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Quantum Entanglements, Part 1 | Lecture 4

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Quantum Entanglements, Part 1 | Lecture 4
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Lecture 4 of Leonard Susskind's course concentrating on Quantum Entanglements (Part 1, Fall 2006). Recorded October 16, 2006 at Stanford University. This Stanford Continuing Studies course is the first of a three-quarter sequence of classes exploring the "quantum entanglements" in modern theoretical physics. Leonard Susskind is the Felix Bloch Professor of Physics at Stanford University.
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theorem but that's OK we do next week I hope we get to entanglement right but we were in the midst of still entanglement has to do with what happens the simplest case of it is when you have till electrons with their Spain's forget the fact electrons to move around just worry about this news entanglement of what happens when you have more than 1 system Burma it's we kind of interesting quantum states that can be made for systems which have parked them more than 1 part where still struggling through for trying to understand the simplest quantum system the 1 Bates warning quantum bits system spend the two-state system where the 1st set up because it will come in very central way all of things with Europe talking about about the simple system last time we talked about what happens if you're interested in the component of Spain's along some taxes which is neither 3 tore 1 borrow on some arbitrary axis and arbitrary axis can be represented by a unit vector a unit pointer a unit pointer are that has components tree and 1 and 2 and 3 and incidentally on OK so here's my a quarter space I guess this 1 1 this 1 2 of us were rainy and Eimile laps every now and then the calling 1 x tool why and 3 z so our car keep murmuring but I'll try not to or try to keep a consistent rotation and my notes but I do have a habit of some firms going back and forth from 1 . 3 x exporters column and then if you have any ordinary vector in space not this unit vector in particular but some are the better a right on-site intended it to be right in there but I can tax I think so but I most of my shoulder was broken on freedom it doesn't matter where George it does matter Brett birth is a ride in the reason not what was left out of place some of them have MY going into the boards I don't have that why going into the board right there get why yeah yeah aid broke my brokers shoulders a glimmering Murray orientation right now you have some physical vector thing the measure and in particular things that we represent as having components Sigma Sigma X saucer once signatories of this is supposed to be some physically measurable Will magnetic moment or spin vector it has components along with 1 2 and 3 taxis we can call them serve wants to maturity of 3 we have called absurd norms 3 laws are observable quantities and we represent matrices but for the most forget that matrices are if we're interested in the component of not all along the arbitrary access then what were interested in the bar Proctor of said the area and the indicating unit records conventional growth little have on top of it to live vector then let's just right back coupled equations and 1 square Ross through squared off and 3 square is equal to 1 of their best revolutionary Rebecca and say not in this is the ordinary surbase opera not be a problem the 2 abstract vectors representing states but just ordinary Bob Probert space maybe back there but we can measurable school comments are measurable quantities and the unit vector and this is by definition and warn so you want us to stay with us and rainy said authority but now each other singers is itself a matrix it is numbers correction call signs of summer action work they are direction call signs of some direction and Our about the the same as on matrices and so this whole thing is itself matrices or operators of a warm be really technically correct our linear operators the voters think of them as matrices and then remembering what's what's right down the Matrix areas OK so soon after remorse in the 3 years it's the Matrix will 1 and minus 1 solved that means we would multiplied by 3 and put in 3 here and my and three-year are single 1 in Paris a 1 over here and a world here so we would put him in 1 over here and in 1 over here and then Sigma chill has on May and plus I so we were right plus in the debate bigger and 1 plus I in into and then Darren here it would be in 1 or starter might decide you're BN 1
minus fast quite and to remind us and 3 over here Norris 3rd term agenda bag or Missouri the off bag moments of complex conjugate to each other limited emissions and I'm going to just make a little notation may or may not use it for a March but standard notation right In 1 minors and 2 0 is equal to and my best and and warned plus I and 2 0 is equal to and plus the definition of half the writer another interesting simple equation if you multiply and plus 0 0 then we can just right this is equal to 0 and simplified expressions and 3 miners Cross minors 3 that's the matrix saying Mumbai area and it's supposed to be observable something we can measure we measured by putting electron or suddenly turning on a large magnetic field Oriental an axis and then checking whether it does or doesn't make a photon it doesn't mean that a photo it has positive component of stigma Byron OK not 1 thing that we check last summer believe which are brusque arm was that elite is still the box and squared is equal you could check that either by multiplying the matrix by itself and then using the fact that in 1 square presenters were square % Group 3 square equals 1 or you can check it from the fact that each 1 of the Bali checkered from properties of the stigma matrices missing matrices Sava think I would not last time young follows from the fact that any 1 of the singers times anyone other sectors is equal on minus a following Oracle to Jerry ROC mourns Roberto was minus admit to signal 1 and you can check that easily the squares now on a matter of drove this right now it's a very very active broke every single little detail of everything we do do take forever so here and there if I say something that you don't know usually it usually means something that takes about a half hour wine to probe you could progress our figure a homework assignment of saw a similar not in squares warned wealth number squares equal the warned that means that the numbers you the the plus or minus 1 per ever matrix squared cyclical warned it means that Eigen values a plus or minus 1 so synod and in fact has to offer any alien has 2 eigenvalues 1 plus 1 and 1 from Argos worn as you might expect corresponding to 2 possibilities your new measures longer you part my soul was Latin squares a global 1 that's 1 fact another little of algebraic fact which is kind of neat multiplied and all stroke off and class plans and my that's equal to an 1 square Boston 2 squared all notice and plus and minus a complex conjugate to each other so in last and my books and classes and my is equal and 1 squared plus In square a rubber statement right let me get this posture and 3 square and then subtract and 3 square I haven't changed when I see that and plus and minus cyclical 1 my and 3 square 1 minus 1 minus Iowa square the whole trainer we want square in 3 that's so that serve a useful well algebraic track that if you go the homework which I'm going to assign for use all over and over again works to serve its Her rather making deep just to Breyer track OK here's the question about the alternately the question I wanna ask is a far surpassing naive winters spend and I preparers and by putting it in a large magnetic field waiting for it to decay on Arctic force it into a particular law particulary of bars for us at the BDR along an accessed then I measure the component of the spin along with different Baxter's for score would be an axis Uzi over here What's the probability that I make this begins with this spring not only an axis and then made sure it alone end the Baxters I will find a colossal Longley an axis of course the opposite probability of my Osoria this is the central kind of questions quantum mechanics deals worth air from these simple half of a simple Spain's electron spin this is about be only interesting question to permit questions are you could also of course ask supposing your along the axes and measured on an axis by symmetry it's pretty clear that they should be equal to each other and furthermore you might even expect that the rotational symmetry of the world should mean that the install strongly depends on the angle between him and there would be a embarrassing discovery package from the answer depends on the angle between them would be part of your homework assignment but how you know it rocket so we have to go back there the basic who are probability Our posture with about quantum McCarrick so that what we 1st duo is calculated be eigenvectors of and the eigenvectors assume about him now we 1st find the eigenvectors signal about in the corresponds to spend point being definitely along with this axis and then we take In the product with the eigenvectors stigma about EM corresponding to spin along this direction before we correct a sap let's make an operation that question not yes I want to be bothered him as that Sasha as well it it's not single expect this the hall accumulated during the are it was a whole lot of
