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WKU PHYS 480 - LECTURE 9 IN QUANTUM MECHANICS

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What Does It Mean?Da Big QuestionRealist ViewOrthodox PositionAgnostic ResponseEPR ParadoxEPR Cont’dMovin’ On Up to the Big Time, the Deluxe Apartment in the Sky!The 3 views againEPR SaysBell’s ThmThe GedunkenThe Results of the ExperimentHoweverWhy?Let’s do some mathSoIf c is an other unit vector then|P(a,b)-P(a,c)|<= 1+P(b,c)But What does it Mean?Supernaturally SuperluminalTwo types of influenceHere, kitty, kitty ….The Cat ParadoxIt’s ALIVE! It’s Dead! Etc. etc.Out of the ParadoxMy Answer to All These Difficult Dilemmas1What Does It Mean?From the “Afterword” of D J Griffith’s Introduction to Quantum Mechanics2Da Big QuestionDid the physical system “actually have” the eigenvalue in question prior to the measurement? (REALIST)OR Did the act of measurement “create” the eigenvalue (constrained by the wavefunction)? (ORTHODOX)ORCan we completely duck the question? (AGNOSTIC)3Realist ViewIf the realist view is true, QM is an incomplete theory because:Even if you know everything that QM has to tell you about the system, you STILL cannot determine all of its features!4Orthodox PositionMeasurement forces the system to “make a stand” helping create an attribute that was not there previouslySince repeated measurements yield the same result, the act of measurement collapses the wavefunctions.This is strange but not mystical5Agnostic ResponseI refuse to answerI ignore these problems6EPR Paradox1935- Einstein, Podolsky, and Rosen came up with the EPR ParadoxDesigned to prove that the realist postion is the only possible oneA simplified version goes like this:Assume that a pion at rest decays into an electron and positron0->e- + e+The positron and electron fly off in opposite directions and the pion has spin=07EPR Cont’dSince pion is spin 0, then electron and positron are in singlet configuration |00>So if the positron is spin up then electron must be down or vice versa.QM can’t tell you which you will get, but that they will be correlated 218Movin’ On Up to the Big Time, the Deluxe Apartment in the Sky!Now move the electron far apart (pick one)10 meters10 light yearsNow say you measure electron spin down, then IMMEDIATELY you know that the positron is spin upBTW, this is an INSTANTANEOUS knowledge!9The 3 views againRealist: This is not surprising, the positron was always up since the decayOrthodox: Hmmm… electron was neither up or down until measurement. The measurement caused the wave function to collapse and that “knowledge” was transmitted instantaneously across the gulf to the corresponding particle.Agnostic: I don’t have an opinion10EPR SaysOrthodox view is “spooky action at a distance”Ergo, says EPR, the realist have it correct and the spins were predetermined at the decayOf course, this is all predicated on the argument that nothing, not even information, can go faster than the speed of light– called the principle of locality11Bell’s ThmEPR did not doubt that QM is correct, just incompleteSome other “hidden variable” is needed to completely quantify the system.The hidden variable could be a single number or a whole collection of numbers; it doesn’t matterJ. S. Bell proved that ANY hidden variable theory is INCOMPATIBLE with QM12The GedunkenInstead of having the electron and positron detectors along the same direction, allow them to be rotated independentlyThe first detector measures the component of the electron spin in the direction of unit vector a and the second along the direction of bFor simplicity, we will record the spins as +1 (up) and -1(down)a be-e+13The Results of the ExperimentBell proposed to calculate the average value of the product of the spins called P(a,b) If detectors are parallel, then we have original EPR configuration so always +1 and -1 and therefore P(a,a)= -1 (and so is the average)electron positron product+1 -1 -1+1 +1 +1-1 +1 -1+1 -1 -1-1 -1 +1… … …14HoweverIf they are anti-parallel, then P(a,-a)=+1For any arbitrary orientation thenP(a,b)=-a ·bThis result is IMPOSSIBLE for any hidden variable theory15Why?Assume, that the hidden variable is called kk varies in some way that we neither understand nor control from one decay to the nextSuppose that the outcome of the electron measurement is independent of the orientation (b) of the positron detector. b is chosen after the decay but before measurement of electron and thus is hindered by speed of light (locality condition).16Let’s do some mathSo there is a function A(a,k) which gives the result of the electron measurement and B(b,k) which gives the result of the positron measurementA(a,k)=+/- 1 and B(b,k)=+/-1 When detectors are aligned, the results are perfectly anti-correlated A(a,k)=-B(b,k) for all k17So dkkbBkaAkbaP ),(),()(),(Where rho is the probability density of kNow let’s eliminate B(b,k) by using our correlation function  dkkaAkaAkbaP ),(),()(),(18If c is an other unit vector then),(1),(),()],(),(1)[(),(),(0)],(),(1)[(1)],(),([1),(),()],(),(1)[(),(),(:1)],([)],(),(),(),()[(),(),(2cbPcaPbaPdkkcAkbAkcaPbaPkcAkbAkandkbAkaASincedkkbAkaAkcAkbAkcaPbaPkbASincedkkcAkaAkbAkaAkcaPbaPBell’s Inequality:19|P(a,b)-P(a,c)|<= 1+P(b,c)It is easy to show that P(a,b)=-a·b (the QM prediction) is incompatible with Bell’s InequalitySuppose all three vectors, a, b, and c lie in a plane with c at 450 to a and b (a perpendicular to b)Then P(a,b)=0 and P(a,c)=P(b,c)=-.707Obviously .707 is not greater than 1-0.707 (.293)20But What does it Mean?If EPR is correct, then QM is completely WRONG!On the other hand, NO hidden variable is going to rescue us from the nonlocality that Einstein considered preposterousMany experiments were performed to test Bell’s inequality: the results were compatible with QM and incompatible with Bell’s InequalityIn other words, the realists are wrong and there is spooky action at a distance Or in the lingo: there is the possibility of superluminal influences21Supernaturally SuperluminalA causal influence that propagates faster than light is bad newsBecause relativity says that anything going faster than light is going backward in time!Faster than light things: any geometric pointWe don’t


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