Quantum Entanglements, Part 1 | Lecture 2

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Quantum Entanglements, Part 1 | Lecture 2
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Lecture 2 of Leonard Susskind's course concentrating on Quantum Entanglements (Part 1, Fall 2006). Recorded October 2, 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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climate we haven't talked about quantum mechanics we've talked about classical physics in a world where will we can call the state space space states is discreet with talked about hopping around from 1 state to another and now our information is counted in Bates how with time you could jump from 1 state to another and so forth that's all classical physics or physically based on classical logic
classical logic means that the space is
a bully space available in space spaces not all that means is a bunch of points some space and that each point represents a state and all of the logic classical logic classical logic of Aristotle understood that the Newton would've understood and so forth where quantum mechanics makes use of entirely new kind of logic which is in a very basic way different told illustrated I like to start with basic idea 1st of all of this single quantum bake Q we talk about the classical bait which is just heads tails type distinction here we can error I have my prof already this is different than and point up that's 1 state of pointed bound that's another state that's the two-state system her tourist it nothing in between the song nothing in between that lets the basic classical bit you grow up or down with nothing in between Our nuclease many gets West fund you get many bits to bits 3 bids for whatever and pretty much all of physics anything in physics can be represent there are at least approximated by a system of of classical bits and think classical physics quantum physics the concept of it is very very different the concept of a bit of information here that so we wanna talk about today the basic simplest example is the speed of an electron now we don't have to know very much about what's been means does it have something to do with spinning or not this is not very important what it is important is that every electron has associated with it a vector now we really get the big trouble because there are 2 things that a wind-up calling vectors 1 of them is abstract vector spaces which come to a little while and the other is that dude In the ordinary space that there I did this I took with me just illustrate the idea that there are you know order that is an ordinary space that's an object Elaine than the direction and I put nail in the end he it they indicate the arrowworm so it you know which way point fingers are pointing at me pointing up a quarter of a Roman that's your naive elementary physics concept of a vector of things would never space that has a direction of course it also has components for example the vector here has a component in a horizontal direction as a component of the vertical direction altogether if pointing at 9 direction so that's 1 concept of a vector a point in space with Wayne and the direction now there's another thing we recall vectors which is a more abstract concept and is not necessarily in general will not be reforming the things In space at abstract math I just wish it was a different word for it really talking about vector spaces I think sometimes when I teach quantum mechanics and make opened word for 1 or the other of them Polish nectar or something an hour but it never really works so we have to keep In mine when I use the word sometimes maybe I should just make the word yes that's a good idea let's just call this an arrow you know how long that last before cycling that there are not too long refers an now we spin on an electron is AT & Harold was Goh said Dexter I can't help but now I cannot I can't do it at inequality Becker recalled that there can help myself Alaska less spatial there have a special special that door means 1 ordinary space so my sister's electron overheated electron has attached to it a mathematical vector it's not a rod-like theirs but a sense of directionality really what it is is the direction twist but just think of it as every electron has attached to it a vector which can point to you might think in a new direction space some sense can point to a new direction space in a fact that there could also be thought of as a magnet it can be thought of as a bar magnet with loss poll and South Pole is the North Pole with a mail the South Pole was over here every electron is a little mad that and could make the amount the strength of the magnetic which is measured in some units which I forget what they called strengthens magnitude of arms His particularly number so that means all electrons there space vectors that go with that in the year of the of the core the magnetic moment the magnetic moments are all the same late may point in different directions but they're all the same way we can simply imagined that they are our objects all the same rate or common line of different direction is now let's talk about ordinary classical magnets I won't talk about the concept of preparing the state and detecting a state we still thinking very classically for the moment everything explained he was completely with classical logic so here's our electron has attached to it and Arel with the North Pole and a southpaw