FOUNDATIONS OF QUANTUM MECHANICS

74 CHAPTER III. THE POSTULATES
III. 6. 3. 3
CONDITIONAL PROBABILITIES
In chapter VII we will also need to know, again in case the particles are in the singlet state, the
probability for the spin of particle 2 to be found in the direction ⃗ b, given that the spin of particle 1 was
found in the direction ⃗a. This conditional probability is, by definition,
Prob ( ⃗ b · ⃗σ2 = 1 ∣ ⃗a · ⃗σ1 = 1 ) = Prob ( ⃗ b · ⃗σ2 = 1 ∧ ⃗a · ⃗σ 1 = 1 )
Prob ( ) . (III. 172)
⃗a · ⃗σ 1 = 1
Here the joint probability is
Prob ( ⃗ b · ⃗σ2 = 1 ∧ ⃗a · ⃗σ 1 = 1 ) = | ( ⟨⃗a ↑| ⊗ ⟨ ⃗ b ↑| ) |0, 0⟩| 2 , (III. 173)
with |⃗a ↑⟩ ⊗ | ⃗ b ↑⟩ the direct product of the eigenstates of ⃗a · ⃗σ 1 and ⃗ b · ⃗σ 2 having eigenvalues +1.
Again choosing ⃗a and ⃗ b as in diagram III. 3, |⃗a ↑⟩ = |z ↑⟩ and | ⃗ b ↑⟩ equal to |⃗n, +⟩, (III. 138), we find
for the direct product
|⃗a ↑⟩ ⊗ | ⃗ b ↑⟩ = |z ↑⟩ ⊗ ( cos 1 2 θ ⃗a, ⃗ b |z ↑⟩ + sin 1 2 θ ⃗a, ⃗ b |z ↓⟩) . (III. 174)
Therefore, with (III. 165),
( ) √
⟨⃗a ↑| ⊗ ⟨ ⃗ b ↑| |0, 0⟩ =
1
2 2 sin
1
2 θ ⃗a, ⃗ , (III. 175)
b
and we see that the joint probability is
Prob ( ⃗ b · ⃗σ2 = 1 ∧ ⃗a · ⃗σ 1 = 1 ) = 1 2 sin2 1 2 θ ⃗a, ⃗ . (III. 176)
b
Likewise, again using (III. 173) with ⟨ ⃗ b ↓| equal to |⃗n, −⟩, (III. 139), we have
Prob ( ⃗ b · ⃗σ2 = − 1 ∧ ⃗a · ⃗σ 1 = 1 ) = 1 2 cos2 1 2 θ ⃗a, ⃗ . (III. 177)
b
This yields for the marginal probability
Prob ( ⃗a · ⃗σ 1 = 1 ) = Prob ( ⃗ b · ⃗σ2 = 1 ∧ ⃗a · ⃗σ 1 = 1 )
and we see that the conditional probability (III. 172) is
+ Prob ( ⃗ b · ⃗σ2 = − 1 ∧ ⃗a · ⃗σ 1 = 1 )
= 1 2 sin2 1 2 θ ⃗a, ⃗ b + 1 2 cos2 1 2 θ ⃗a, ⃗ b = 1 2
, (III. 178)
Prob ( ⃗ b · ⃗σ2 = 1 ∣ ⃗a · ⃗σ1 = 1 ) = sin 2 1 2 θ ⃗a, ⃗ . (III. 179)
b
◃ Remark
By definition there is no correlation between the two results of measurements of spin if
Prob ( ⃗ b · ⃗σ2 = 1 ∣ ∣ ⃗a · ⃗σ1 = 1 ) = Prob ( ⃗ b · ⃗σ2 = 1 ) , (III. 180)
which is the case if ⃗a and ⃗ b are perpendicular. ▹