data about as you could do this experiment all over and over and over again and eventually you were discovered but the probabilities there are probabilities that's an experimental facts of the probabilities and given by the formulas were right now and that was something that was dose by guest was guessing was both guest of however guessed it it wasn't any single person there was a buildup from the work force or starting what Einstein who was the 1st to realize the quantum theory of all probability His born Heisenberg act so the Caesar steps in which all of this was eventually discovered nobody nobody has ever understood what ahead of hell Heisenberg was thinking when he invented matrix mechanics nobody understands or what own mystical experience or what he was smoking a what her about her game and I think is simply have to take quantum mechanics ads the fine by matrix mechanics as an experimental fact is that it can be derived from anything it's not something you could derive from our from anything else server empirical facts which is summarized by this calculus of probabilities OK so outlets let's 1st inventor notation 1st along with inventing patience for the eigenvectors of sin and about and equals plus 1 quite that they're the eigenvectors obsolete Nevada end with Eigen value plus 1 What does it satisfying satisfy stigma that an equals 1 if your apply saying my enter way you just get the same back-to-back this hall elaborate thing here it just it was just aren't a statement on talking about the eigenvectors signal about in William value plus 1 if I apply Sigma got into such an eigenvectors I just get 1 times the same OK laps that's the eigenvectors corresponding to a state we could court we caught Eigen state the word state is also use over time and this is the Eigen state Our Sigmund Bob and such that stigma then if you measure it definitely equaled the course what we could do the same for about Soon Nevada and on the state stigma about em equals Warren also equals about him this sequel here Islam equals is just part of the rotation labeling go you have elected I could use some former patient but Seoul how where we have to do the 1st step is to construct these vectors to find out what the eigenvectors of Sigmund by an oral and written about him and then leave done that you construct was called the probability amplitude 1st step was the probability amplitude in a product between the prepare state N B final out calm woes probability your interest there were 5 and equals 1 practices the probability amplitude that fuel create an electronic with along the annexes and a new measurement along the an exodus that you get what you get past due course 1 this is not the proper abroad this is the probability amplitude together probability this is a complex number in general use square at you multiply by complex conjugate which is labeled by tow bar where the absolute value of it squared classify Lurie quite probability Back IRI Ross out it's a probability that as about end is equal to 1 and then you measure singing about if you'll get 1 as opposed to plan that's the kind of thing the quantum mechanics calculates that's exactly what the corner can expose In every single case questioned what as the probability that you get a certain how calm about a particular variable given that you prepare the state in some way perhaps involving some honorable way given that software the offer what OK so we could be could then right probability the stream of Bob and equals Warren van about him cause warned of probability in general will be less than 1 what what sort of figures cases In more equal obviously the probabilities warned an MR opposite directions and that's just the question What's the probability of suspenders and now rules that must be 0 so we should fight back about 90 degrees a half anywhere is a plane is in the plane perpendicular and we would expect to get probability being a half and strong about how he how would justify however we calculated Solheim take you through the calculation of the eigenvectors you'll Arden your whole and take their inner product topic a product you'll square and boil it caucus stamp on him cry it's a it's an ugly little calculation present 1 outside after a hard look at me about 15 minutes by user lock tracks which means the lot larger longer take your bag of tricks OK so let's find the eigenvectors not when you start looking for eigenvectors will try and solve the problem said he had not area where is it and 3 miners and 3 and minus and plus bastard Nevada and I were going to look for a vector Alpha Beta such that we get half of back now on the 1st round and not going Tawarri about normalizing effect later on we can normalize the vector Our normalizing it needs multiplying it by some constant so the sums of squares and the 1 but for the moment let's not worry about that if I don't normalized Walsall and other things don't think I've ever mention all the physical quantities in quantum mechanics are insensitive from multiplying the vectors anyone of the vectors by a phase fate is means of complex
number of magnitude 1 and the reason is because all the physical quantities you always complex conjugate Inman Square at the complex conjugate our a number of was magnitude is worn normal Ikeda the I favor it will not enter into any probability that you ever calculates of take a vector and you multiplied by the number equally I figured you have changed nothing about were properties they cost you always wind-up multiplying quantity by its complex conjugate so that's mini keep in mind that however you should normalize the vector by multiplying both the top and the bottom he about the same Ito the RIA data doesn't change a thing right but that the upshot of Abbas we can arbitrarily for example choose How could be 1 if we don't care about normalizing the vector and we don't care about phase there was no loss of generality and just choosing offered to be worn on the way and later we'll come back and normalized so what happens if we choose our words she was after we won and the mark although Lauren treaty years why get back warns question is what the question is what Izzy have to be worked as a numerical values 5 so that off work out the equation but top component and 3 times 1 class and minus frenzy and 3 firms Warren plus and minus frenzy that has to be equal 1 frustrating component of the left inside study of equal to top component of the right side now you could do the same for the bottom from Bonn you get exactly the same equations Missouri restoring Bava we obviously have enough equations here Missouri we don't need the right the 2nd equation so it's just sovereignty would subtract and Bahraini we get and DE is equal for 1 miners and 3 provide by and writers away have found now found eigenvectors policy will not and here it is our right about now it's equal warned 1 miners and 3 all were and might no it's not normalize it's not normalized normalized we have multiplied by some number some some number of scored 3 hour of and multiplied something so that this so that we of the sums of squares the 1 now where we actually have to do to be more precise we have to make in a product of his vector with itself be equal to 1 let's see what a product of a vector with itself must 1st write the rollback of the corresponding vector 1 1 miners and 3 now sore right in my Estonia can't you a can't use and class that's a row vector of our multiply this by recalled veteran 1 1 miners and 3 over an hour and my nest general get his warned harms warned plus end 3 squared off Iran's plus and minus but I do have a right for a South Bridge off for it yacht birds Ross work out work it out Laura what's multiplier won by and plus and minus of this becomes a plus and minus plus 1 homeowner square this saga warned posture and 3 squared minus twice in 3 years a right right there were half-timber back right and plaster discuss take your little ways into this calculation now waters and plus and minus that were to rehab and plus and minus as 1 might 3 squared I surplus Century Square this just gives us and brought minus is worn minus and 3 squared and 3 squares cancel B 1 is a head thing you get to minus twice and Over and plus and minus direct thinkers Shrek now symbol 5 0 a mess now is 1 one-liners and 3 square downstairs on either the wall the algebra underskirt take you away we're back what I would they have got or was to outside vendors or 1 minus 2 and 3 In the numerator and then the that's 1 miners and 3 squared what the 20th 3 squared which is 1 minus three-time 1 3 record story Co on my has sent 3 times warm applause and 3 in or if you don't enough on was are and how we can so 1 hr act we get answer the answers tools provided by and 3 what was that that was and that was the EU product of a vector with himself we don't want that 1st but we have normalized where we have to deliver the vector to normalize are we have the multiplier not by the square root of this but as weird worn over years want to kill we want to cancel To a divided by war and 3 and so we have the multiply the vector the eigenvectors by the inverse square wrote so he is normalized eigenvectors bear the hallmark of courses to check this and a check was normalized by guiding vector over 2 times 1 1 miners and 3 that's the normalize eigenvectors corresponding for measuring the spin the annexes the same thing is true along the Amex's except where see and you put em whatsoever
