260
H Solutions to Problems
of the spin components, products of noncommuting operators must be fully
symmetrized. For the f z 3 function, this is the case for the functions 3zx 2 and
3xy 2 , which are parts of 3zr 3 . As an example, the operator analogue of 3zx 2
reads
3zx
2 → ˜
S z ˜
S x ˜
S x + ˜
S x ˜
S z ˜
S x + ˜
S x ˜
S x ˜
S z
One then has for the operator equivalent of 3z(x 2 + y 2 ):
˜
S z ˜
S x ˜
S x + ˜
S x ˜
S z ˜
S x + ˜
S x ˜
S x ˜
S z + ˜
S z ˜
S y ˜
S y + ˜
S y ˜
S z ˜
S y + ˜
S y ˜
S y ˜
S z
= 3 ˜
S z
˜
S
2
x + ˜
S
2
y
+ i( ˜
S x ˜
S y − ˜
S y ˜
S x ) = 3 ˜
S z
˜
S
2
x + ˜
S
2
y
−
2 ˜
S z
where we have used the commutation relation for the spin-operators:
S x S y − S y S x = iS z
The octupolar spin operator will then be of type
H f =
μ B
3 g f B z
˜
S
3
z −
3
5
˜
S z ˜
S
2 +
1
5
2 ˜
S z
+ B x
˜
S
3
x −
3
5
˜
S x ˜
S
2 +
1
5
2 ˜
S x
+ B y
˜
S
3
y −
3
5
˜
S y ˜
S
2 +
1
5
2 ˜
S y
In order to identify the parameter correspondence, let us work out the action
of this operator on the quartet functions. As an example for a magnetic field
along the z-direction, the matrix is diagonal, and its elements (in units of μ B )
are given by
±
3
2
H f
±
3
2
=±g f B z
3
2
9
4
−
45
20
+
1
5
=±
3
10
g f B z
±
1
2
H f
±
1
2
=∓
9
10
g f B z
By comparing these elements to the results in Table 7.8 we can identify the
parameter correspondence as
J f =−
3
10
g f
(13)