let's say I want to prepare that electrons in a configuration with its Matt where it magnetic moment so that pointing his North there doesn't stay that Stanford's direction it stands of the North Pole I want to have the North Pole I want a magnet to be appointed vertically upward or prepared that way how I do it OK what you do is you simply create a large magnetic field for example if you put a new create a magnet the pair could create a magnet for as an electromagnet if you like and current make an electromagnet with an offer India South Pole over here and electron are supposed to be in the magnetic field and what will happen to them will bar magnet well what actually will happen too little bar magnet if you know anything about bar magnets will magnetic fields is it will preset so it will not just jump up interim right position now West Berkshire instead what it will do is real process like this around magnetic field everything imagines in a vacuum because of friction or anything like that electron will process around like this that's what that's will bar magnet with durable but still unrelated you will get rid of that processional energy anybody know how it gets rid of the processional energy bill radiates somewhat radiation radiates a little bit of of electromagnetic radiation or a lot depending on the circumstances and the size of a magnetic field and so forth will radiate away its energy and common best energetic configuration which is pointing straight up with the North Pole magnet is pointing toward the South Pole but with the North Pole of electron is pointing toward the south pole of Our of magnets only which part of the draw so that's the preparation and would do it with a very very big magnetic field who you radiate that energy quickly and very quickly come through equilibrium with 3 North Pole of electron pointing vertically upward if you like if we don't get was use were entangled but and brought over the question of where the electron there's just imagine in your head that I hold the electron absolutely fixed in position so the only thing they can do anything interesting is the EU is the direction of the magnetic field OK so after a small amount of time that after that electron will be pointing in the art direction we could if we likely could prepare the electron and the different
configurations by rotating the magnetic poles for something like this South North we could prepare the electron initially saw that for Bingham et direction so we start with 8 electrons pointing in some direction and now we want and measure we want to do our measurements to find out which direction the Scissor Sisters this is a 2nd kind of experiment it's the detection was 1st was a preparation then only do something silly electron whatever it is and next we want to find out who wanted be head which direction electrons pointing it is pointing in that direction is pointing and some of the direction mainly we could ask what's its angle relative the magnetic field and the answer that question we do it exactly the same thing that we had to to be part of the prepared electron we stick it in the large magnetic field and wait for it to radiate some radiation now if this is all a physics for classical and this will really a perfectly classical set a classical composer two-year instead of a quantum mechanical electronic Lyndon B. of again be electron will eventually point way itself but 1 measure of the Englewood be how much radiation needed for example was almost perfectly upward to begin where before almost perfectly upward they only has a little bit of extra energy perfectly upward as the energy when North post pointing the South Pole That's the minimum energy of that they're a little pointer rotated away a little bit you give it a little bit of magnetic energy costs hefty put into the lifting let it come to rest again it does so by giving off that little bit of energy if you are more extreme situations where you point of electrons in some horizontally and by the time it got vertical it would be made even more energy finally he pointed straight down it would limit the maximum amount of energy in the form of radiation but and so you could measure at least angle relative to the magnetic field by measuring how much radiation comes out and the answer would be a nice continuous functions Of the angle of electron so that when they hang Sony electron was pointing up no energy comes out of angle horizontally energy emitted if the angle 0 that's where pointing straight up no energy FE angle is downward maximum energy and then as you turnover always backed 0 energy himself forth so you will measure them the same way the EU prepared you measure the spin the spin direction of electron by letting it amid metaphor by letting and make some radiation Excuse me and camping up the America radiation now I've told you completely the wrong story for a real electron this is not the way it works this is not the way it works a genuine or something else happens so you know what it is and it is very weird really does illustrate the weirdness of the quantum world 1st of all no matter which way you create electronic here I won't even try it give a classical pictures I try to give classical picture I will produce story you do something of electrons and you put it over here with every year due to the electron but 1 thing you could do to the electron is initially prepared in some funny Engel born earlier due to the electron you put over here you turn on the magnetic field and 1 of 2 things happens only 2 things can happen 1 is no photon no electromagnetic radiation the other is that 1 . 