22.2. BELL’STHEOREM 347
Nowifboth(A)and(B)aretrue,itmeansthatbeforeanymeasurementtakes
place,RisneitherinstateβzR(asinstateψ 1 ),norinstateαzR(asinψ 2 ),because
thiswouldcontradict(B).So,supposewemeasurethespinofLalongthez-axis,and
itisfoundtobeinspinup. Atthatpoint,weknowthatRisnowspindown. But
wehavealreadyconcludedthatbeforethemeasurementtookplace,Rwasneither
spinupnorspindown. Thismeansthat,althoughRwasnotinaspin-downstate
initially,itmusthave”jumped”intoaspin-downstateafterLwasfoundinspinup,
evenif,atthetimeofmeasurement,LandRwereveryfarapart! Buthowcoulda
measurementofL,whichperhapsisinPhiladelphia,influencethestateofR,which
ismaybeinDetroit? ThisistheEPR”paradox,”whichledEinstein,Podolsky,and
Rosentoconcludethat thequantummechanicaldescriptionof natureissomehow
incomplete.
22.2 Bell’s Theorem
Quantummechanicsmightbewrong,butsofarthereisnoevidenceofthat. Soin
what way coulditbeincomplete? Onepossibilityisthat,contraryto everything
saidsofarinthiscourse,thephysicalstateofaparticleisnotcompletelyspecified
byitswavefunction. Perhapstheparticlealsohassomeother,”hidden”degreesof
freedom,whichdeterminewithcertainty,ratherthanprobabilistically,theoutcome
ofanymeasurementoftheparticle.Theprobabilisticnatureofquantummechanics,
accordingtothisview,issimplyduetoourignoranceoftheactualvaluesofthese
”hiddenvariables.”
Ifthehiddenvariableideaweretrue,itseemsthatwemightbeabletowriggle
outoftheEPRparadox. Denotethehiddenvariablebyμ,andsupposethefollow-
ingvaluesimplythefollowingresultsofspinmeasurementsalongthezandx-axes,
respectively:
μ= 1 leadsto upz or upx
μ= 2 ” upz ” downx
μ= 3 ” downz ” upx
μ= 4 ” downz ” downx
(22.15)
Then wecouldexplain(A)and(B),forexample,by sayingthat thetwoparticles
comeoutoftheapparatusonlywiththehiddenvariablecombinations
(μL,μR)=(1,4) or (4,1) or (2,3) or (3,2) (22.16)Ofcoursethisisnotyetatheory,onlyanideaforatheory,buttheideaitselfwas
squelchedbyacelebratedtheoremprovedbyJ.Bellin1964,whichessentiallysays
thefollowing:
Bell’sTheorem