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The Theoretical Minimum | Lecture 6

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after it was over there was an issue that is not the last time I didn't understand the question was questioned the kept coming up and I didn't understand where waters until the very end of the year class III who was an interesting and good questions so I wanna go over it because it is important that had to do with the bigger the magnetic field emitting a and on maintaining Mr. photon remember the these Norway we set things up we said Let's take the Hamiltonian the O may go over to move let's say said fast so I give very whose plus 0 may go over to some regular over eigenvalues score even warned the energy the law of the tow eigenvalues it is minus may go over to the Opel warned here's Of course he minors indeed class is plus so that's all energy levels of rejoice energy level diagram energy we would have to eigenvectors be 0 here but that would not be 1 of them we would have won over here on may go over to make I know what happen on the steps we are not we the study this process yet but talk about 4 moment given to states like this it is possible for Adam Moss the Aurora system of energy levels like this to ease a photon emitted photon and go from the Opel warrants a 1 if it's in the opera ever so obviously cannot be far apart doesn't have the energy to do it the upper stated photon and that will have the energy the difference between the upper and lower state so all have the energy of frequencies or energy omega South Africa's year will be frequency but more for your energy of Omega Bond missing setting age Oracle 0 nor before warned by either setting age particle the 1 that's really bar Omega but continue to say Oracle 1 because we're all grownups we are not afraid of setting its particle 1 right West I would move which was nevertheless yes it was 0 what that's correct OK so we drop picture of a full-time her a transition photon being emitted now some said what if this was not pointed or down was 0 . 2 views z-axis Wavell appoint lesson this way along the x-axis and then it into a photo would not he a photon of half the energy namely Omar may go over 2 years no it would not be part of energy may go over to let still could discuss 1st what would happen classically classically if we had a broader a speedy rotary magnetic field that had an energy which are always the typical energy would be called proportional to a cosine of the angle between these axis of the spin axis of the is a magnetic field off the energy of that rotating object would depend on the orientation of the spin wrote the magnetic field and you would be minimum down there were gradually increase that No. 1 No. tool classically and quantum mechanically classically the actual value of course mechanically the expectation value of this spin processes around accessed with frequency omega but also happens classically it's In these classical mechanics notes In porcelain racket section where we studied exactly question was called spilled it was called angular momentum a spinning top car was exactly the same thing is that spilling really is a little electromagnet our magnetic system rotating it will radio in the process of Rome going around with frequency omega it will emit radiation I frequency a leg of the related mail frequent radiation and how much radiation will announce classical it's not emitting photons submitting classical radiation will gradually as it emits radiation until it is there'll be no more radiation how much energy would be great altogether and is said depends on the angle started at a resort up to limit altogether and the amount of energy use proportional essentially Omega if it's pointing down it will no energy pointing horizontally the limit let's say may go over to Soviet continuous possible value on emission of radiation at the same frequency quantum mechanically it can only amid off photo armed with where energy that photon hairs the answer is a definite answer related but different answers if he makes it will mix of photon now energy all omega on the other hand the probability of whether units is not 1 necessarily the fact let's suppose but the initial state His somber how far off course beta down this Alpha and Beta are equal that corresponds to being Orient along the x-axis Alford is 1 and beta 0 pointing out if Alpha 0 and bathe 1 pointing them and up electrons or not speedy and Willie e-mail of photon over may go on May frequency omega of energy omega author was just the upstate yes indeed it would admit Omega was the best this state that's nothing start in superposition states them with probability out star Alpha mix with probability Bader Starbase doesn't it what is it average energy given that as a probability after star offer that he admits with uh well for kind of frequency omega and your program and data Salome that it's nothing What is the average energy and the allergy energy is the same as the classical the average energy is the same as a quest but each individual units a photon on either either image no photons or emits a photon of energy make this was less a question somebody arrest question that a as rough Betsy upshot learned the sources Colonel consistency Kwan McCarrick subclass to really . 9 50 year old child here but not with me yes yes yes what's grow out precisely what goes on at least 2 days direct pens on a life partner of the state which in turn depends on how strongly coupled the yard depends on depends on the magnitude of the dipole moment the stronger the dipole moment magnetic dipole moment Fassero in the festering theatergoers short of a lifetime of the excited state will be its 1st life which sets this scale of exactly what you're asking me precision with which you have to be at exactly cause was a spread in the energy levels spread of energy levels or uncertainty in energy levels and the uncertainty in energy levels In the 1st the on the lifetime of the state so right you're absolutely right is a spread Her its governing body the energy time uncertainty principle were whether we get there from OK is not full-time vision was to greet have apparatus to British Adriatic Sea that's 1 way that's 1 way to do it that's 1 way to do it I decided
not to you think about that as an apparatus because it is more like it didn't satisfy the ruled that Sharma that if you measure a component of spin you will leave the system with exactly that for a fact that the thinking about it I decided that could not qualify as aging you warning love measuring apparatus so we're come with Kanter measurements these measurements a little while but there are assume moved to half an hour primary From OK if there are no more questions I want to move a subject most entanglement I had a few words to say about the collapse of a wave function let's let's just talk about for a moment the idea of collapse on the way we talk about how quantum states involved according to the Schroedinger equation and that is the way they evolve but that's the way they evolve please the events where somebody or something detector or apparatus would just some of the system Nextel system interacts with our the original qualms forever quantum system we have other things in the world including observers including apparatuses end just including or other things here we take away all the other things the system might be left and some state and the system will then evolve according to destroy equation and only maybe bringing back the other thing the other things might be an apparatus with apparatus go apparatus of hiding but there could be our apparatus or something else but let's say it's an apparatus and apparatus measures some specific they were happens