to write 1st actually put square money inside square root of cost and really over tool and 1 and read this bookstore carpet with this more complicated than it really has to be but 1 plus 3 square root of our that's normal hours I know I'm not 3 other half of the calculations the other half of the calculation is the other vector the eigenvectors of about him and that's easy or I have to do is replaced and my and the and complex conjugate solve the appropriate role looks very similar square root of 1 class and 3 and the 2nd entry form minors and 3 hours better Emma my ears will put and plus because complex conjugated and square root of 1 applause and 3 over to where we do it as unless we take the inner product theater product I God do it priority answers will multiply this 1 by this 1 over the right to become would multiply this 1 by this 1 and we end this 1 firms is what it looks like an unholy mess by ifyou euros rowboat of algebra and you just go through it are brute force who would take it 10 minutes if or tricks it'll take a 15 minutes so Our and you just use these is over and over again the fact and plus and minus 1 Meyers 3 squared you feel good I never tell what theaters not the there's not for the summit's out yet to the 1 more thing what's Aleksandr Brewer process why is at White this is Emma what does nothing of viral American if this way and this would be a vector times we started yes mn and with the same field probably 1 so we took care to make sure of none of normal Faeroe will join them will argue with a fast swear they get funded complex conjures it kind complex conjures so we take this enterprise we multiply out of big mess but it's just it's it's just the mess and we picked off they get it and we multiplied by its complex conjugate they also said they knew calculated calculated you calculate new our beginning driver role on the computer and see what you get using mathematical owes something order you like how would you like to do it but the answer is incredibly simple after I have yet to to be able defined a one-line weight growth could demonstrate the answer I was happy to do it go through the whole thing Our I'm absolutely convinced that was a warm or argument but never find answer is 1 plots in 1 and cost and tool and too fast and 3 M 3 provided by 2 Ed class and . AM or through that's howitzer after you finish it as NZ members all of the components Place and it's simply boils Barham eventually this expression was and I am anonymous that's the cosine of the angle between them it's a cosigner scoreless played in the M B angle between me and axis the interior axis so this is just 1 plus cosigned our faith in him provided by To now 1st of Wall 1 interesting thing about it is only depends on the angle between them and them that means you get the same results if you rotated the whole experiment that's an interesting facts are important fact chorus 4 has to do with rotational variants of physics that taxis the matter father rotated the whole experiment you would get the same answer it's also symmetric in and Joey depends on the angle between them the angle between him and the matter with you right in the 1st hour and 1st it's exactly the same answer if you interchange in and that's interesting but let's see what no I gives various situations 1st also posing in in the same direction then them on the same direction cosine they later 0 misses the Angola 0 sigh of 0 1 1 plus 1 too weak to divided by 2 was 1 the 1st thing it says is that if you prepare electron pointing along an axis and then you measure it along an axis you will definitely find it off you made it up you measure it and you find out that what about it and in MOR opposite to each other about perpendicular composer and later a 180 degrees cosigned 180 degrees minus 1 1 plus minus 1 0 0 that says that if you will prepare the electrons along the indexes and then look and ask whether it's spinners along the Midas and access to conceal a probability of course pointing that way it's definitely not pointing back way it's a good thing was that is 90 degrees but that doesn't matter not this doesn't matter In this next M and anywhere in space as long as it dangles bricks so could be calm but many cases their fate is 90 degrees were you get course I know 90 degrees 0 hour karzai 90 degrees a 0 answer you just get 1 half which is what we expected but now we actually know the interpolating formula of formula that's good for many years ago and this is the most primitive but basic quantum mechanical calculations that uh that require mechanics by so enormous that we think that this is a remarkable achievement we have some strange theory in where whenever we measure the space along any given access its up class on and nothing in between and yet there's something about this system which doesn't care about the act you can orient the spin along any axis and measurable on any other axes and then getting fractionally answers you get fractional probabilities the result is a fractional from brewery OK other any questions up till now the plot was fortunate to layers of Pancho angle at about 180 degrees goes 0 0 degrees warned I watched the 180 degrees over here it just looks like that saw what we normally insurmountable any question that meaningfully about a single quite a bit but were just how 1 electron we have of over
opt for a loss at while we we we both a cavity impotent some sealed walls of prevent anything from going and going out there we we do find that electrons in there we can measure the electric charge by measuring the or outgoing electric flocks we find it change so now we can measure the of of electric charge for example and the created squad the what if the Paris at that point if we do the same thing to everyone of electron sorter young that's right and then in practice we can 1 of 2 2 things you don't measure a problem but the measure of probability you have to do an experiment repeatedly carry you have to do the same experiment over over you could do it in 1 of the ways you could do was take the same electron and continuously repeat the express so you put your elect try to put a big magnetic field along an axis lineup that electron measure long actress and when you've finished you put back you big electric field along an axis a line up again the experiment over and over and over and over again you have you could do that old or you can get yourself a whole bunch of electrons and who are million identical experiments doing 1 experiment on each electron you should get an you do get the same answer thing you do get the same answer for the probability indicated together million electrons a probability simply means the ratio of the corrupt pounds or within the margin of error of course so with margins up to did the lack of you want but you could you can do an express which you take a million electrons lying them all up along an axis you're pretty Frost lined up along an axis of big magnetic field each 1 and then measures each ones Spain along the Amex's some will be plot some will be Margaret's but the ratio of the number of pluses and minuses will measure the probability for across we all have the same why not OK but if you really want me I'm right but that that's that's yes that's true but let's be revoked Serbia will be purist about said we do the same repeated experiment over not but get your right now what yeah noise awaits and people out there and was numbers and those numbers as the energy Fabian by turning on the magnetic field when you turn on the magnetic field if the spinners in the wrong direction and you have to do a little more work on magnetic field benefits and