1 photon of electromagnetic radiation of very particular frequency it's a particular frequency that corresponds to energy of jumping from Ban thought wherever that energy is you commit a full of just exactly that energy and no other energy so this seemed to be it's almost as if the electron had only 2 possible configurations you jumbo up you don't know what you've done to it you stick it in here and it seems is only 2 things that can happen 1 is that it behaves as if all pointing up which case it gives off no photon the other is that it behaves as if a war pointing downward and gives off the and those are the only 2 things and that they can happen but depending on the use of that seems a little weird because supposing you prepare elect charmed by putting it into magnetic field that would have the effect of freezing it into any direction let's say a 45 degrees are even better at 90 degrees relative worse horizontal then we would think for ourselves what I electron is now pointing in some other direction now when I stick it into the vertical four-year-old you would think that the response would be something different and would just plain up or down but not the response is always just 1 of 2 responses it does give off a photon or doesn't always of the same energy so it seems that in some sense there only 2 states for the electron either it pointing up was pointing their but later met where about a few put the electrons into Haiti on magnetic field pointing in a different direction many of those defined that they're all these 2 states for the electron the pointing this way appointing that way some things very confusing states the electron actually have only always all only upward could you make a point in some other direction and if you don't make a point some other direction how come when you stick it in the lead up that a detective that detects direction the up their direction but you only get 2 possibilities how come there isn't a continuum of possibilities in between that's a puzzle that's 1 of the puzzles applied mechanics that behaves as if the only 2 states up or down when you measure it but you can nevertheless prepared that electron pointing in any direction by using a magnetic field and the appropriate action that the 1 thing go that is true is if you started with electronic pointing upward but suppose we created the electron pointing upward by putting it in his very strong magnetic field shut off the magnetic field for a while and then turned it back on the answer is we will get no photons electrons pointing up supposing we 1st create electron in some other way directions pointing along a 45 degree axis here and then we do the vertical experiment we turn on the vertical magnetic field we will even discovered that its upward down for the probability distribution it will not be definitely opt for definitely down we can do this experiment repeatedly all over and over and over many many times identically sometimes we will find a photon emitted sometimes we walk the more we electron was initially pointing vertically upward the less the probability is that we will admit that single photon maximum probability for forbidding for forum will be if we were originally pointing down somewhere between the probability will be a hat for Opel can't be admitted Our fact that's exactly the
case if the the original electron were prepared horizontally by a horizontal set up and then put into the vertical field it will be a probability of a half that emits a photons and a half that it doesn't far can't sell the something extraordinary going on here in some sense it seems the only tow configurations every time you measure that electron you will see the finds out boards Darren nothing else nothing in between but there are situations where it will be probabilistically if you do the same thing many many times you'll find as a probability distribution and that probability distribution our which has some memory of which way you originally created the faux pas so quantum bid is a confusing thing they will say has only 2 states up or down but then you could make a point in some other direction our so that it looks like this other states possible again when you measure it you only 2 possibilities as too confusing entity quantum bit Alta any questions of our now said if a man like that not that another man megabits a patch of electron back but all yet I read you but not yet beef while the right as we press with small but I know I am I would at some of this stuff they are but let's still wonder time and then will come to the 2 electrons system to elect on system the 1 that's really interesting because that's where entanglements happened so let's what's so not try that when you get those steps will forget when questioned Is the he added duty Alistair differ while do there were a lot of God is no way that we won experiment did you can accumulate a probability distribution yeah right now jet fuel up here and there were fears that other terror in that yes yes it Yukon prepare a lot of electrons at once but there but the question or something like that buzzing you prepare it this way and then measures and then measured the same 1 again and then measure the same not once you discovered this was not the 1st time it will be up