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Multiple choice
A spin-1/2 particle is in state
∣
ψ
⟩
=
cos
(
π
/
6
)
∣
↑
z
⟩
+
sin
(
π
/
6
)
∣
↓
z
⟩
|\psi\rangle = \cos(\pi/6)|{\uparrow_z}\rangle + \sin(\pi/6)|{\downarrow_z}\rangle
∣
ψ
⟩
=
cos
(
π
/6
)
∣
↑
z
⟩
+
sin
(
π
/6
)
∣
↓
z
⟩
, i.e.
α
=
3
/
2
\alpha = \sqrt{3}/2
α
=
3
/2
and
β
=
1
/
2
\beta = 1/2
β
=
1/2
. The expectation value of
S
z
S_z
S
z
(in units of
ℏ
/
2
\hbar/2
ℏ/2
) is
⟨
σ
z
⟩
=
∣
α
∣
2
−
∣
β
∣
2
\langle\sigma_z\rangle = |\alpha|^2 - |\beta|^2
⟨
σ
z
⟩
=
∣
α
∣
2
−
∣
β
∣
2
. What is its value?
🔬 try it before you answer
0.5
0.866
0.25
1.0
Check answer