doing the chorus Of these interaction with the apparatus is a measurement takes place and the system is left in an ideal state corresponding to the particular How come so for example we might start but suppose we want to sales and our system after a while might be in state some summation or what are a bunch of complex numbers alphasub times eigenvectors of eigenvectors of old where there are with eigenvalue and right OK what's a probability for the outcome of the experiment the system Garcia by involving using assuring equation which started out with although a Schroedinger equation he is all we got to the end of the day and now we make a measurement of how we get some specific the value of warm random them will always be but still be someplace let's say it's warned the Senate measure said they after all right after that the it is eaten state land 7 the cost of the operation of measurement is not governed by the Schroedinger equation of this system is governed by the nature of these measuring process is simply tells you the you come out with 1 of the Eigen values as a measured value and the state vector will be just 1 of these which want ran from random depending on these measures that's funny because you might think look after all would give the terminology terminology is the wave function has collapsed around 1 . is collapse around 1 single Eigen value you can't predict which 1 but all of the structure here with all the other ones becomes irrelevant and only 1 state pops into existence was by now what followers what on got at exactly nothing route to correct description of the measurement process but what does very unsatisfying about it is at door apparatuses are just quantum systems why do we have to rules for the way systems involved what destroying equation and won this arbitrary collapse of the wave function if is but With each if we want to sink quantum mechanically about the whole system of system plus apparatus that's fine we could do it and I hope will get to it tonight in this Schroedinger equation but we can not separator systems and to desist from an apparatus we have to think about the whole quantum system as being system costs apparatus that raises the question immediately right now how you combine systems together with someone who 1 system it could be electron spin and the apparatus we have 2 systems now imagine we have a good quantum theory of the apparatus the apparatus is a physical system quantum quantum theory governed How Willy described the combined system where we take Cook components once a 1 apparatus putting together how we make it into a single quantum system and how it with single point system the we described the process of measurement that is a very confused generations of physicist partly because Boro was very stubborn about really quite right and the firms it was a nice man he was a very nice and very glad I've never met a value was a nice man but the thing is he was also a stubborn man and birth and this particular thing you he was most mostly for height but that he remains of wrongdoing was apparently was ripped off to be around for a while as they will try to explain gloom over and over again you have to think of the combined system and you have to think of the combined quantum system if you want danced in the quantum mechanics of measurement Our you believe that anyway combined system we have 2 systems marrow it's just not I worry about what they are they are 2 systems will get more specific than a moment but let's given names but score 1 of the systems an AT&T a does not could stay in the apparatus but it doesn't have to is just a system a no-cost basis states of system a S sub a S so they could stay the system may but now I have a mean the space of quantum states of the system now we have another system somewhere call a part at some other system with its songs basis states I saw this system a hazard state which are labeled by a basis of states you chose a basis estates in the system a new label them a Aiken go from 1 whatever the number of states Of the many years whatever dimensionality of the space states a system i.e. as states by any for any state of System a if you didn't have would be a superposition of these bases a say the system or I would be a superposition of these basis states I would put the systems together make a single quantum well on their tell you the rulers of your cookbook rule but it's pretty compelling away real things work thank you build a space of states Of the system a body which means a and the RIA together both of them that space of states was called mathematically is what is called the Consumer Product rhetorical Europe means in detail but tens of product of the through space states A. This is a symbol for cancer product tests are now what does cancer probably well there is something very simple it means you construct a no space no mathematical space whose vectors are labeled by a value of they have a value of all another words you pay off the bases vectors all possible pairings 1 from anyone from high and you construct a basis which is labeled a fight if this let's suppose a dimensionality of this place is these AT and the dimensionality of
this phrase is these of all eyes these are the biggest capital a I want these capital a decent capitalized another was the number of orthogonal states each 1 separately then what's the on what these dimensionality of these surprise these 8 times DID you get that 1 from 81 1 from Chinese restaurant 1 from 81 from drink and how many other altogether D 8 friends that's why it's called a product because the dimensionality a product of this was a prime br invention hourly These are the basis states for the tensor product came here labeled with 2 industries now you think of the dozen or so normal basis or so long the following if you take till such vectors they are unless where is another 1 of the bases vectors of a N J is another 1 of the what seal a product between let's take these to be or stick is to be normalized vectors What is the inner product between them and the rule for a tensor product is the inner a product between this and this of course we have to Baker Rector Avatar Norberto tour in order to pick its product but if we made a abroad actor out of this the goods in a product of this 1 rule is that is a product is 0 on unless a disabled the B and i is equal to J. Georgia must match if they don't match these might be a observably different for example of this be was the same as this a but I was not the same as JP than 1 half of the system here would be would erode be who who would be observable this differences associated with observed that differences in the subsystem are sought and then have an overlap be must equal aII J. must equal on observably the same that we are through enters a visibly factors is ours would yes you could take any also normal basis for this 1 and of normal basis for this 1 we are not Paducah begin also normal basis from this 1 many normal basis from this war and put them together this way you get the same result was a matter which basis to pick prepared for a 4th pick basis to be concrete was databases if a are not the same as both BJP theater Prada 0 Bottom right equation for DJ a R is equal to Delta crocodile Delta baby crocodile RUJ if I is not equal to J war a is not equal the this 0 always nonzero of Arias equal to J. & a is equal to be sold these vectors formerly by construction by construction which means you simply assigning values the things now but In a way that Siew for what they these vectors form by definition or by construction also normal basis for the combined system this is and a define bases I would like to have as a defined it let's put it this way the words depend on the basis the construction