a right direction energy which is effectively calm hour automated so not limit the number of times you could do experiment 5 years obviously 1st yes that followed but that's what makes me back have pearls bubbly misses and I of course shortly are fresh the idea is that 1 of these days are modified the trick his calculations are not River users there was a lot better off if they are ownership of this way there's another way to prove that answer only depends on the angle between them but that will take us into unitary matrices and things will get beer and symmetries interesting things that we haven't got into the act but I couldn't do we just a simple example I could for example take in the Gobi op suppose it took in br op let's see what we get if we take into br OK so far if it is op and dozens EM eigenvectors stays the same right amateur some arbitrary direction but let's take in eigenvectors me off then this is just 1 0 now the answer is easy it's just this arms their bosses this 0 right server probability amplitude thing use Squire it is just the square root of one's boss M 3 D Portillo we square-rigged get we 1 pause and me all over town but I'm sorry is just the cosine of the angle of end of him would a 3rd axis of 23 years and is direction co-star in this case the direction gross on him so it actually is just co-starring of Wrangell so it will take particular directions and particularly of some of the directions or long access 1 reacts to use our are ministers reversible those which is not a unit matrix that 1st Aussie matrix here this was not a matrix this was absolute value square it means pint complex conjugate where were yes but the truth life yes yup yup 1 of the matrices was just 3 by itself which has eigenvectors 1 0 0 0 1 Rice chores my argument that particular wish to check that that really is the eigenvectors opposing in the 3 years plus born supposing plus 1 and 3 B plus 1 just means that enters along the 3rd axis pointing out 1 sensory would be too old to a divided by 2 was 1 of us just becomes 1 of its kind Warner's RPS but 1 minor send me that would be 0 defend 3 is 1 as would be 0 factor for the special place where in 3 is along with the relaxed there's the eigenvectors just 1 0 was the eigenvectors band of just plain also SINGAPORE go so or were well known on audio you write your right by the door was nothing from the start the guarantee that we will be able to find AER arm find rotational and variants problem so 1 of the things we proved by Billings calculation is that the answer is rotation there if we had the independent reason to believe that cancels rotationally invariant and we just pick an axis of well I just picked this crazy said a free said matrices when I get from yacht yes yes Red Beaulieu were would touching on the question of symmetry namely the symmetry of a system under a patient ended with you and there's a whole subject of symmetry and quite mechanics which will counter which is in interim which is interesting out on that at the front of the that discussions about picking taxis until we of the we tho we have a language to speak about it at the moment are at the moment but just take it as a bong here are a gift from God that the answer came out to be only dependent on the angle between the euro between me and them yes other
books are you here but a of patrol if you go out to as of Nov what the book that I'll tell you with let's wait a little while what's with a while but it is the question of symmetries and water and how symmetries working quite mechanics and Va it's a step that I don't want to do right now because I don't want to get to entanglement getting entangled in their good questions the very good questions but I but I would prefer just to say at this point just lock Walker a draw would pick the right matrices right on his 1st of all for you can we took our grammar will will just difficulty outer layer of Somoza off fee are yacht another important idea Brown there were so mentioned but never really focused part is when you measure a thing you get an answer but then you will leave the system in some new state or the new measure something you go into the system you interact with a new change its stake in general in particular if you measure a particular quantity and you get an answer that answer corresponds summer Eigen value you leave the system in and eigenvectors With that I give you stop if you if you measure the spin-off along Amex's to be up you leave the system in the state with along and axis is up you star with state along an axis you measured it by by putting into the slot magnetic field annual leave it in the state corresponding to the measurement that you made so that's important idea not only is the measurement of the crime filter which like like a polarizer the polarized photons are you measure the polarization of photon by sending it to a polarized but at the same time your also leave it In the state of polarization corresponding to polarize the same idea here that however the electron is less than Eigen state of whatever it is you measured corresponding to the value it 2 measured OK but say all is another interesting sphere on how their can you measure new door single measurement or simultaneously to to measurements that measures both 1 ensued with simultaneously can you go a measurement which measures both of them Ehrenberg doesn't screw up some and the answers the answers were so the reason is very simple the eigenvectors if you measure said no 1 and you get an answer as a signal to you go sir remember the rulers that you leave the system in the 19 state corresponding to add measurement and does not get what you've got here a year not problem there are nor will I get there and no eigenvectors commerce between signal and say they have different eigenvectors they are not commerce are therefore you cannot measure them both simultaneously because you cannot leave them in the state which is both an eigenvectors about and about and that just on all nor eigenvectors which which they share so no it is not possible to measure but the interesting question is what is conditioned the mathematical condition that you could measure to quantities simultaneously and the answer is that they had better how old the same eigenvectors of the eigenvectors of 1 of them had better be exactly the same as the eigenvectors yellow Melbourne because of you measure them illegally Our over the answer has to be I and I give value the Eigen values correspond to distinct eigenvectors and the only way that you could ever measured 2 quantities simultaneously be sure that you get answers is here we share for the same eigenvectors if so matrices share all the same eigenvectors they have a property the property is they commute from this is again something I do want to get to where we're going tonight let me right that was a theorem on the blackboard where South Korean air there were measured while how you how would you measure Sigma axes of How do you measure say next but putting in people want to make few along the x-axis you measure single y by putting even tho magnetic few longer axis well you could put little magnetic field which is both along its axis in the axis but then be measured would you be measuring not X awesome Hawaii but along a 45 degree angle you measuring something different so forward the answer is you'll never think of a way of measuring it but that's that's good because the mathematics says you shouldn't be able to rule on the condition if I have to matrices but they have all common eigenvectors right condition end even look up proverb religious offer really is a one-liner this really is a 1 or maybe 2 lines most of 2 matrices and they have all the same eigenvectors core and how fast are the various eigenvectors death that's our right bland robbery Ms. I eigenvectors this on other major exactly the same argument crews but perhaps different died in use of common knew you are crimes are far from or is the same set of our office are the eigenvectors of both makers then you can measure them simultaneously to court if they have the same eigenvectors there are and how under those circumstances there's a fear that a one-liner researcher says that 80 times B is equal to beat arms or are mathematical terminal World War 80 miners B be minds BAT 0 order that you multiply matrices import multiply matrices support if the matrices Camille that means it doesn't matter which order you write them and then all of the eigenvectors but simultaneous eigenvectors revert to operators that's theorem necessary insufficient conditions for all the eigenvectors to be said to be the same other matrices commute commute means you can interchange fee or another way of writing the same thing mathematical notation miners BAE Mr. just write racket about racket commutator score commutator means it a be minus operators Camille Yukon measurement simultaneously course they have exactly the same ideas as a theorem and as a said it really is a wonderful warning film you know how Aiea acts too small for number and they don't be
