Biomedical Engineering Reference
In-Depth Information
and has dimension three. It is instructive to calculate explicitly the
T
2
,
2
1
1
0
matrix
necessary for the calculation of the triplet blocking states and the associated
blocking states. The states in the 1
0
doublet and in the two times orbitally degenerate
triplet 2
1
are labeled and ordered as follows:
⎧
⎨
|
2
1
,
=+
h
,
S
z
=+
h
|
2
1
,
=+
h
,
S
z
=
0
1
0
|
1
0
,
=
0
,↑
|
2
1
,
=+
h
,
S
z
=
−
h
,↓
,
2
1
.
(7.45)
⎩
|
1
0
,
=
0
|
2
1
,
=
−
h
,
S
z
=+
h
|
2
1
,
=
−
h
,
S
z
=
0
|
2
1
,
=
−
h
,
S
z
=
−
h
γ
ασ
matrices that compose
T
2
,
2
1
1
0
The elements of the
have thus the general form:
γ
ασ
(
S
z
,
)=
,
=
S
z
|
S
z
,
1
0
0
,
d
ασ
|
2
1
,,
S
z
.
(7.46)
By orbital and spin symmetry arguments it is possible to show that
δ
S
z
,
S
z
−
σ
(
√
2
te
γ
ασ
(
S
z
,
)=
h
φ
α
S
z
,
δ
S
z
,↑
.
+
δ
S
z
,↓
)
(7.47)
1
0
,
=
where
t
=
0
,↓|
d
M
↑
|
2
1
,
=
1
,
S
z
=
0
.
The subscript
M
labels a reference
dot and
φ
α
is the angle of the rotation that brings the dot
α
on the dot
M
. The explicit
form of
T
2
1
,
2
1
1
0
reads:
⎛
√
2
e
−
i
2π
/
3
√
2
e
+
i
2π
/
3
⎞
0
0
0
0
⎝
⎠
e
−
i
2π
/
3
e
+
i
2π
/
3
0
0
0
0
e
−
i
2π
/
3
e
+
i
2π
/
3
0
0
0
0
√
2
e
−
i
2π
/
3
√
2
e
+
i
2π
/
3
0
0
0
0
T
2
,
2
1
1
0
=
√
2
e
+
i
2π
/
3
√
2
e
−
i
2π
/
3
t
.
0
0
0
0
e
+
i
2π
/
3
e
−
i
2π
/
3
0
0
0
0
e
+
i
2π
/
3
e
−
i
2π
/
3
0
0
0
0
√
2
e
+
i
2π
/
3
√
2
e
−
i
2π
/
3
0
0
0
0
(7.48)
The rank of this matrix is six since all columns are independent. Thus
C
2
,
2
1
coincides with the full Hilbert space of the first excited two-electron energy level.
The blocking space
B
2
,
2
1
,
1
0
reads:
T
2
,
2
1
1
0
,
B
2
,
2
1
,
1
0
=
P
2
1
ker
(
T
S
)
(7.49)
