit interaction breaks the degeneracy of a given LS level into levels
with different J values. It may be observed that the average of the L.S interaction,
summed over all the states of a given LS level, is zero, i.e.
Σ Σ
⋅
S
L
M M
L S = 0
(5.78)
(this follows from the fact that with a given orientation of S, for every term with
a given L, there is another term with –L). Now, the summation over the states
can equally well be carried over M J and J, which implies that
J
J M
Σ Σ L.S = 0
(5.79)
For a given J, the expectation value is the same for all M J values, so that this
relation is equivalent to