acts also multiplies by number so you could easily figure out what the product of a and B is on any vector on any eigenvectors and from that you could include you could make you conclusion so I ate also the proverb because of the because restraint but where do I get to OK there's 1 what you're 1 more about the simple spare and it's a very interesting Iran important fear this what is really quite important it's a Purim about the singles says is no matter what state your right there were vector of awards any better be right Alpha Beta where presumably office star are proposed by strawberry equals 1 normal hours is no matter what state you right barrel is always some direction of space such lack of electrons in this space directed its electrons in the state then the spin along back direction definitely plus another words sector that you right down isn't eigenvectors of somber stigma give an alpha and beta you can find a directions space such that this is an eigenvectors what's right Alpha Beta equals our from Baker any Alpha Beta in this case the problem was not finding the eigenvectors given a you could look for eigenvectors misses reverse problem given the state of an electron can you find the direction along which that electron is definitely not and the answer is yes and the simplest way to prove it is the count in the think of this as equations often they'd off fixed numbers a fixed complex numbers this is an equation for the components in 1 in to win are the simplest way to prove it is simply to count the the number of equations and the number of variables if you have a number of equations is the same as the number of variables basically there are 2 equations 1 for a cop component to mourn for bottom component how many components of an R yes and say now yup knows only too this still variables told real variables because Annex clippers and white square Parsons is quoted as you could 1 another's enters is a unit vector and so the number of variables takes described it ought to angles longitude latitude are sphere so Soweto variables 2 equations 1 Jones an and a number of unknowns number of equations is I check that so there's always end was always in given any that so it will let that says that if you take a few easily you might do the experiment you may electron in some definite state but did not sure what they're actually corresponds to assure you did something with electron end you aren't clear arose somebody you you will you know somebody put magnetic field army and lined up along some or YOU nor whether they did something and they left electrons some state and we would like to find out if there is a definite direction along with 2 spinners are so easily we 1st of all make a whole bunch of replicas of the same Express so we have our hollow bunch of electrons all identically prepared all identically prepared now we measure now we use them and how many of them are 100 zillion we're going to use the 1st 100 million the find out which direction the spinners point and then will check will arrest them and check and make sure that they really are all pointing in the same direction so what you can do is you can imagine the are the first one along some of our our arbitrary axis measure the 1st 100 of them along that acts if we all point up along the axis it sounds like a good bet about there are 4 faxes and then you measure all arrest them and you find more but you might find that a few measure along the z-axis you'll find some optimism that if you measure the 1st 100 of them you find some muffins on change direction change the direction of your Magna will be and check another 100 of them if you still find there was some Arpent some damaging the direction magnet again keep changing the direction of magnet on steel you find the direction where all funded in a test electrons are definitely off then when you found that you will find all the rest they are a thousand may not be a big enough numbers use 10 thousand armed for that's the idea there Is there any state of an electron how you prepare to prepare carefully there will be a direction along which it is definitely pointing in that direction now yes We Can has 2 degrees of freedom right group eager source as for it sounds like for right we have normal we have normalization but we also have freedom to change the we also have the freedom to change the face of his nor the phrase drops out of the equation also born well looking were looking for a were looking for their actions such that the idea that was 1 was searching for some direction values warned his younger here at exactly the right questions attracting variables equations or mail as as many equations as Iraq components of Alpha Beta there are 4 or complex components this equation reduces rainy but then the fact that that Riyadh the overall phase of the wave function Whittaker Malta that our doesn't matter in this equation either was multiplying by both sides the aisle data doesn't matter so really there wrongly tool degrees of freedom In writing bound ADR 80 complex vector from electron 2 degrees of freedom to 1 or 2 with 2 variables he added to independent interesting variables here 2 of them here and so the number of equations is the same as the number not but they say that the overall phase of the wave function was not have or this is called a wave function the overall phase of the state vector does not have a physical significance why not because so because whenever you calculate any physical probability you always want the multiplying it by a complex country multiply by a compote conjured a phase goes away so the have serious there are 2 0 0 variables physically interesting variables that go into the findings of the estate of an electron and then one-to-one correspondence with the tool physical variables that define of directionality in space so you could make a correspondence between the between the components here directions a space that's true of single electrons everything right now I am going to continue because well integrity verify referred
outlets let's try to get entanglement today Bob we get
entanglements we have to think about that is something I went to a school a middle school where I talk about 10 year olds about string theory an extra dimensions of it was kind of interesting teaching them was very much like teaching you guys that how turn-on all was it are the water was then unafraid they ask questions the boys boisterous on the beach it's the people on the move undergraduates who were just shot by rescue question does not on each 1 KSB a brilliant question said the neutrinos is always neutrinos nobles of things it could meet headaches my are really has a say we critic rim again now studying pairs of electrons get the the basic idea and tangle with all used to electrons but does following that I Wannamaker a table table over the action of the Sigma matrices the far-right stigma 1 quarter said market said is the matrix which the changes up narrative it off bad loans and means of interchanges takes out Soon warned on bound equals ought to tool water to adorn well send the 1 of very similar except as an idea Maren my side so I would say with torn up doesn't know what is your a get down again but it gives you damn where I am from consumers who are down gives you might ears firms are we reject that just from matrices and finally stigma 3 worsened with 3 efforts off that what discuss at is an eigenvectors 3 what Eigen value plus 1 voters get up and referee on down years minus down because Trigon vector with might aside this is over this issue abstract are representation of the signal matrices from Abstract Concrete prepared it has the same content the Europe metric structure now let's come parable electrons Ross think about a pair of electrons once again I'm not concerned with that positioned imagine that the nailed down nailed to the wall and the only thing with freedom varied thin spread of electron magnetic moment that carries and now we have to what have 2 of them were interested in the space of state possible configurations of tombs again I can measure was removed for example along with 3rd axis in you lose the measurements along the axis of the 2 spaces calm I could choose the same access just for simplicity I will choose the measure them along restrained access and label them very similar to the way I label the single electron but now we need Till labels suffer example is fact we can write down over state from Iraq on possibilities we have In the 1st entry he will correspond to the 1st electrons and the 2nd entry will correspond the 2nd electron so this means electron No. 1 is up an electron number tours I will never get confused about which electrons which 1 of