every time afterward and so what you will discover in the 2nd experiment the 3rd experiment What experiment no full so wants once the the experiment has fixed it To be up that's it it has no memory anymore of love of its no-hitters young there is waiting you want to think that the interests as well as well later as 1 way of preparing for the usual way EAA about itself well on talking about an lecture perhaps which is in some potential which nails a position we don't upward of moving with just would just concentrating now purely on the spin of electrons so imagine driving and rule the heart of the electrons and Darren nailing it to the blackboard it still has the ability of fruits Pinto do you want is a letter you sit sit he also is it cost of both have rockets were rare the experiment so a taken electron which is there originally when I there anywhere like without the the magnetic field being now want to measure that electronic so I vary quickly turned on our very large magnetic field how you do that emits an electromagnetic very easy you flip a switch and all of a sudden there's a large electric field star magnetic field pointing upward that electric that magnetic field was large enough it will very quickly cause electron radio OK so I would say would not wanna be thinking about turning on the field to do measure what to do euro's food preparation I the bed then the high end of the leash then got a better use direct turning mobile device you start Our South the bottom where South belongings and north in the top when North belongings than the electron point down eventually now you remove the magnetic field you remove the magnetic field but suddenly reversed this would just turn off the magnet and reverse the polarity you could do that by by reversing the current and electromagnet so that suddenly the South Pole was on top of the North Pole was on the bottom of the electron it was bad loans Emery short amount of time it'll flip up and give off a photo was at the question of how a man and a probability of as 1 yup the probability that it will give off yes in that circumstance the probability that will give offer for cars 1 if on the other hand is for perfectly aligned and Europe's and caddie wipers angle and the probability would be less than 1 and sows probabilities which constitute the experiment would not be as large as it is a long shot that doesn't get electrically charged has electric charge turning now turning on turning on me on electric current 3rd going through these magnets you'll have to do a little more the poor little more energy into a magnet because because electrons another words original the energy comes From the energy that you are but you used to turn on a view on what is he is a plus point that's our the language is probably wrong is what I can tell you if you make the magnetic fields sufficiently large the electron will very quickly and the end of our time now to correct answer in general is that the photon will be they'd ever more or less random time in the same way that a radioactive Adam our will be case it decays with a half-life it a case of half life so if you put the electron into the magnetic field in the wrong direction it will with a certain half-life became became means give off of photon right itself in the other direction but if you make the magnetic field is large enough that half life will be very short answer will vary suddenly for a lack of G-8 theory is vision for a learned in quantum mechanics it is to ask questions only if we know how to make the measure if you could tell me in detail how to measure away in the warned electron that on the contriver answer it I would say that's 1 of these questions that is not supposed to serve as a thank all the energy time uncertainty principle if you know the energy of electrons New York uncertain about the time but let's let's can't Thorn certain principles later life sensitive like you do what you will cat life yes it is a lot more than you gave a rare but so what dish at 4 don't the more but
that Little Piggy magnetic field as you put in more quickly below that right absolutely look more quickly in the sense of a small I have quite right OK so 1st of all we're dealing with the theory that only has problem that has a probabilistic description our end we will see ultimately that there is really problem this probably no way around the back of the description has 3 probabilistic our wrist at some level but the mathematics of the state space over got concept of a state state different in quantum mechanics is clearly different was were clearly different it's a system with only 2 states is think but then again it seems to be able to be oriented many direction be mathematics Oh of quantum mechanics and the states In particular the states of a quantum bit on not the mathematics of said of 2 points not bully mathematics of a set of 2 points the things which describes the mathematics prescribes a quantum state is a vector space now I use theory in a different sense it is no longer an arrow in ordinary space mathematical abstract vector space character you come to understand if you persist then and that persist that will bet you will begin to understand what this vector space it is and how it characterizes the possible configurations of the possible states of electron and how it can be consistent with this fact he OK on I'm going to spend a little bit of time now just doing some abstract mathematics very