quantum mechanics

8.8.2 Violations of Bell’s Inequality

We outline now two physical schemes for testing Bell’s inequality (8.48) against experimentally con?rmed predictions of quantum mechanics.


1
Entangled spin- 2 particles
Consider ?rst two spin- 1 particles A and B in the entangled state
1 A B A B
|?2) = ? ( | ?z )| ?z ) + | ?z )| ?z )) . (8.49)

In a real experiment, these particles could be, e.g., Hg atoms. Then the en- tangled state (8.49) could be realized in two steps: First, a Hg2 molecule is
photodissociated into the singlet state 1 ( | ?z )| ?z )? | ?z )| ?z )), whose two
? A B A B
2
constituent particles (atoms) A and B ?y in opposite directions, parallel and
anti-parallel to the y axis, respectively. Next, a longitudinal magnetic ?eld is applied to one of the particles to rotate its spin around the y axis by angle ?, realizing thereby the ?y transformation that results in state (8.49), to within the trivial overall phase factor ei?/2 which can be omitted.


Stern?Gerlach apparatus
? z
y x

Fig. 8.12. Spin- 1 particle passing through a Stern–Gerlach apparatus rotated by angle ? with respect to the z axis.


Let us ?rst determine the probability of detecting a spin- 1 particle in the spin-up state | ?? ) along the axis rotated by angle ? with respect to the z


axis. Such a measurement can be realized by letting the particle pass a Stern– Gerlach apparatus and detecting its upward de?ected component, as sketched in Fig. 8.12. This amounts to projecting the state of the particle onto the state
| ?? ), which can be obtained by applying the rotation operator Ry (?) to state
| ?z ) (see Sect. 8.4),


| ?? ) = Ry (?) | ?z ) = e?i??y /2 | ?z ) = cos


?
2 | ?z ) + sin


?
2 | ?z ) .

The probability P (?) of detecting the particle in state | ?? ) is then given by the expectation value of the projection operator ?? = | ?? )(?? | , P (?)= (?? ). If we now have a system of two spin- 1 particles A and B, each analyzed by its own Stern–Gerlach apparatus rotated by the corresponding angle ?A,B, the joint detection probability PAB(?A, ?B) is given by the expectation value

of the product of two projection operators ? A

= | ?A

A
)(??A

| and ??B =

B B A B
| ??B )(??B | , PAB(?A, ?B)= (??A ??B ).
We can now easily calculate all detection probabilities for a pair of particles in the entangled state (8.49). For PA and PB we have
1
PA(?A)= PB(?B)= 2 (8.50)
for any ?A and ?B, while for the joint detection probability PAB we obtain

1
PAB(?A, ?B)= 2 cos

2 . ?A ? ?B .
2


. (8.51)


Equations (8.50) and (8.51) show that the rotational invariance assumed in the derivation of function S(?) is indeed satis?ed in this case. Choosing the

four detection angles as ?A = 0, ?B = ?/4, ?r

= ?/2 and ?r

= 3?/4 yields

? = ?/4. From (8.48) we then obtain


S(?)=

cos2 . . cos2 . .


1.2 ¢ 1 , (8.52)

3 ? 1 3?
2 2 ? 2 2 c

which clearly violates Bell’s inequality. Hence, the predictions of quantum me- chanics, which have been veri?ed in many experiments, contradict and thereby invalidate the hidden variable theories based on Einstein’s local realism. This leads to the inescapable conclusion that quantum mechanics is a nonlocal theory.