258
H Solutions to Problems
Multiplication table for the double group D ∗
3
D ∗
3
ˆ
E
ˆ
C 3
ˆ
C 2
3
ℵ ˆ
C 3 ℵ ˆ
C 2
3 ℵ
ˆ
C A
2
ˆ
C B
2
ˆ
C C
2
ℵ ˆ
C A
2 ℵ ˆ
C B
2 ℵ ˆ
C C
2
ˆ
E
ˆ
E
ˆ
C 3
ˆ
C 2
3
ℵ ˆ
C 3 ℵ ˆ
C 2
3 ℵ
ˆ
C A
2
ˆ
C B
2
ˆ
C C
2
ℵ ˆ
C A
2 ℵ ˆ
C B
2 ℵ ˆ
C C
2
ˆ
C 3
ˆ
C 3
ˆ
C 2
3
ℵℵ ˆ
C 2
3
ˆ
E
ℵ ˆ
C 3 ˆ
C C
2
ˆ
C A
2
ℵ ˆ
C B
2 ℵ ˆ
C C
2 ℵ ˆ
C A
2
ˆ
C B
2
ˆ
C 2
3
ˆ
C 2
3
ℵℵ ˆ
C 3 ˆ
E
ˆ
C 3
ℵ ˆ
C 2
3 ℵ ˆ
C B
2
ˆ
C C
2
ℵ ˆ
C A
2
ˆ
C B
2
ℵ ˆ
C C
2
ˆ
C A
2
ℵ ˆ
C 3 ℵ ˆ
C 3 ℵ ˆ
C 2
3
ˆ
E
ˆ
C 2
3
ℵ
ˆ
C 3
ℵ ˆ
C C
2 ℵ ˆ
C A
2
ˆ
C B
2
ˆ
C C
2
ˆ
C A
2
ℵ ˆ
C B
2
ℵ ˆ
C 2
3 ℵ ˆ
C 2
3
ˆ
E
ˆ
C 3
ℵℵ ˆ
C 3 ˆ
C 2
3
ˆ
C B
2
ℵ ˆ
C C
2
ˆ
C A
2
ℵ ˆ
C B
2
ˆ
C C
2
ℵ ˆ
C A
2
ℵℵℵ ˆ
C 3 ℵ ˆ
C 2
3
ˆ
C 3
ˆ
C 2
3
ˆ
E
ℵ ˆ
C A
2 ℵ ˆ
C B
2 ℵ ˆ
C C
2
ˆ
C A
2
ˆ
C B
2
ˆ
C C
2
ˆ
C A
2
ˆ
C A
2
ˆ
C B
2
ℵ ˆ
C C
2 ℵ ˆ
C B
2
ˆ
C C
2
ℵ ˆ
C A
2 ℵℵ ˆ
C 3 ˆ
C 2
3
ˆ
E
ˆ
C 3
ℵ ˆ
C 2
3
ˆ
C B
2
ˆ
C B
2
ℵ ˆ
C C
2 ℵ ˆ
C A
2
ˆ
C C
2
ˆ
C A
2
ℵ ˆ
C B
2
ˆ
C 2
3
ℵ
ˆ
C 3
ℵ ˆ
C 2
3
ˆ
E
ℵ ˆ
C 3
ˆ
C C
2
ˆ
C C
2
ˆ
C A
2
ˆ
C B
2
ℵ ˆ
C A
2 ℵ ˆ
C B
2 ℵ ˆ
C C
2 ℵ ˆ
C 3 ℵ ˆ
C 2
3 ℵ
ˆ
C 3
ˆ
C 2
3
ˆ
E
ℵ ˆ
C A
2 ℵ ˆ
C A
2 ℵ ˆ
C B
2
ˆ
C C
2
ˆ
C B
2
ℵ ˆ
C C
2
ˆ
C A
2
ˆ
E
ˆ
C 3
ℵ ˆ
C 2
3 ℵℵ ˆ
C 3 ˆ
C 2
3
ℵ ˆ
C B
2 ℵ ˆ
C B
2
ˆ
C C
2
ˆ
C A
2
ℵ ˆ
C C
2 ℵ ˆ
C A
2
ˆ
C B
2
ℵ ˆ
C 2
3
ˆ
E
ℵ ˆ
C 3 ˆ
C 2
3
ℵ
ˆ
C 3
ℵ ˆ
C C
2 ℵ ˆ
C C
2 ℵ ˆ
C A
2 ℵ ˆ
C B
2
ˆ
C A
2
ˆ
C B
2
ˆ
C C
2
ˆ
C 3
ˆ
C 2
3
ˆ
E
ℵ ˆ
C 3 ℵ ˆ
C 2
3 ℵ
7.4 The action of the spin operators on the components of a spin-triplet can
be found by acting on the coupled states, as summarized in Table 7.2.A s