constructively and burst into applause this is however keeper mind that these bases vectors or not all of the sectors in the tensor product you can then take did he a combination of these basis vectors and there are also a the sector in the tensor product so the general vector and tensor product is of the form some of eye and a powerful it's called some of why a Y a general linear combination with coefficients that depend on what's inside year times the vectors times they suspected that the answer is if you were to have chosen another veteran the basis most of Macy's you have gotten except exactly the same vectors except they would have been written differently it would just been be written in a different basis of the combined system changing bases for the individual systems corresponds to a basis change for a combined pockets let us go where are the idea of a cancer product and will give you a concrete example mount a concrete example is the next most complicated quantum system mainly because there's Morris debatable whether it's the next most complicated the next next most complicated the reason I say that is because toast Britons have 4 basis vectors you could imagine a system with only 3 bases factors such systems exist but but since we're trying to learn how a combined systems let's take toll systems and combined them each of them means a spare sorely have we have on electron nailed down this position over here and has a spring which will call Sigma has 3 components stigma x y z soul singer ex-wife have another 1 here with me in the infrared light colored for awhile of calling this once while this was true but there were many you do it to me in indices have signal 1 x 1 Weiss in the back so I decided to give this 1 another Maine and the Greek letter that comes as the city is town item of excellent it's won English alphabet S 2 yachts from Max former government is how and I simply call this 1 comedy components as how I look how was also always been so also has 3 components at I suppose is not right was for the basis vectors for each of these the op and now basic factors words these states In which said was Z is up or down and the states in which z valve then Our base is that there is a stigma and that are bases vectors for a trial exactly the same there may be in order to indicate wrath readers Our vectors for the system and these electors for the system stigma let's use a slightly different kind of bracket notation Hewitt's user could choose a curly brackets on developed very shortly I just wanna I just want to point out the use of different from news and well jet Burke was the basis vectors for the combined if we put the systems together what are the basis for this the combined system there are 4 of them through time stores for more Ave they are states and with both spins are up Warren 1 I always take the 1st entry here to correspond the stigma and the 2nd entry to correspond to tap of Darragh bound up and down Maryland Basel 4 basis vectors these bases sector
carrying full labels Meryl us stigma label tolerable has the idea said this this is a very reasonable idea yes slows To those who say that she should vote is vector stay the same but the linear combination of bases vectors of describing the system will change so any state in this place can be written as a how up plans are up class how far up down up down costs offered bound up down off costs and how far barrel down down that's Ross general thing Rico right at all Of course I guess it feel question is if you start out with a particularly a combination let's say for example just how far up up both opens up new let them interact Billy Joel happens is course officials will change in fact happened that after a timeout prop up might go out of their marriage the words through spins off might interact such a we just the warning that I would not be a change in what the basis vectors are there would be a change or the Alpha OK so hard watched for another still questioners recommit incidentally 1 although the won't criteria for doing this correctly is that if we take that combined system but then we only experiments on 1 of the half of it go on doing whatever is doing and they're well separated from each other but suppose a well separated from each other how we do experiments on only 1 of their is that we get should be the same that we got it when we didn't even think of it as a combined system if we have a fury olive sigma which we talked about now for 5 weeks I think we all about it not experiments are and now we think of it as half the system composer Sigmund how soon and how far from each other with among their acting with each of substantially we do exactly the same experiment so we are talking about how we should give the same answer so we should put these things together in a way which makes sure that earlier issues the earlier description of either subsystem is still correct now I about proving that I think you get the idea and you'll see how works and drove away we are putting things together does guarantee that OK knowing went to direct we can talk about space states or we can talk about the observables away from the box and the bar profit of our coming toward talk about space estates and they won't talk about with on basis I know I was there to back from a bike why not because of Rio's we left it was as if all they are said a are young supposedly did I suppose and we want the state which was a left for stigma up the so this will be a fixed this would be a state which would be prepared we have till apparatuses Merrill uncomfortable Woodward apparatus was due to be apparatus odd but I don't think that's right with apparatus as 1 that we apply to sit on the other that we applied How will you don't take our apparatus 1 apparatus assume apparatus weak point lead that say the ex-wife of along the X axis and the other 1 that we point along the z-axis and we aren't wind-up discovering that singles along the x-axis with plus 1 and how was along the z-axis also with plus 1 so that the seal was up but Howell was right I will make a state which is so the answer is yes you outside your do but then we will eventually Have a mathematics Paris city is definitely off but How is a linear superposition of easy just take up up class up bag square of square you can think of this on another way you could say
it was think of it as a product of trial of art sorry simmer there right but how now woes right right is up plus battle right right of course so this could also be written as also heightens for a big square bracket around it squarely bracket plus bound but I want normalize innocent Queiroz to downstairs so we just think of it as a product of single are and how along the right direction but lesser saying things yet laughter we're square to pop up Mrs. 1 and woes were of 2 of busses what much of the sensitive question or not figure probably partly due out this kind of state is called a product state as a prostate in which you know exactly what's going on with Sigma mainly the up direction and you know exactly what's going on with how it's in the right direction is a prostate and it's simply 2 separate things which will prepare independently on a product state is by definition a product that's right the most general products state the most jobs this is now Sersen uppermost general products most general products that could be read as follows thought up set down there that's the city north of towel but scored Badeau what's called local systems from Bedford towers call beta made up squarely was made now was us mainly this way single was prepared in this particularly a combination here by holding you detector and some funny angle and this 1 is a response the holding the how detector at some other angle and