22.1. THEEPRPARADOX 345
affectthespinoftheotherparticle(”R”)ontheright,andyetweknowthat Ris
spin-down,itmustbethatRwasspindownevenbeforewemadethemeasurementof
L.Therefore,thespinstateoftheparticlesbeforeenteringthedetectorsmusthave
been
ψ 1 =αzLβzR (22.4)
whereαz,βz referto spinupanddown states, respectively, along thez-axis. For
futurereference,letsrecallfromLecture 13 that
αz=[
1
0]
βz=[
0
1]
(22.5)and
αx=1
√
2
[
1
1]
βx=1
√
2
[
1
− 1]
(22.6)Now,bythesamereasoning,iftheparticleontheleftwerefoundtobespindown,
thenthe2-particlestatewouldhavebeenbe
ψ 2 =βLzαRz (22.7)evenbefore the measurementwas made. According to (A),in half the runs the
particlesareinstateψ 1 ,andintheotherhalftheyareinstateψ 2.
Thisreasoningseemsprettysolid,butnowatroublingdiscoveryismade. After
manymoreruns,itisfoundthat
B.Whenbothdetectorsaresettomeasurespininthex-direction,thetwoparticles
areneverfoundtoeachhavespinup,oreachhavespindown.
Theproblemisthat,iftheparticleswereinitiallyinstateψ 1 ,thentheprobability
thatameasurementwouldfindbothspinsupinthex-directionis
|<αLxαRx|ψ 1 >|^2 = |<αLx|αLz ><αRx|βzR>|^2=1
4
(22.8)
Likewise,iftheparticleswereinitiallyinstateψ 2 ,theprobabilitytofindbothspins
upinthex-directionwouldbe
|<αLxαRx|ψ 2 >|^2 = |<αLx|βzL><αRx|αzR>|^2=1
4
(22.9)
Theniftheparticlesarehalfthetimeinstateψ 1 ,andhalfthetimeinstateψ 2 ,the
totalprobabilityforfindingbothparticleswithspinupalongthex-axisissimplythe
averageofthetwoprobabilities,i.e.
P(upup)x−axis =1
2
|<αLxαRx|ψ 1 >|^2 +1
2
|<αLxαRx|ψ 2 >|^2=