QuantumPhysics.dvi

b→−a, which is consistent with the fact that the spins measured byAandBshould be perfectly
anti-correlated in the state|Φ〉.

Are the Bell inequalities obeyed by quantum mechanics? To find that out, consider the com-
bination

P(a+,c+) +P(c+,b+)−P(a+,b+) (17.51)

The Bell inequalities require that this combination be always ≥0. The quantum mechanical
computation gives,

P(a+,c+) +P(c+,b+)−P(a+,b+) =

1

2

(

sin^2

θac

2

+ sin^2

θcb

2

−sin^2

θab

2

)

(17.52)

It is easiest to analyze the expression on the right hand sidewhen the directionsa,b,care coplanar.
We then have

θab=θac+θcb (17.53)

If theθabangle is taken to bethe smallest angle subtended between the vectorsaandb(i.e.
0 ≤θab≤π), then the above planar relation between angles requires that

0 ≤θab≤π

0 ≤θac≤π

0 ≤θcb≤π (17.54)

In this sense, the directionclies betweenaandb. Using

sin^2

θab

2

= sin^2

θac+θcb

2

=

(

sin

θac

2

cos

θcb

2

+ sin

θcb

2

cos

θac

2

) 2

(17.55)

and combining with the other terms, we get

P(a+,c+) +P(c+,b+)−P(a+,b+) =−sin

θac

2

sin

θcb

2

cos

θab

2

(17.56)

Given the ranges of the angles, the combination on the right is always negative or zero. Sakurai
quotes an even more special case whereθab= 2θ andθac =θcb =θ. The above formula then
reduces to

P(a+,c+) +P(c+,b+)−P(a+,b+) =−sin^2

θ

2

cosθ (17.57)

which is manifestly negative when 0≤θ≤π.

Thus, quantum mechanics gives a radically different prediction for these combinations of prob-
abilities. The measurements can actually be carried out on pairs of linearly polarized photons,
produced in cascade decays of atoms such asCaorHg, excited by laser pumping.^16 The experi-
ments clearly confirm quantum mechanics and thus invalidatehidden variable theories.

(^16) A. Aspect, P. Gragnier, G. Roger, Phys. Rev. Lett. 47 (1981) 460; Phys. Rev. Lett. 49 (1982) 91.