an example, where we have added the electron labels 1 and 2 for clarity:
S x |+1=S x
|α 1 |α 2
=
S x |α 1
|α 2 +|α 1
S x |α 2
=
2
|β 1 |α 2 +|α 1 |β 2
=
√
2
|0
S y |−1=−
i
√
2
|0
These results can be generalized as follows:
S z |M S =M S |M S
(S x ± iS y )|M S =
(S ∓ M S )(S ± M s + 1)
1
2 |M s ± 1
The action of the spin Hamiltonian in the fictitious spin basis gives then rise to
the following Hamiltonian matrix (in units of μ B ):
H Ze
|0| + 1| − 1
0|
0
g ⊥
1
√
2
(B x + iB y )g ⊥
1
√
2
(B x − iB y )
+1|
g ⊥
1
√
2
(B x − iB y )g || B z
0
−1|
g ⊥
1
√
2
(B x + iB y )
0
−g || B z
We can now identify these expressions with the actual matrix elements in
the basis of the three D 3 components, keeping in mind the relationship be-
H Solutions to Problems
Multiplication table for the double group D ∗
3
D ∗
3
ˆ
E
ˆ
C 3
ˆ
C 2
3
ℵ ˆ
C 3 ℵ ˆ
C 2
3 ℵ
ˆ
C A
2
ˆ
C B
2
ˆ
C C
2
ℵ ˆ
C A
2 ℵ ˆ
C B
2 ℵ ˆ
C C
2
ˆ
E
ˆ
E
ˆ
C 3
ˆ
C 2
3
ℵ ˆ
C 3 ℵ ˆ
C 2
3 ℵ
ˆ
C A
2
ˆ
C B
2
ˆ
C C
2
ℵ ˆ
C A
2 ℵ ˆ
C B
2 ℵ ˆ
C C
2
ˆ
C 3
ˆ
C 3
ˆ
C 2
3
ℵℵ ˆ
C 2
3
ˆ
E
ℵ ˆ
C 3 ˆ
C C
2
ˆ
C A
2
ℵ ˆ
C B
2 ℵ ˆ
C C
2 ℵ ˆ
C A
2
ˆ
C B
2
ˆ
C 2
3
ˆ
C 2
3
ℵℵ ˆ
C 3 ˆ
E
ˆ
C 3
ℵ ˆ
C 2
3 ℵ ˆ
C B
2
ˆ
C C
2
ℵ ˆ
C A
2
ˆ
C B
2
ℵ ˆ
C C
2
ˆ
C A
2
ℵ ˆ
C 3 ℵ ˆ
C 3 ℵ ˆ
C 2
3
ˆ
E
ˆ
C 2
3
ℵ
ˆ
C 3
ℵ ˆ
C C
2 ℵ ˆ
C A
2
ˆ
C B
2
ˆ
C C
2
ˆ
C A
2
ℵ ˆ
C B
2
ℵ ˆ
C 2
3 ℵ ˆ
C 2
3
ˆ
E
ˆ
C 3
ℵℵ ˆ
C 3 ˆ
C 2
3
ˆ
C B
2
ℵ ˆ
C C
2
ˆ
C A
2
ℵ ˆ
C B
2
ˆ
C C
2
ℵ ˆ
C A
2
ℵℵℵ ˆ
C 3 ℵ ˆ
C 2
3
ˆ
C 3
ˆ
C 2
3
ˆ
E
ℵ ˆ
C A
2 ℵ ˆ
C B
2 ℵ ˆ
C C
2
ˆ
C A
2
ˆ
C B
