FOUNDATIONS OF QUANTUM MECHANICS

Likewise we calculate the following probabilities
∑ ∑
and
σ 1 ,σ 3
∑ ∑
σ 2 ,σ 3
VII. 3. WIGNER’S DERIVATION 149
τ 1 ,τ 2
p (σ 1 , +, σ 3 , τ 1 , τ 2 , +) = 1 2 sin2 1 2 θ 23 (VII. 40)
= p (+, +, −, −, −, +) + p (−, +, −, +, −, +)
τ , τ 3
p (+, σ 2 , σ 3 , τ 1 , +, τ 3 ) = 1 2 sin2 1 2 θ 12 (VII. 41)
From (VII. 40) and (VII. 41) it follows that
= p (+, −, +, −, +, −) + p (+, −, −, −, +, +).
p (+, +, −, −, −, +) 1 2 sin2 1 2 θ 23 and (VII. 42)
p (+, −, −, −, +, +) 1 2 sin2 1 2 θ 12, (VII. 43)
respectively. Consequently, we have for (VII. 39), the probability for σ 1 and τ 3 to be both +1,
1
2 sin2 1 2 θ 23 + 1 2 sin2 1 2 θ 12 1 2 sin2 1 2 θ 13, (VII. 44)
which, using sin 2 1 2 θ = 1 2
(1 − cos θ), is equal to
(1 − cos θ 23 ) + (1 − cos θ 12 ) (1 − cos θ 13 ). (VII. 45)
This is, in essence, the same as inequality (VII. 10); rewriting (VII. 45), realizing that 1 − cos θ 0,
and comparing E(⃗a, ⃗ b) to − cos θ 12 etc. yields
1 − cos θ 23 | − cos θ 12 + cos θ 13 |. (VII. 46)
n 2
n 1
ϕ ϕ
n 3
Figure VII. 7: Violation of the Bell inequality again
With θ 23 = θ 12 = 1 2 θ 13 = ϕ as in diagram VII. 7, (VII. 45) becomes
1 − 2 cos ϕ + cos 2ϕ 0, (VII. 47)
and using cos 2ϕ = 2 cos 2 ϕ − 1 we see that
cos ϕ (1 − cos ϕ) 0. (VII. 48)
Since 1 − cos ϕ 0 for every ϕ, this inequality is violated for every acute angle.