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Multiple choice
A spin-1/2 starts in
∣
0
⟩
|0\rangle
∣0
⟩
and is rotated about the y-axis to the state
cos
(
θ
/
2
)
∣
0
⟩
+
sin
(
θ
/
2
)
∣
1
⟩
\cos(\theta/2)|0\rangle + \sin(\theta/2)|1\rangle
cos
(
θ
/2
)
∣0
⟩
+
sin
(
θ
/2
)
∣1
⟩
with
θ
=
π
/
3
\theta = \pi/3
θ
=
π
/3
. The expectation value
⟨
S
z
⟩
=
(
1
/
2
)
(
p
0
−
p
1
)
=
(
1
/
2
)
cos
(
θ
)
\langle S_z\rangle = (1/2)(p_0 - p_1) = (1/2)\cos(\theta)
⟨
S
z
⟩
=
(
1/2
)
(
p
0
−
p
1
)
=
(
1/2
)
cos
(
θ
)
, where
p
0
=
cos
2
(
θ
/
2
)
p_0=\cos^2(\theta/2)
p
0
=
cos
2
(
θ
/2
)
and
p
1
=
sin
2
(
θ
/
2
)
p_1=\sin^2(\theta/2)
p
1
=
sin
2
(
θ
/2
)
. What is
⟨
S
z
⟩
\langle S_z\rangle
⟨
S
z
⟩
(in units of
ℏ
\hbar
ℏ
)?
0.5
(
=
cos
(
π
/
3
)
)
0.5 (= \cos(\pi/3))
0.5
(
=
cos
(
π
/3
))
0.25
0.75
0.43
(
≈
(
1
/
2
)
sin
(
π
/
3
)
)
0.43 (\approx (1/2)\sin(\pi/3))
0.43
(
≈
(
1/2
)
sin
(
π
/3
))
Check answer