electrons nailed the blackboard over here the other 1 nailed over here court 1 and To be no confusion about which electrons which Mrs. or draw local electron throughout and then as a state in which electron mourners downed Toronto was up and there was another 1 in which electrons for electron was now in the last found them that's all it was all possibilities that we can get for possibilities to punch through these are before a possible states who's so it evidently dealing obviously with four-dimensional vector space now with 4 basis vectors our and tool sets of quantities 2 sets of observables now to set of observables are of course the spin of the 2 electrons How am I gonna label with 2 spins of electron I can't call them Signora tho because I've already used more with 2 the something else for the 2 components of the spin warm electron so I need another means the first one I will continue to call 6 the 2nd 1 simply going to give another name Howell follows stigma and somehow Barraler photo label the 1st electron spin and just quiet stigma has 3 components as it had before more with doors with 3 the other 1 on discoidal Howell also has 3 components tower 1 tower 2 How do we the spin operators for the 2 electrons how they operate and what did they do to these vectors let any limit teacher now how you think about 1st let's take the 1st electron what does the spin operators in 1 with 2 wins in the 3 each village of these 4 state directors how that act on these 4 state and he is easier answer if you're interested In the 1st former left here you simply turn a blind eye to the 2nd 1 you don't look at it passes the operator sigma which refers to the 1st and has nor what action on the 2nd electron it does nothing so similar acts in mail may not fully the speed of the 1st electron but leaves the 2nd 1 completely on Cheney that's the rule now this will of course is an empirical rule but sounds reasonable that these the 1st electron should only act on the 1st electronic Monday on a quarter of 1st or try or for example to a toward of examples of what what does more rule when acts on the state are down just as arbitrary example well you just turn a blind eye to down to the fact they turn a blind eye I mean we will not change the down for the 2nd electron both look the 1st electron exactly the same thing that said 1 bid on the 1st electron would do it flipped it down solved turning a blind eye to the 2nd electron soon formed makes found makes makes the 1st electronic ban has done nothing the 2nd electron was completely passive signal 1 that ought store on another example just 1st Hussein born on downbound while again it does nothing to the 2nd or charm and puts the 1st electron up they you see I
mean by saying the 2nd electrons passive are now let's ask about what howled now the operators associated with the 2nd electron they also do something of a state vectors this time you turn a blind eye to the 1st charm and allow the operator to act exactly the same way he accepts substitutes how exactly the same way but as it applies army before I do that but we don't want more cases what I've said rainy are at stake and arbitrary case pop up back where's is kept Watson the real not meet up with plus 1 so it does nothing up and sector mourners passed stays where whereabouts in the on back bound for a local here discharge city symmetry 3 takes down down but with reminders so it gives you might Baron another words the 2nd electron goes along for the ride what about the was pals act on the 2nd Elektra argued and the 1st electron goes along for the ride selects the warmer 2 cases let's take tool and act on top that 5 of the sea let that as we go the charter were acting on the 2nd electron down so we go over signature 1 down and gives us miners I find off salt there's just give us minus I plans up up the 1st electron was passive nothing's happened to it the 2nd electron has responded exactly the same way to Powell count to as dictated by this chart yes that's a set of definition of SOsub definitions of what stigma and pound war if you know what stigma and how do all of the various factors independent vectors but you also know what that what they do to any linear combination of so this is a complete specifications how 1 and in the and entire wall components of them act by many vector got a big laugh laughed and laughed a lot the case is not a plum multiplying vector Brian because our warriors for our 2008 was no mean putting it inside are you could multiply a vector by a constant but the constant that's a rotational Forum patients then the patient leave the consul outside effect on he's not just a yard that's a label right right that's right as a labor what's at just labeling the configuration it's not that doesn't make any sense multiply the label by by a number of those on the outside but deal you are the I mean you wouldn't be able to tell from this equation along from this equation along with his body was because of this article about but local various possibilities your fears his power through a trial various things have to condole us restore up up castle tools but see that we have to see count to rule on up killers White times barrel that vii crimes down players I saw for cleaned off the list of em the point is we know what stigma and town due to every 1 of the vectors and so we know completely with the nature of these operators or we don't bother writing down matrices we know what they do that's enough for us ah but now we could invent matrices describing their habits are necessary for the moment may be labeled not so it's just not well it's just multiplication i not physically too much just multiplication of America by a number on it doesn't represent any any rotation in real space that's escape was not represent the rotational space it's just the action of these matrices or matrices do they just the same as they were done at Bill label mark in their are sock now how many the duel over here this over here remarked before Our about the number Ava variables that it takes to specify the States electric and are was to theater was truly independent real variables a report and specifying the state of electronic maybe think it takes a specified 2 electrons the please thank would what book to account was a four-dimensional vector space you can wear Ave pop up down up down there what states could you build you could multiply each 1 of these by a complex number and them altogether that's the most general state of 2 electrons it you could write down the linear combinations of arbitrary arbitrary coefficients subject to the constraints that the sum of the squares Oracle the warm and each phase doesn't matter nominee or to Khairpur grow how jerk 8 for complex numbers yet we we have removed to pressure pressure by war complex numbers that's aid variables a real variables remove prove that gives you 6 right so I'm here there's this interim what possibilities you might have guessed had used said each electron itself convenes on State Beach electron wasn't some stayed each electron requires 2 variables the virus physical state you might have guessed beer source for an armed it seems that these spaces states must somehow be richer it must contain the things which are which cannot be described simply as each electron having its own state because there are just too many variables to many possibilities as the reporter 606 variables 6 0 6 independent real numbers characterizing the state photoelectron system this is angle but 1st let me tell you what corresponds to a pair of electrons each with its own state each with its own state are that you were recognized as Our support state of being electron No. 1 force opposing we got electron No. 1 in the state How far data bound and phone 1 he had to indicate that this is electron warned that's electron 1 state of electron 1 now about electronic to try to rule on the right out tool kinds up because to finds back on
price state of 2 state neither of these are really electron states tho electrons states have have to industries what if electron voters in this state electronic who was in this state then the chill electron state you right down the following year you right as a prop house for 1 hour for work out for 1 hour for Spadea 1 now Troy use which grow bracket Our tool cost data 2 0 now and then that off from there so the following object how format that you find both electrons are our class how awful warned they had 2 0 from the 1st oil from up this accruing from now they pay 1 hour
2 fear 1 firms tool cars down our if you're violated cause Bayer 1 beta till 0 Barham Baron this school electron state this to electrons state which was obtained by combining tool 1 electron states correspondence
