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Two spin-1/2 qubits are in the state
∣
u
p
,
u
p
⟩
|up,up\rangle
∣
u
p
,
u
p
⟩
. Using
S
2
/
ℏ
2
=
S
1
2
+
S
2
2
+
2
(
S
1
⋅
S
2
)
S^2/\hbar^2 = S_1^2 + S_2^2 + 2(S_1 \cdot S_2)
S
2
/
ℏ
2
=
S
1
2
+
S
2
2
+
2
(
S
1
⋅
S
2
)
, where each spin gives
S
2
=
3
/
4
S^2 = 3/4
S
2
=
3/4
, the
S
1
z
S
2
z
S_{1z} S_{2z}
S
1
z
S
2
z
term gives
(
1
/
2
)
(
1
/
2
)
(1/2)(1/2)
(
1/2
)
(
1/2
)
, and the ladder terms annihilate
∣
u
p
,
u
p
⟩
|up,up\rangle
∣
u
p
,
u
p
⟩
, what is
S
2
S^2
S
2
(in units of
ℏ
2
\hbar^2
ℏ
2
)?
0.75
1
1.5
(
=
3
/
4
+
3
/
4
)
1.5 (= 3/4 + 3/4)
1.5
(
=
3/4
+
3/4
)
2
Check answer