you had only 1 of these the other you just written this the a written started together you write down proper vessel rules course alternately these rules are the rules because they agree with experiment but pretty plausible rules as a product which also arrived how far they are kind of art class How far off base their of that going to right the rest of the world class house down data class then Barrow now right now of course this it is a form of a general vector but something special about the coal efficiency of the core over coefficients factorize into it and therefore this is a certain sense Special to some extent less Eveleigh can quantify how special now there is less hesitant tensor product this is an element of a tensor product its attends a product of this vector without that so vector drawn a composite system as it a product of a vector describing water system vector describing the strictly speaking attends a product is usually used for the whole space but then you would say this is a product state which is products based announced scorer pence approach or products is general and to see how general areas we can count parameters Gil a general system with general state vector homily parameters independent parameters was described as a hazardous is for complex numbers how far onslaught down of band on I'm not assuming there has a special form just general for numbers for complex numbers it a real numbers but I whether that that's a little too much because we should assume that the state is normalized so that's 1 constraint and will also know the overall phase is not important so you can't parameters you always subtract who won for the old normalization and 1 from phrase used 8 miners was 6 6 real parameters describing a general state of coups formerly here well the state describing the stigma particle best parameters four-wheel parameters alpha opera offered then but then we subtracted same thing here this is normalized state and overall phase doesn't matter this is normalized state its overall phrase doesn't matter altogether four-wheel parameters of the same for real parameters that you might have guessed if you just have to to spend each has 2 parameters necessary write them stake to wield parameters promise could have been me polarization directions of the new axis along which is oriented or so therefore planned this year 0 6 parameters here that's simply proves it that not all states are products that most states are not from states with talk about among products they soon enough Botha this is an awesome products and the prostate we can really think of separate states for each of the particles for each of the subsystems but from more general state you can't really divided Theodore state for 1 subsystem state for system there and tangled entangled in a more elaborate than more Gaza on what forward is entangled entangle the more entangled away where discussed marriage important concept of course OK so that's 2 Spain timely parameters 6 vs. for their entangled right we talk about products states and that's all we know about space estates of this system except for the describe it well with 4 parameters of the product state prostate summer ones for which you could separately say was a estate for 1 system and stayed for the other systems and the stars at him now let's talk about observables now we can't go observables what are the observables for this composite system while they must be the same observables asked for the separate systems by themselves another Lawrence Cigna x y and z and pal ex-wife Missouri where are the things you could measure with 2 0 apparatuses on the same card as we used in our experiments on 1 spirit we 2 them so we have sit next to the wines and how works toward housing let's see if we can figure out how operators what I have operated on the state of the combined start the UConn still gets I'll spell out but he is a rule let's take X unless applies to a state Of these things must stop obviously along with the stigma system reply stopped it gives down over applies to down gears up you could check that that's a fact stigma X is equal to 0 0 1 1 0 this off diagonal character means it takes up down in down this Lewis stars narrowly if we only had 1 subsystem is what does on that combined system stigma X on any vector which begins with pop and has a thing or 2 Utah
which 1 was AU which 1 was on life cycles I Colin what I tell towers can solve similar X where acts on book parties all at Is it up all the 1st thing here and it's completely passes with respect to the 2nd index another warrants it doesn't even notice a 2nd exit leaves it the same doesn't change it but it does whatever it should do on the 1st entry likewise stigma X on their own are a course up are you just take what I did for a single part the single off spin space and you just think of other index as a kind of spectators does goes along for the ride was right thing on the other hand also 1 more soon 1 more Sigma Sigma Z I thought by what was Lizzie daughter gets aware XRR so rosy is 1 minus 1 0 0 it just is backed up with Barry water Seretse shows up buying more about singers eons back 4 Susie now mind this barrel of the skiers minus Maryland How much signal Watterson more similar why does something very much the same stigma exits at Rosaline site but the numerical complex number on a throws around because of the X Mississippi Missouri Sigma X was 1 might sworn and signal was equal to minus IRA 0 0 signal Wyatt also flips opted down and down but frozen with tourists you're OK so that's that's what hours now that's how well can all does exactly the same thing X drive the 1st entry becomes a spectator and only the 2nd entry gets gets acted on so let's Our works through before Powell X on ADD brawl they could be a visionary can be up-or-down was not politics on a Op is equal the AD fans down tower X on a a that zinc auto a cop Leyritz touring cows East pause may opt it was a art parents how's why a barrel is a quota minus a Dallas and you could figure out the actually are over arm of the white components of this straightforward with this definition is quite clear but if you're talking about an observable associated with Sigma that that's how degrees of freedom are completely passive money happens to them the eigenvectors of the components of certain simply products states of the eigenvectors work times any vector for power source for example the eigenvectors the eigenvectors the eigenvectors eigenvectors of let's take Sidnaw y ordinarily the eigenvectors of foreign 1 are for 1 Myers I can't remember which 1 has argued value plus 1 minus 1 Oreos are that means that means the linear combination up close by barrel that's an eigenvectors of signal why if you want the eigenvectors Hawaii what we're talking about the composite that's exactly the same except to put them out and I guess I inherited he say same are sold for example it could be plus Bahai down up or are bound bound by the RIA as a spectator it just goes along for the ride anything which was an eigenvectors of a single system remains an eigenvectors from a single system except you have in the index for the other particles from over a from subsystem that's solos toyed with these definitions as I said it's pretty clear the separate subsystems behave independently quite independently as long as you don't let them interact and and feel each other or to something interesting whichever was wars there are separated caucus terms request what those forces in there said there's which got she should however was normal Segner X steady farm yes all was just or by matrices sailor X is 1 1 0 0 Fox is 1 Lopez 1 0 what happens this terms this is wrong this terms this is warning salt but has gone down likewise if we put 0 here no 1 here the this will become more ones 0 so it just takes 1 0 and flips Sytos those 0 1 and the back-and-forth like until cost you would be right there for this particular our case or just 2 to bar with a off a