correspondence through a state in which each electronic comfortably placed state sought independence traffickers if you wore for example could take such a state you would find that there it is some direction you signal 1 is definitely up remember any electrons state that you could make with laugh and a beta there is always some of us on the direction in which stigma news to Nevada and is up this state has a property With this seemed just using for 1 May 1 and the same way your find a direction space such that signal 1 is definitely positive it'll be the same direction as you would've gotten by just looking at the 1 electron system by itself same thing for the tour electron the 2nd electron there exist sums of some direction of different direction general different directions in which 2nd electron is also pointing along with positive axis you could check these 2 states have the property but this was definitely a lonely an axis of this was definitely along the MX Baxter's then you could check that the thing that you've composed in this way has the same property with respect to more complicated or chances remember a single 1 only acts on the ups passed on a 1st electrons soon with the power to enact 1st electron power on 2nd electrons and you could check that uh electron state has all properties you expect if you simply have to distinct electrons prepared completely separately never talking to each other independently not necessarily in the same state to different states for example we just electron 1 in magnetic field pointing this 1 a chance to win magnetic field pointing this way can't arrest them up and give off the ballpark of tar and that's what this kind would be called up proper state it's a simple props state although always has a pair of directions all long war which each electrons polarized that's the word to say the electron has it definitely up along some polarized along the our axis knowledge of constructor you a state which obviously doesn't have that property which does not have the property some directions along which this is art and it is and tangled state it's 1 does not have the property that you could think of it as 2 separate electrons which is prepared independently each 1 has song state somehow electron somewhat intricately entangled with state-of-the-art charm From born intimately in an interesting week and Pangle this is a very simple 1 electron us art world watch up the other electrons Barry the yield is the right 1st electron bound from from Top Of course Jacopo over square tool to normalize it back said moreover squirrel was not from any role his estate and this state has a property 50 what it means it has the following properties that if you measure electron number tools and find it down electron No. 1 must be up and if you find electron No. 2 opt electron No. 1 was down this state has a funny property video if not state if you have another state the combined system then he would learn something about the 2nd electron by measuring the 1st elect this August A. has a property that if you measure the 2nd electron to be down in the 1st electron was up and if you measure the 1st electron Safran electron opt the 2nd was Our 1st for your drupelet wake her when brought something to you when they profile that every 1 of the matrices as well as How matrices has averaged value 0 In this state land I calculate the average value use em which the operator director steered between the state and state on the same state on 1 side or the other I suppose it would draw supposing the expectation value of all 3 of the figures were 0 that I will also be true of the expectation value of any signal not in will be 0 if the expectation value of sitting in the back by 0 that means that as a half a probability for it to be up along that direction and a half a probability for it to be so if all the expectation values of all 3 psig Missouri Cooper 0 there could be no direction along which certainly is definitely op because along any direction the probability or half a half so let's it would improve that let's take the recent murders and check 1st of all we have to make a broader best all the coefficients are reels of no complex conjugation led with the pillows Querida was like a play any interesting role here but the way must make a corresponding brought back over corresponding brought record also is the sum of 2 charities and there are found paths Barry R. and that's shown up at a security soon warned send and then saying state except the cat version of it Our Baron cost Barak broker nasty Woornack for said attacks on the 1st entry here and it changes related points up and but applaud buzzer changes after that on the 1st entry in America are so similar 1 mom the stayed here gives us our course down down down and Texas down makes it up with precise hero from nothing drop-down across Barak for what seal a product of raw there were Maryland for to show you for the answers 0 because physically different situations where 1 electronic which was up over here is dour here that always means there are far is means a far different labels always correspond up down up up down down down our war mutually orthogonal which our budget assumptions by but it goes together with the idea that these are measurable distinctly different configuration stopped We have a shut down the back up over here but not down not down up over here so these must be orthogonal everything here is a father everything here the inner product is just plain 0 you only get something if something common between the left between the brawl vector kept only where you get something on 0 SC expectation values In 1 What about the expectation value of sigma too well similar to it does about the same thing 1 except that throws in some eyes a minus sign but still resort down here and now not here was only down are so it's still orthogonal this is still final purveyors the expectation value of sigma told 0 expectations by a certain ones equal to 0 the expectation value of sigma is equal to 0 the only 1 left the stigma parade peso it's checks in the 3 what saying
Mercury while gives up does nothing was this want my Assad so just multiplies is formed by minus now let's look at the inner product was still a product of up down up down or 1 stay a product abrupt Alabama the different different situation 0 Paramount blew up down 0 up down which are bound up with Bam Bam up as warm because a minus sign so we found on all 3 expectation values are equal to 0 we've encountered a situation which is impossible if every electron had its own state in the same sentence that would have a state are as if it was just a warm electrons from any elect 1 electron system there was always some actors such that about musical toward the expectation value of Sigma musical 1 intensively in any idea how many eigenvectors to the expectation values just that like sewers always some direction that signal and is able to warn but all 3 components of sitting have 0 expectation value in every city not in expectation value is equal to 0 In every single directions space the probability is a half Hey hey after finding the your drawn the Mr. something no concentrate our single electrons and prepare them a magnetic fields on how we pray magnetic field we found directions where the electron spin pointing in that direction now only have to electrons in this interesting state where measuring 1 tells you something about the other we find something new rules we find that there was no direction for every direction the expectation value of sigma physical 0 equivalently any direction but you measure spent along your fired 50 probability that it's either up or down no bias put up this Obama's along any axis whatever which is quite different than you might have expected if you just have independent electrons prepare them with magnetic fields this is something something that cannot b if the only possible states were the ones that were like just having tool independent electrons called entangled state a set the one-way equally author Pangle is from the fact that you could emerge warned you learn something about the op OK now How on a phenomenon through With larger sublime but this is our 1st experience saying quantum entanglement you were Friday identified for right measure vital for 4 1 gives off but you start off by saying he was not aware of this the measure of of the site 1 the size of the guy I thought was variety that plenty left year statement about 100 back was from so if not system that no matter what direction you put the magnetic field on electron 1 forget electron Norma or what direction you compose the magnetic field you will always find 50 50 probability this is something that would not happen had you created electrons bound independently by the wrong field was not question how he'll create electrons and part of the state over here nor can't that bar the moment apparently in mathematics there are a collection of state vectors are collection of a quantum states which have properties that not all where now they don't have