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350 CHAPTER22. THEEPRPARADOXANDBELL’STHEOREM

TherelevantexperimentswereperformedbyAspectandhiscollaboratorsinthe
1970s. Thetwoparticlesweretwophotonsproducedby thedecayofPositronium
(anelectron-positronboundstate). Allofthequantum-mechanicalpredictionswere
confirmed. Themysteriousnon-localbehaviorofquantumtheory,inwhichamea-
surementofparticle 1 somehowcausesthedistantparticle 2 tojumpintoastateof
definitespin,cannotbeexplainedbyalocalhiddenvariablestheory.

Problem- Giventhestate(22.19)andtheO 2 eigenstatesineq. (22.18),derive
theprobabilitiesshowninFig.22.3.

Problem - Supposethetwoparticlesareelectrons, andO 1 = Sz is thespin
componentalongthez-axis,withthegreenlightflashingforsz= 1 /2,andthered

lightflashingforsz=− 1 /2.AssumethatO 2 measuresthespincomponentSe=S%·%e
alongsomedirection,specifiedbyaunitvector

%e=





a

b

c



 (22.20)

Again,thelightflashesgreenforspinupalong%e,andredforspindown. Withthis
information,andeq. (22.18),findthedirection%e.

22.3 Entangled States for ”Quantum Radio”?

TheEPRparadoxandBell’sTheoremrefertoexperimentsperformedonparticlesin
entangledstates. Itseasiesttodefinesuchstatesbywhattheyarenot.
LetΨ(x 1 ,x 2 )denoteatwo-particlestate.Thestateisseparableifhastheform

Ψ(x 1 ,x 2 )=ψ(x 1 )φ(x 2 ) (22.21)

Ifitdoesnothavethisform,thenthe stateisinseparable, or ”entangled.”Now
supposewearegoingtomeasuresomeobservableAofparticle1,witheigenstates

A ̃φn(x 1 )=λnφn(x 1 ) (22.22)

and,forsimplicity,supposetheeigenvalues{λn}arenon-degenerate. Thenwecan
alwayswriteΨ(x 1 ,x 2 ),whetherseparableorentangled,intheform

Ψ(x 1 ,x 2 )=

∑

n

cnφn(x 1 )φn(x 2 ) (22.23)