shaft of study this doesn't say our but this is a 1 0 0 0 1 0 eigenvectors of X far from survives far from eigenvectors you could get when you Iike listing attacks on this when you get a completely orthogonal state so this is not the rule says this is not the equation that 0 1 0 1 0 0 eigenvectors off eigenvectors is a vector which is kept in the same direction by action of the operator this is as far as we get from the operator I give you something orthogonal and that's OK right after the afterward about this is part of less prior years mutants right right if Burg Crepps grip that case or class so is there any physical fighting operators were not better operation most of the firms that's just the auxiliary air operations in some calculation are over on well also particular special physical meaning but In this particular case what it tells you is 0 1 1 0 or about as far from ideal that was as to get away tells you is firm if you measure Signetics in I state are was this is random as a committee as random as a committee but will go forward young this will be the weather will have year however he used to be what I did what I had a call for it to be Mr. next firms X bread Gore that out less joyously it's little fight was later in the Little right now since were added fight the X cow X then from what vector you elect applies to up start Auerbach does go up where was How duel with how I keep forgetting numbers for the the Wagner after sitting right there just once where wants tower a solid towel and with theirs was do it operates on this 1 only uses formal start this year so had the header on the hour that operates on this 1 we've this well right OK so reducing the X have not act actor with that at times that's how X flips as foreign source are Baroness FOGEI however said Nextel yes Fluxus swan saw that just goes down narrow warned but that's where vows West so the officer order but state How X and acts on Bob whether they had those exactly the same thing saying backs that's why reason some and then our comes in as the other 1 foot solved went how X prime stigma X actually do exactly the same thing as soon next expand how X can't get that hour of sigma X with how x equals 0 no difference between the order of operations let's try for stigma X cows eat to it for certain acts housing and see what we get all I have not proved this because I only exhibited for 1 particular state In order to argue that same X terms how X Eagles cow X stigma X your have to solve that they give the same thing when acting on a basis that it's enough to show that they give the same thing on basis vectors opera that's pretty easy Clark doyou Kanawha on any basis vectors and you find out that the order of operations doesn't matter and that the same is true of you take Sigma Texas Kansas State Housing let's do it again so I take a different that their take some other vector pop down West go down has to work hours and they will go in the opposite order are what does Tauziat duel with suspected here only acts on the 2nd entry E. component so does his plus warned on the pop right so this just gives stigma X thereon off so how's EER Hughes art Strong Writers was right through them mortar Cigna X knew what they're this gets a bit woebegone probably go at this point where Kong starts rebelling what does sit next to work with its towering its described it is slips bound top vote-getters pop up right were still in the opposite order housing sooner X on bound up my 1st stigma X bathrobes down shop so it gives us firms up and then housing he just gives worn with CRP as it is the same thing again this is OCY was acting
on a on a separate indices here it doesn't matter which way order this is very general doesn't matter which states take it doesn't matter which component of a component of figures commuters with any component of towel check that just by knowing how they act right concise saying man anything or Corleone next year said earned an alarm clock variety you desire for something of our signal and soon learned how can they is in in commute the mean of that is that if you have till separate subsystems which are combined in this way these are observables for 1 and the observables for the other simultaneously measures it that secular systems and far apart from each other and you treat them as a single quantum from you expect that measuring while doesn't interfere with measurement of the other 1 and that's what this says that measuring our stigma not preclude measuring of and vise versa Freddy components over so that's since they behave like independent systems to make any measurement on stigma and any measurement on how that observables life or this would be a virtual lock or that is it is not as as it was before you with the of some yes yes a request although observables which are somehow more intimately connected with having both from it yes for example said what kind power we're kind pal next time how Exocet next time how wide that sort of thing from our observables that somehow no more intimately about the combination of both and they know in particular about entanglement if it was a joke but are out all our arms so we at we have always done up till now is all the ingredients which tell you that long as you do step experiments separate operations on the 2 subsystems that they will be Hayes exactly as they would have had the other systems I missed what about father and more on more In particular interstate tour of the system is the combined system the Summerpark straight sets up Probst straight so that each component has its own state vector then the quantum mechanics of each 1 of them is separate from the other they don't interfere with each other they don't know about each other in ways he has worked as prescription insurers the woman not entangled yet laughter parts know we found out is the the products they of 4 parameters we found that also that the full space states has 6 parameters so there are apparently states which are not products Bozo the states which have varying degrees a degree of entanglement of Reich album most entangled state however examined and what weights are sure Volvo and that of their many highly entangle states maximally entangle states are growing more 1 particular 1 my sampling minus heaters are just because I have no better a particular has a particular property which is not all-important at this stage perplexed of the miners my aim is bound up whoa squarish so of course when I was quoted too firms a whole thing now this day interested it has a property dish if you well it has a correlation if you know how was down signal must be out on the other hand if you know the Carroll is up the signal must be there now that was only mathematically going to find out exactly what mathematically this is not about state there's no way that this could be read as a product of 1 state as a somewhat or our differences this case of 2 products that it not proud of how poorly tell how could we tell so but let's consider the expectation value in a state of the components of Let's go back for the 1 subsystems who want system quantum mechanics ones Marxist where we know about that the secular I'm interested in expectation values are in a state figure state and take the expectation value of all 3 components this is the average value From my multiplier rank and is considering them individually can they always Iran the all simultaneously 0 4 for an ordinary state of the art the ordinary 2 parameter
