346 CHAPTER22. THEEPRPARADOXANDBELL’STHEOREM
in totalcontradictionto the reported fact (B)!Conclusion: no experimentshould
reportbothresults(A)and(B).
Andyet,asEPRpointedout,therearequantum-mechanicalstateswhichleadto
preciselytheseresults. Supposethatthe2-particles,beforereachingthedetectors,
areinaspin-0state
ψ =1
√
2
(ψ 1 −ψ 2 )=
1
√
2
(αLzβRz −βLzαRz) (22.11)ThentheprobabilitytofindLwithspinupandRwithspindownalongthez-axisis
P(updown)z−axis = |<αLzβzR|ψ>|^2=1
2
∣∣
∣<αzLβzR|ψ 1 >−<αLzβzR|ψ 2 >∣∣
∣2=
1
2
(1−0)^2
=
1
2
(22.12)
Likewise,theprobabilitytofindLwithspindownandRwithspinupinthedirection
is
P(downup)z−axis = |<βLzαRz|ψ>|^2=1
2
∣∣
∣<βLzαRz|ψ 1 >−<βzLαzR|ψ 2 >∣∣
∣2=
1
2
(0−1)^2
=
1
2
(22.13)
Thesetwoprobabilititesareexactlywhatisreportedin(A).Thenwecomputethe
probabilitytofindbothLandRwithspinupalongthex-axis
P(upup)z−axis = |<αxLαRx|ψ>|^2=1
2
∣∣
∣<αxLαRx|ψ 1 >−<αLxαRx|ψ 2 >∣∣
∣2=
1
2
∣∣
∣<αxL|αLz><αRx|βzR>−<αLx|βzL><αxR|αRz >∣∣
∣2=
1
2
(
1
2
−
1
2
)^2
= 0 (22.14)
whichisthesameasresult(B).