2
ˆ
C C
2
ˆ
C A
2
ˆ
C A
2
ˆ
C B
2
ℵ ˆ
C C
2 ℵ ˆ
C B
2
ˆ
C C
2
ℵ ˆ
C A
2 ℵℵ ˆ
C 3 ˆ
C 2
3
ˆ
E
ˆ
C 3
ℵ ˆ
C 2
3
ˆ
C B
2
ˆ
C B
2
ℵ ˆ
C C
2 ℵ ˆ
C A
2
ˆ
C C
2
ˆ
C A
2
ℵ ˆ
C B
2
ˆ
C 2
3
ℵ
ˆ
C 3
ℵ ˆ
C 2
3
ˆ
E
ℵ ˆ
C 3
ˆ
C C
2
ˆ
C C
2
ˆ
C A
2
ˆ
C B
2
ℵ ˆ
C A
2 ℵ ˆ
C B
2 ℵ ˆ
C C
2 ℵ ˆ
C 3 ℵ ˆ
C 2
3 ℵ
ˆ
C 3
ˆ
C 2
3
ˆ
E
ℵ ˆ
C A
2 ℵ ˆ
C A
2 ℵ ˆ
C B
2
ˆ
C C
2
ˆ
C B
2
ℵ ˆ
C C
2
ˆ
C A
2
ˆ
E
ˆ
C 3
ℵ ˆ
C 2
3 ℵℵ ˆ
C 3 ˆ
C 2
3
ℵ ˆ
C B
2 ℵ ˆ
C B
2
ˆ
C C
2
ˆ
C A
2
ℵ ˆ
C C
2 ℵ ˆ
C A
2
ˆ
C B
2
ℵ ˆ
C 2
3
ˆ
E
ℵ ˆ
C 3 ˆ
C 2
3
ℵ
ˆ
C 3
ℵ ˆ
C C
2 ℵ ˆ
C C
2 ℵ ˆ
C A
2 ℵ ˆ
C B
2
ˆ
C A
2
ˆ
C B
2
ˆ
C C
2
ˆ
C 3
ˆ
C 2
3
ˆ
E
ℵ ˆ
C 3 ℵ ˆ
C 2
3 ℵ
7.4 The action of the spin operators on the components of a spin-triplet can
be found by acting on the coupled states, as summarized in Table 7.2.A s
an example, where we have added the electron labels 1 and 2 for clarity:
S x |+1=S x
|α 1 |α 2
=
S x |α 1
|α 2 +|α 1
S x |α 2
=
2
|β 1 |α 2 +|α 1 |β 2
=
√
2
|0
S y |−1=−
i
√
2
|0
These results can be generalized as follows:
S z |M S =M S |M S
(S x ± iS y )|M S =
(S ∓ M S )(S ± M s + 1)
1
2 |M s ± 1
The action of the spin Hamiltonian in the fictitious spin basis gives then rise to
the following Hamiltonian matrix (in units of μ B ):
H Ze
|0| + 1| − 1
0|
0
g ⊥
1
√
2
(B x + iB y )g ⊥
1
√
2
(B x − iB y )
+1|
g ⊥
1
√
2
(B x − iB y )g || B z
0
−1|
g ⊥
1
√
2
(B x + iB y )
0
−g || B z
We can now identify these expressions with the actual matrix elements in
the basis of the three D 3 components, keeping in mind the relationship be-