Lecture 16 July 20, 2009 Is there a problem or isn’t there? Double slit calculations Local Realistic Theories and Einstein Locality The Einstein-Podolsky-Rosen Experiment Feynman’s Path Integral Formulation of QM Is there a problem, or isn't there? This thing is completely characteristic of all of the particles of nature, and of a universal character, so if you want to hear about the character of physical law it is essential to talk about this particular aspect... It will be difficult. But the difficulty is psychological and exists in the perpetual torment that results from saying to yourself, "But how can it be like that" which is a reflection of an uncontrolled but utterly vain desire to see it in terms of something familiar. I will not describe it in terms of something familiar; I will simply describe it. There was a time when the newspapers said that only twelve men understood the theory of relativity. I do not believe there ever was such a time. There might have been a time when only one man did, because he was the only one who caught on, before he wrote his paper. But after people read the paper a lot of people understood the theory of relativity in some way or another, certainly more than twelve. On the other hand, I think I can safely say that nobody understands quantum mechanics. So do not take the lecture too seriously, feeling that you have to understand in terms of some model what I am going to describe, but just relax and enjoy it. I am going to tell you what nature behaves like. If you will simply admit that she maybe does behave like this, you will find her a delightful, entrancing thing. Do not keep saying to yourself, if you can possibly avoid it, "But how can it be like that?" because you will get "down the drain" into a blind alley from which nobody has yet escaped. Nobody knows how it can be like that. Feynman 1964 We have always had a great deal of difficulty understanding the world view that quantum mechanics represents. At least I do, because I'm an old enough man that I haven't got to the point that this stuff is obvious to me. Okay, I still get nervous about it... You know how it always is, every new idea, it takes a generation or two until it is obvious that there's no real problem. I cannot define the real problem, therefore I suspect there's no real problem, but I'm not sure there's no real problem. Feynman 1982 If you do not detect them Interference? Yes!!! If you do detect them Interference? No!!! What if you detect some of them? Classical Particles Interference? No!!! Classical Waves and Photons Interference? Yes!!! EXAMPLE 1 EQUAL SLITS With Slit 1 open 100 photons/second => A1=10 With Slit 2 open 100 photons/second => A2=10 HOW MANY WHEN 1 AND 2 CONSTRUCTIVE? A = A1+A2 = 20 A2 = 400 photons/second HOW MANY WHEN 1 AND 2 DESTRUCTIVE? A = A1+A2 = 0 A2 = 0 photons/second constructive => 400 photons/sec destructive => 0 photons/sec with only slit 1 open => 100 with only slit 2 open => 100 with both open no interference = 200 photons/sec 200 with both open = average of interference 0.5 (400 + 0) 500 400 300 200 100 0 EXAMPLE 2 A1=2 A2=6 With slit 1 open 100 photons/second HOW MANY WITH ONLY SLIT 2 OPEN? 2 (A1) = 4 4*(25 photons/sec) = 100 photons/second 2 (A2) = 36 36*(25 photons/sec) = 900 photons/second HOW MANY WHEN 1 AND 2 CONSTRUCTIVE? A = A1 + A2 = 8 A2 = 64 64*(25 photons/sec) = 1600 photons/second HOW MANY WHEN 1 AND 2 DESTRUCTIVE? A = A2 - A1= 4 A2 = 16 16*(25 photons/sec) = 400 photons/second constructive => 1600 photons/sec destructive => 400 photons/sec with only slit 1 open => 100 photons/sec with only slit 2 open => 900 photons/sec with both open no interference = 200 photons/sec 1000 with both open = average of interference 0.5 (1600 + 400) 1600 1000 400 EXAMPLE 3 put a partial detector behind slit 1 for a total of 100 photons/second thru slit 1 50 photons/second are not detected => A1 = 7.07 50 photons/second are detected => A1d = 7.07 no partial detector behind slit 2 => A2 = 10 100 photons/second not detected thru slit 2 FOR THE PHOTONS NOT DETECTED: HOW MANY CONSTRUCTIVE? A = A1 + A2 = 7.07 + 10 A2 = 291.42 photons/second HOW MANY DESTRUCTIVE? A = A1 + A2 = 10 - 7.07 A2 = 8.58 photons/second FOR THE PHOTONS DETECTED: NO INTERFERENCE 2 (A1d) = 50 photons/second with only slit 1 open => 100 photons/sec of which 50 photons/sec are detected and of which 50 photons/sec are not detected with only slit 2 open => 100 photons/sec not detected constructive => 291.42 photons/sec not detected destructive => 8.58 photons/sec 341.42 200 58.58 EXAMPLE 4 equal slits put a partial detector behind slit 1 3/4 of the photons/second are not detected A1 1/4 of the photons/second are detected A1d 2 2 (A1d) = 1/4 (A2) => A1d = 1/2 A2 2 2 0.5 (A1) = 3/4 (A2) => A1 = (3/4) A2 The total intensity is proportional to 2 2 I ~ (A1 + A2) + (A1d) The Intensity Contrast = Imax/Imin is given by 2 2 ((1/2)*A2 + A2) + 3/4 (A2) divided by 2 2 ((1/2)*A2 - A2) + 3/4 (A2) So, the Intensity Contrast = 3/1 Quantum Entanglement EPR: The Einstein Podolsky Rosen Paradox “If S1 and S2 are two systems that have interacted in the past, but are now arbitrarily distant, the real factual situation of S1 does not depend on what is done with S2 which is spatially separated from the former.” EPR is based on three premises: 1) REALISM Observed regularities are caused by physical reality independent of human observers. 2) INDUCTIVE INFERENCE Consistent observations produce legitimate conclusions. 3) EINSTEIN SEPARABILITY (LOCALITY) No influence can propagate faster than light. LOCAL REALISTIC THEORIES (LRT’s) obey premises 1, 2, and 3 LRT and QM disagree QM agrees with experiments => We must give up 1, 2, or 3 !!! 1) The universe exists without us 2) The scientific method 3) Locality So, we give up locality. The EPR Paradox source D2 D1 Measure X2 Measure P1 => know X1 => know P2 Measure both => violate Δ x Δ p > hbar/2 The Bohm-EPR Experiment Measure the polarization/spin Source P2(θ) P1(θ) Measure P1 Measure P1 zero total momentum and => know P2 => know P2 zero total angular momentum if you if you measure it measure it along the along the same axis same axis Bell’s Prediction S(θ) Curve predicted by quantum mechanics Bell’s Theorem => all local realistic theories stay inside the interval [-2, +2] Aspect’s Results Local realism is violated by 22 sigma !!! My Personal Resolution of The Bohm-EPR Experiment Source P2(θ) P1(θ) Both particles are in both detectors, so measuring in either detector collapses the state vector for both particles. QUANTUM- ~&CHANICS Each photon in a pair knows what its diimt partner does, and does the same thtng. Quantum Spookiness Wins, end,ithasaninfluenceonwhathappensonthe"EV~if you change a [setting]oaly on one Einstein Loses in Photon T~s on at a &stawe, in the sensethat what happens at one detector has some ''I cannot seriously believe in [the quantum theory] because it cannot be reconciled with the idea that physics should represent a reality in time and space, free from spooky actions at a distance," meEinstein to the German physicist Max Barn in March 1947. Einstein Einstein loses was particularly bothered by quantum theory's oddball claim that the states of two particles or ' photons can influence each other no matter how far apart they are. Despite Einstein's misgivings, researchers have gone on to dem- onstrate the reality of quantum spookiness, and now-just 140 kilometers as the photon at a distance." tons acquire a particular state only The new result, am-.' when a measurement is made on one of the pair, instantly determining the -phone state of the der. Rame.Aphoton Shimony adds that the result is "pretty source (on definitive disproof that entanglement falls plugged inta Ifre Gene,,a telephpne off with distance," con* to proposals by (farm, some, including the late British theorist N, conip David Bohm. Indeed, it hithat quantum tated photons to two events in a far corner of the universe might neartY MHages (~1.influence events here on Earth. ties persist over distances Gisin points to more down-to-earth im- of up to several kilometers. Some theorists and his colleagues in Paris in 1982, have plications for telecommunications, implica- have speculqted that these correlations would backed the quantum claim. tions presumably not lost on Swiss Telecom: weaken with distance, says another quantum With a little help from Swiss Telecom, "If these correlations hold over very long mechanics expert, John Rarity of the De- Nicoh Gisin and his group at the University distances .. then they could be exploited for fence Evaluation and Research Agency in of Geneva have now gemonstrated quantum a variety of applications, especially quantum Malvern, United Kidom. But in the Swiss action at a distance on a large scale by turning cryptography." Contrary to Einstein's fears, result, "we've now got to 10kilometers' sepa- the countryside around Geneva into a giant quantum correlations can't be exploited to ration, and quantum mechanics is appar- quantum laboratory.