a property work among MobilCom will come back it supposing you measure electron 1 and find it in some state you learned anything about the electron Soviets will be yes there'll be just as you might expect as as a set of measures like trying to you will find your try warned down how orders this from a simple probability point of view well In some sense it's not that all consumption supposing they have no opinion and dying and I take a penny of the dying well suppose for the moment that by the Pentium dimer exactly the same size where can I what difference with my him taken a shake them up and without looking I grab warned Maghreb Yarborough and think far guess what if I look at this 1 I find that a binary instantly know this was a penny likewise if I look at this warned firefighters and some from the bar without any year where Grammy Time Warner's so that's call classical version of entangled probabilities this now although not power it or at 1st sight it doesn't seem terribly are different but they will come forward next time were the differences there is a difference are more CompuAdd but just on a simple probabilistic point of view let's say just measuring the off this bound electrons at 1st sight it seems no Otter van the situation where I have a penny and by a Shaker Marvin Allocca may ask about the probabilities work take a bite your wife 1st of all be 50 50 probability that's Upward down you want them murder which y whatever we look at 1 of them but if I look at 1 that I instantly nor the other 1 only if I'm sure that I bought exactly and the corresponding fingers through here only if I'm sure but prepare to electrons system in this state only if I'm sure that prepares to electron system in this state but what that is we him although he said that with the latest measure for both of us all got his hair Web back followed old Europe that I have ever part that would be part of a Yosef Tuesday got a fire are the product of the whose figures warehouse good example forces a right state OK votes yes but the but would use at Armed award with shares Roark are where TO what was that you said you said that it was part of the or both In 1 case a little more borrowed in 1 measurement you will find their opposite but you could say the yacht the measure them both simultaneously you could measure 1 of them over here and your friend now may have a your your friend them measure the other 1 and then you could communicate menu Yukon come together and talk about what you discover himself for what you can't assure that you discovered a penny you Commissioner when your friend comes over and it tells you what that she discovered there will be a dime on unless something intervene but had what what was said it is a lot has their native of are but Rory yeah I write what will be out of the world but also asked just In short Of all well aware there's not
a lot of last year's at the yard right now I got along wants I can get back for a long time they are not always the or using a lawyer laws of large numbers in many cases it will using the laws of large numbers but at the moment I'm not but in many cases when I say the expectation value of something the expectation value was something that you only measure by doing many many repeated experiments and gathering data but in this very simple case of a penny and I need only do 1 experiment if I do 1 experiment and I find that any final the other person will find find you may want to do this Over and over again you may wanna collect data so you may want do the same experiment over and over and check really works every time I find an hour penny she finds a buyer every time I find dying she finds a penny so we would perhaps like to do this experiment over and over and over again and check that statistically this correlation call a correlation correlation exists between what she measures more I measure it would be necessary to do over an over again and collect statistics about a mile or we should find that every single time we do it it will be a week or and correlation between what 1 person season would be of a person's so entanglement of probabilities is not an artist here that just means you know something about the combined system and that that's something doesn't get disturb our our perhaps as you seperate things the odd thing is you could seperate showed 2 people 1 with a penny in with buying a car off office and Terry with picks for years to get information from 1 of the other nevertheless we instead I looked at my head I know what Oh my friend out there who wore forum she doesn't she lost prior or how does us classical act as Bell's theorem films about it between the situation I've just described and the corresponding quarter but he had a different matter for Our Gabel Rio make it did this you Nevada might sign him with on change everything nothing will change half put from signs are here but there may still have exactly fresh art that's better state of Bahia say make it remember that each electrons itself a magnet or play with the state corresponds to correspond to these magnets being oppositely oriented box were married by without a preference for which direction and so we store are of whichever way Magna's pointing your was poor thing in the opposite direction and that actually happens if you bring to electrons about his electrons together because bringing 1 electron into the field of the other electronic they naturally of 1 electron wants to respond to the other electron and make its magnetic moment the opposite direction so if you brought electrons together and some long configuration is would amid a time and jumped or opposite configuration so all you have to do is bring the 2 electrons together leave them for a while Our photon deal will be emitted and after a while they will be in a state of being in the state because 2 magnets will try or by like each other but does not particular preference for entire life this way this way this way and store where are your have something which is seen in every direction but no matter what direction you measure the space you find the other 1 is in the opposite direction the yacht really them and that's right bread and is bring them together and leave for a while ago going to this so-called Stiglitz state as saying what state of collect after a while no at the moment is no exclusion principle are we gotta think about the exclusion principle for the reason that we definitely put electrons at different places in space if they're in different places in space then there's no issue about them about Our exclusion principle a healthier if you bring them so that they away from here number songs in the Alabama where are you not to bring them on top of each other you just have to bring them close enough with magnetic fields interact with conceal each other and then if the electrons are parallel that's like to magnets which a parallel the north poles repel each other and has more energy than a theory is that parallel and sorrel happened with resume amid a photon somehow come out come out with opposite were polarization and this is what will happen to the most Electra so it's really just using 1 electron as the magnets which with Joe a polarized the other electron flows from the dead there is viewed much on legal or van electron there I are you have to I missed electrons are far apart from each other in the electron beam electrons are very far very far apart God joyous electrons of electron beams of very very Di Luca thing Our electron electrons are very very far apart by comparison with goalie scales inside electron losing E for across or if hit each other what rapture of Tom Williams or triads observers they still don't get all lie close to each other they still will electron a single being pulled still good approximation they don't interact with each other very much U.S. some call repulsion between them but there's very little spin spin interactions between him to form part of deal said the are OK thee magnetic moment has units of white the magnetic moment a magnetic charge pions distance between them a dramatic moment as you're soaring man that's the link which sets scale about how close electrons should be stored lowers the should be within that roughly with dissidents and then in a fairly rapid markets question time scales no matter how far Park electrons are they will eventually settled down to the ground state which is the opposite eventually they were it's a question of how long you willing to wait for the of electrons a far harder than the difference between the energy of 2 parallel warns into to an arrow warns is so tight that the time to do such a long way with photon just very very very long time for that process perhaps so Temple electron beam it just isn't enough time for a fully electromagnetic moments the feel each other in March of 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