family of states of a single system to reap bigger voters register is a dollar cannot all be 0 or here's a reason on well the reason is given any state there is always a direct in space three-dimensional space the direction for which similar in that direction component of stigma in that direction is definite and equaled the plus 1 given any side as Oracle in the beginning of a lecture or maybe before lecture Gillard any state there is some linear combination of signatures which is definitely a coca plus 1 In that case of I find that particular combination of sooner and extra boxers in fact particularly combination its expectation value can be 0 0 because it's definitely Opel almost that direction so the answer is given all state always some direction scores Nevada and that depends on sigh it depends on sigh about and for which this is not only a 0 but Siegel the plus 1 the average average I'll sit about it is of course soon about it Best and that's not only has no ambiguity about the averages just equal the value of the however is not the be so there was always some end price for any sort there exist and and such bad this is true therefore I cannot be sure that all of these concerns previously 0 for a state of a single spins the you can measure more simultaneously but if I give you a state I tell you the state then I can lend you couldn't tell me what direction of space should you oriented detector so that the answer comes our plus 1 could have Americans give you off OK so what direction if I gave you are what their actions said new pool oriented z-axis what I gave you plus down where square of what usage your interim detector make sure you get plus 1 along the x-axis right where I gave you are plus I down From some of the some leave the class warrior this wife they'll freer which 1 give it any combination there are not directions but you could find the direction for which made in that direction is definitely plus 1 and therefore it it cannot be that all of these expectation values were all 0 White because you know the stigma back in his equal to plus 1 unless of course just a linear combination of these focus those asserted it's impossible if we have just a state of war and it's been isolated by itself and prepared somehow are however it prepared there exists a direction in which spins definite and not equal to 0 in on or about the state what about the state by now let's take a stigma and asked whether any of the components of cinema Have a non-zero expectation value in this state why we're shocked really answer right now we are still there all 3 components of stigma have Zero expectation value in this state that means the state cannot possibly be product a proxy is 1 in which each component behaves as if it were just a state of the subsystem by itself this is going to behave in a way that no simple props possibly behaved the findings so let's do it let's do it there OK let's put the square root of 2 eras for that that's not important thing we have on their Myers back up we take the expectation value so that means we have to make a sandwich the sandwich which will have up down minors bound up and little here would a process for did the same with how it's a mail so let's put saying busy 1st West puts it was here and see what we get 1st with act was listed Missouri State Lizzie give went that them it just use up down right right of this just gives up down or was was a key 1 of its discover here neither is bound up with uranium miners surplus the uranium miners miners killed here and that's the enough product for spurts some racket around the city in a product with of down minors off OK let's calculated what's down up with their own 0 0 0 what's down but without 1 so yeah 1 was down up without them sums of Ron Arad earlier this should be a
paradise of them focus was go back up down up down as 1 of them down up a 0 but were bogged down dropped for the down 0 or bound up without merit 1 but as a Simitis sign here this is orthogonal this year product 0 and the expectation value Of said Missouri is equal to 0 removes it is state not state but in this state 0 let's try stigma X let's try similar acts like Cecilia Placidyl XES Gotsis build our sandwich again assigned here know what happens with sigma x equals or Brown who of them Is it flips the ban say Exs Flipper and referred the 1st entry years of 4 of the 1st entry to entry drop we're taking me a lot prop With his vector over here so let's say are the user anything in common over here with something in common you know so little weight given a product bound up with Paramount 0 etc again this is 0 so segment Exs equals 0 expectations OK what I'll signal wiII host Cosimo more so why it is also a flip-flop operator this puts up the Down down thought although I I don't remember which will the be Igor was it fewer Sierra are here some summary or fat however does flips this up to down get down own and puts the stand up so you're not again no in a product 0 so there we are all 3 components have 0 expectation value but that's funny because we cool well we didn't prove by told you have you I think any any new state a brief singles there is never the case that free swimmers have 0 expectations about what lesson the lesson is there are states of the composted system that can in no way he ever be thought of as separated states the 2 subsystems that's what entanglement ah sir example of entanglement Jong will is surely thought that if you do that us that Luo said part the year that's correct that's correct well few Charlotte for the singles spending by itself and the other index just goes along for the ride so however derived its 401 spin by itself you do exactly the same derivation except for a resource spectator index the ride along with so sold E-II it's obvious that 0 1 subsystem myself there will be true for a composite what's not obvious is that it's true for one-stop system or young vessel exercise at an exercise the shell the shoulder all 3 components of this cannot agree on expectation values cannot be 0 is always a direction in which its maximum shall not just I'm very natural state are brought to arm if you'll have to spend interacting with each other they will have very young and particularly if their little magnetic moments than they will have an interaction with each other which will prefer to make them line now this is a buyer lined up their own this is in dire lined up and in fact this combination here has the lowest energy this combination was the lowest energy it has less energy than I passed down bring peace to spend together and you wait a while radiate a photon eventually and coming to the state so it's actually novel harder to get this kind of state to have it easy to get described state and our world will discuss that would talk about actions but it's a difficult situation that this is what we our result detectors spins put together and let them Contel equilibrium by radiating a photon 1 and charm 1st or close to bring together 2nd of all how big magnetic dipoles of the spin Our has do with the strength of coupling of electromagnetic field so what ordinary laboratories to Georgia her ill at the electrons are close enough together that they will radiator for cars not too long a time to work to achieve this so-called single expect this is causing what they my you right now for there are 4 races vectors altogether the reason that this was call the same with 3 others which of the triplet he our well and that now at the moment but eventually I'll tell you can't go on if you think of his little angular momenta and put them down my is bound up with total angular momentum 0 if you put them back up plus down the your momentum as 1 man energy is often proportional to angular momentum so war but ruler King Kong about eventual can't so it's a fair question but the hour let's not try answer right minute it chooses is fossils she should basis search 1 basis for you save Associates of part of our state borders table where there are no similar triangle sizzling variant notion the answer is yes yeah suggest is a very definite quantity that you could calculate coldly entanglement entropy which measures the strength of entanglement which is not dependent on any particular bases now I might tell you right now this state OK this state does have a special features abound plus them up wouldn't have supposedly it took this state and this is a good homework assignment on then downed individually from both 1st and power can be rewritten in terms of left and right for example are off his left plus right down to his left miners right whatever square to toward there but you I'm OK I suppose an I write this stage I Pulliam everyplace barely left minus right In the process up left plus right what'll I get out of this and very simple I get left to right minus right left suppose I rewrite it in terms of Elian Alberta where I get in our How
