302
11 Spintronics
σ y =
1
2
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0 −j
√
5 0
0
0
0
j
√
5 0 −j
√
8 0
0
0
0 j
√
8
0 −j 3 0
0
0
0
j 3
0 −j
√
8 0
0
0
0 j
√
8 0 −j
√
5
0
0
0
0 j
√
5
0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
σ z =
1
2
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
5 0 0 0 0 0
0 3 0 0 0 0
0 0 1 0 0 0
0 0 0 −1 0 0
0 0 0 0 −3 0
0 0 0 0 0 −5
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
(11.34)
The direct-product spin-matrices for the two-ion system are given by S (1) =
σ ⊗ I 6 , S (2) = I 6 ⊗ σ , where I 6 is the six-dimensional identity matrix. Spelled out,
these are:
S
(1)
x =
1
2
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0 6
√
5I 6 0 6
0 6
0 6
0 6
√
5I 6 0 6
√
8I 6 0 6
0 6
0 6
0 6
√
8I 6 0 6
3
0 6
0 6
0 6
0 6
3
0 6
√
8I 6 0 6
0 6
0 6
0 6
√
8I 6 0 6
√
5I 6
0 6
0 6
0 6
0 6
√
5I 6 0 6
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
S
(1)
y =
1
2
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0 6 −j
√
5I 6
0 6
0 6
0 6
0 6
j
√
5I 6
0 6
−j
√
8I 6 0 6
0 6
0 6
0 6
j
√
8I 6
0 6
−j 3I 6
0 6
0 6
0 6
0 6
j 3I 6
0 6 −j
√
8I 6
0 6
0 6
0 6
0 6
j
√
8I 6
0 6
−j
√
5I 6
0 6
0 6
0 6
0 6
j
√
5I 6
0 6
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
S
(1)
z =
1
2
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
5I 6 0 6 0 6 0 6
0 6
0 6
0 6 3I 6 0 6 0 6
0 6
0 6
0 6 0 6 1I 6 0 6
0 6
0 6
0 6 0 6 0 6 −1I 6 0 6
0 6
0 6 0 6 0 6 0 6 −3I 6 0 6
0 6 0 6 0 6 0 6
0 6 −5I 6
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