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Multiple choice
By Postulate 3, measuring observable
A
=
∑
k
a
k
P
k
A = \sum_k a_k P_k
A
=
∑
k
a
k
P
k
on a normalized state
∣
ψ
⟩
|\psi\rangle
∣
ψ
⟩
yields outcome
a
k
a_k
a
k
with what probability?
p
(
a
k
)
=
⟨
ψ
∣
P
k
∣
ψ
⟩
p(a_k) = \langle\psi| P_k |\psi\rangle
p
(
a
k
)
=
⟨
ψ
∣
P
k
∣
ψ
⟩
p
(
a
k
)
=
⟨
ψ
∣
P
k
∣
ψ
⟩
/
a
k
p(a_k) = \langle\psi| P_k |\psi\rangle / a_k
p
(
a
k
)
=
⟨
ψ
∣
P
k
∣
ψ
⟩
/
a
k
p
(
a
k
)
=
⟨
ψ
∣
P
k
∣
ψ
⟩
p(a_k) = \sqrt{\langle\psi| P_k |\psi\rangle}
p
(
a
k
)
=
⟨
ψ
∣
P
k
∣
ψ
⟩
p
(
a
k
)
=
a
k
⟨
ψ
∣
ψ
⟩
p(a_k) = a_k \langle\psi|\psi\rangle
p
(
a
k
)
=
a
k
⟨
ψ
∣
ψ
⟩
Check answer