would you so what the structure that this state has major event is actually the same in every basis every basis that looks the same and that would not be true for up them plus down up
down up and there were they were they would there be with Our OK back there we get mixed up with other states that are 3 other states is 1 of 3 others 3 others are a or obviously there were 3 of and they links up with each other when you change the the bases but this obvious is obvious my let's still unless they surely be lasting remarked that progress on lets us now about 1 of these observables which not observable left to be identified with I subsystem but which is somehow got to to do with box and let's look at its expectations are you a very simple example let's take who take the expectation value of the product of city plans powers will do that 1 would lose next time pal X and maybe next time how y where we'll do what will try some examples but said the expectation value of this To do this we make a sandwich and our Our summer Bob bound look sorry but now I see for the square it to and Rizzo set this group analysis on Africa 0 0 Source are really worried at the numerical number Award for us squarish end and here we have up there own Myers that also over squarish do so in fact I can take a supporter of Toulouse and for all factor of war have is the expectation value in the singlet state singlet state we take signal a Z climbs powers let's see what we get 1st of all I would say Housing Act was doing a on August Z doesn't slip it just gives you a song I which is class for this 1 and my ears for this 1 so we could remove the Tozzi just by putting a minus sign here right that right Ms. 2080 Tauziat acts on their own to do backed summed up to give up so has no effect on this turned and it seems is a sign of this thing theater right now not carry my now what I'll Missouri what does it do it flips the sign no it doesn't slip it gives class on this 1 and my on this won't solve changes assigned this 1 OK now we have to put together we have on their own Art Gallery with assigned to warned we have down with their up my Warren cross turns give nothing up down with down up was nothing down not up that gives nothing there are 2 terms you get a tool over to my store Eagles minus 1 What does is this is kind of interesting on the expectation value of sigma X by itself was 0 the expectation value of how X by itself was 0 the expectation value of sigma next tongues power X is minus 1 a ceiling a Y look at the stadium In this component over here obviously those sorry EC as soon as you were talking about Soon we complicit in both contributions he said nosy Missouri for this Peace is minus 1 wires that was from up this was down as soon as it is a plus for this once it was used minus this want a product is negative the product is negative 1 for this 1 also 2nd think about this state is an ideal director of singers NYTimes powers what's the value minus 1 so factor state a state in which a few measured the pro-Nazi Lizzie cows you get minus 1 every single time so the expectation value was minus 1 nevertheless the expectation value of the individual Max surfers disease and housing is 0 so this is a this is beginning to to 0 get some some of the flavor what entangled state years he is simply doesn't behaved like you might expect a separate systems which are just independent of always been independent are other behaved let's try this is unbearably legally boring there were anyway across miners who go back a little salad but instead of Foote's assessing with zeal tools what's through our next our acts are so similar X times how exits easy to see about how X the 2nd entry cigna X flips the 1st entry aren't so this object he simply flips entries without changing any signs of foot no entry all over the this down gap so what does it do it looks like it Hey did Segers these 2 right and the changes is to make us for miners in from class now what seemed a product Bob down with my up down minus 1
barrel not mine is bound up with our Rob minus 1 again and again minus 2 0 0 2 vehicles like this 1 same exact thing as a would you guess about Single White House because nothing special about the Y axis and also has an expectation value equal mines warned so why this again this telling you very clearly this can't be a proxy for a proxy state the expectation value of a product like this would be the product of expectations used out what's going here this is a highly entangled state by was tangos you could get a lot of good play he got there so I thought that was a far cry from Thailand is you in this case remembers no you don't think of not general in general it's just a motor operator if it's her mission some other observable would measure but this case since also figures commute with all tower was you could think about measuring the Bulls simultaneously on this fact is you measures both simultaneously you will simply discovered the obvious fact that they are oppositely oriented but still 1 more let's since I have worked it out and aboard a worker right now let's those power are sitting next Pallas those cities acts of sewer adhere to but still stood with X France Pallas see what we get I don't know the answer of sorts I have a suspicion farther north was book it will find out what happens when Tauziat acts on here how's changes Assad right housing just as valued minus 1 of sup baguette plus 1 so the OIC of them around yes you're right good now OK so how season down here and makes of minors it's easy up here and leaves along OK however said XVO next flips the the 1st entry and so it gives minus down my hair starts flips minors up Brooke we have to take the product With this priority here well nothing over here a match is anything over here so we see that the answer in this case 0 2 0 after the expectation value of X Prize how Z all ADD non equal that spectacle 0 was saying well I'm not there remains the main reason I did it curious what his horse by also did it just as an illustration of the machinery of calculating expectation but these ones tell you clear things what they tell you is that in a very definite stands these Any Given component of stigma and power x component or the Y component orders component oppositely oriented on it's obvious here but the ziti component their oppositely oriented how is opposite this this is opposite to this so when any of the component state vectors here their oppositely oriented but we also found something else out we found that their oppositely oriented in the X places for a white bases anywhere the the bases of you measure any 1 of the springs any component of the Oxford basically it automatically tells you what the other 1 hears this is telling you it's telling you that the product is always equal the minus warned in fact this is an ideal vector of the product of any of these products it tells you if you measure any component for 1 of em back you will get exactly the opposite if you measure up for Ottawa it's a correlation has a correlation statistical correlation there if you measure 1 of the components of spin stigma then that can also be same components for the other 1 of officers doesn't tell you much if you measure the x component of this bill and the Y component of Aspen that doesn't have March but does tell you if you measure the same component of either of them they will come out opposite now the no there was a plot issue was is referred to last revision is a bit of that city's said voters as Clark did not require states Bell dope worth were entangled states uptown runs down exact Ruiz dealt with your lady this is the almost daily interaction between who was board now he is a statistical correlation yes it's a statistical correlation was a synthesis of her book at this level the sum of its a similar as a review of observation of the where phrases I have and they have a penny of buying the pocket the landmark Parker taken out don't look at them without looking at them they want Alison warned a Bob and I Bob take offers a go or in Europe where you were then this could be the case that have Alice looks at her calling she instantly knows what Bob scoring his right she sees if I got by his plenty he I penny by are nuts self is not surprising is nothing in the past they would together some
correlation was established was a correlation correlation was a correlation that if Alice has volume and Bob Harris Penney and vise versa nor the possibilities were possible so was a correlation between the warehouses Corning Bob scorned was by car if call heads and tails plus a lot about plus 1 minus 1 there if Alice's was passed 1 bomb was minus 1 but the opposite is also possibly don't know in advance and yet warehouses far away and barbers four-way Alice looks Kerkorian and she instantly Nova barbers I assure you that those not violate the the special theory wrote and discuss with this column of correlation is the score Einstein Rosen the Boesky correlation all entanglement bombs Is it a quantum version of that same nickel and dime story Penney and story he did not yet know they there was certainly a difference was certainly a difference but just the idea that by looking at something a warm place to instantly know something you didn't know a 2nd ago about the other place that's just statistical correlation to tremendously common thing physics nothing special about the question is what is it that special and different about about the triangle and thus could John Bell found something interesting he did find none locality of talk next time about my 1 9 localities means in physics and what it was exactly the John Bell found on but on the telly right now it was not normal tell it looks like normal cowardly and the reason is because you trying to apply classical reasoning with with John Bell said is you can apply classical reasoning the quantum systems that's what King quantum systems are quite mechanical and if you try to apply classical reasoning Europe defying inconsistencies I can't say it another way of Korea in other ways His away yet on imagine you have a computer when you try to stimulate on that computer the quantum mechanics of a single spent his away with simulated you would Be able to instruct the computer Orient its on apparatus and arbitrary direction words you would put in quart mixture of direction when you put the court this direction the computer would those some calculations appealed for random number generator and spit out a plus 1 or minus 1 who would be an algorithm that would say that if you do it over again immediately the deal spit out the next day number again and if you do it again you'll spell out the same number again but you might say right now let me make it more interesting he is a victim of mathematical magnetic field which allows us process what would you do it would solve the struggle the computer would stall the Schroedinger equation trying now what the state waters after a little bit of time calculate the probabilities for up and down or whatever you decide the measures calculate the probabilities appeal to the random number generator and spit out a number of that as a statistically exactly what you should get on the the average From quantum mechanics somebody who came to this computer and started playing with it doing these experiments rotating is apparatus in different directions our turning on magnetic fields of other directions waiting vary in amount of time and then telling the computer measure this or that component would reproduce exactly quantum mechanics at least to the extent that random number generators of possible the exact possible were doing pretty good random numbers so that would be a quantity that would be a quantum computer that would be classical computer simulating quantum behavior where somebody doing a little experiment could be pulled into think the thinking it was actually dealing with a single electron inside the computer and doing quite express couldn't tell difference now questioners could you do the same thing for a toast bears in particular I want to do the same thing with those plans and I want to do in such a way that 1 of the spins it will this data is located in 1 computer and although data for the others and is located in the other computer makin computers together have wires connecting and so forth for wire store arbitrary things and they cut the wires and separate can I stimulate the quantum mechanics of the 2 particles in such a way that it would appear that 1 of the computers contained 1 particle and the other computer contained the other particles Hernandez are classical computer classical computers and the answer is all the way you could do it would be have wires connecting the computers at all times you could not do it on the last the computers were capable of interacting directly the shortage but that does not need that the system is nonlocal nonlocal means that signals could be propagated from 1 place to another I telling you was you simply can't simulate quantum mechanics in a local way by classical computers the logic of quantum mechanics is different it's simply does not allow itself to be a state by a computer on less that computer is connected that over large distances and everything in the world core mechanics manages to do the same thing without having wires connecting however you could ask the question that constitute what physicists call none locality I you were of cold normal does not not constitute anyway the ability to serve the rest for words former borrower bookable will be not is fundamentally inability or stimulate quantum mechanics per classical computers on us request who brought the among local with more please
visit Stanford Daddy deal
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Metadaten

Formale Metadaten

Titel The Theoretical Minimum | Lecture 6
Serientitel Lecture Collection | The Theoretical Minimum: Quantum Mechanics
Teil 6
Anzahl der Teile 10
Autor Susskind, Leonard
Lizenz CC-Namensnennung 3.0 Deutschland:
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DOI 10.5446/14999
Herausgeber Stanford University
Erscheinungsjahr 2012
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Physik
Abstract (February 13, 2012) Leonard Susskind starts the class by answering a question that arose in the last lecture about photons and the energy at different states and then continues with the topic of entanglement.

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