abstract mathematics and then will seek to interpret it to interpret the state of the electron limit tell you what a linear vector spaces were vector space is a vector space is a collection of objects and here the the rules for an abstract vector space there said do not confuse this with the point there that is a vector an ordinary spaces and abstract completely abstract concept 1st of all vector an object and well label it why back with an object with a set of rules set at a collection of objects it's a collection of objects with a rule festival will he ruled that and said there can be multiplied by can't stand together Novak this gives vector Richard Corbett Court a prime so as a concept of multiplication by constant slowly tell you that the constants were going to be talking about complex numbers not real numbers anybody here if not familiar with the concept of a complex number of sorrow raise your hand and be humiliated score I now you really don't know what a complex that maybe they you know it but by another word now it'd be really don't know about complex numbers look them up office because you need by Trump like remembered the number or ii i.e. in the square root of minus 1 now this 1 has a number of its is no number ordinary number who squared squares minus 1 so there is no square minus 1 any number of times itself was always positive right 3 times 3 is 9 minus 3 times my history is also my but mathematicians many many many years ago invented the abstract idea of an imaginary number the call them imaginary yeah and so let it Be that I squared is equal to minus 1 that the new number we know how use those numbers and not going to move much ought to spend any time added um however is that he and other familiar with eyes equals a square root minus 1 but yeah yes right now you can have complex numbers complex numbers 0 what happens if you take tool to real numbers ordinary numbers but score 18 and be different a plus finally beat another words if you had a real number 2 imaginary number you get a complex number at the hall definition of complex numbers you could multiply complex numbers and all you have to remember you the ordinary rules of arithmetic except you have to remember that I squared minus 1 so for example if you win a World War II as an exercise Yukon square that's what's the square of a plus I'd be accord that's a complex number often character often written Izzy sometimes EU use any levity like but it's a complex number with both the real part imaginary part another concept Is the complex conjugate this is very important you'll need it a very simple it's a I B is complex conjugate Soviet take a we'll have to take a complex number and you change the sign of the imaginary part that's cold the complex conjugate the complex conjugate of the complex conjugate gives you back the original number that these are portend mathematical concepts they're very elementary we give you an example supposing you multiply a busy star by lazy What is that that's oryzae clumsy start married away a plus I'd be kind a minus I did but see what we get disorder multiplication realtor upper brackets we get 8 times which is a square we get 8 times I'd be with a minus sign Martinez a B plus I from this one-time this war and then what about ID times I was I becomes IB give miners be squared because a minus sign here ready so it's minus I times I times be selectors plus squares altogether we get a square Crosby square these pieces canceled so that's an example of using the accompanying the example right value using any parent is just an example of the definition and then using the definition of complex arithmetic to calculate would see stars Norris said Suzzy is always real always positive it's always real and positive a squarely in a real numbers a squared and be squared of both positive East Thorazine is a real number and positive it's is the magnitude of the come of year complex number right sell 1st of all for the kind of vector spaces we going to be thinking about called complex vector spaces and 1 of the rules of 1 of these operations that exists in the complex vector space is multiplication by a complex number now a complex number includes real numbers and so you can see could be tool for worker I worry could be told plus I anything like that is a notion is a vector you may multiplied by a constant and you get a new vectors as 1st operation and only 1 other operations of interest to us it's a adding that there was a rule failing so for so as a rule vector survived 2 vectors and B I can construct an actor called a possibly and that some Novak there which I guess we can call but sees a difference see that appears he oversees
constant misty just stands for no vector which is a plus beat so every pair sectors and and them when a show you show you some less abstract examples but 1st of all the simplest example of a vector space like this is just the complex numbers themselves be ordinary complex numbers allow you to multiply any complex number by another complex number but in particular constant particular complex number and it allows you to edit complex numbers so just complex numbers by themselves are vector space complex vector space that the simplest example but to show you another example the other example something we talked about last time which called column vectors just represent an abstract quantities by a set of components all the components are themselves complex numbers so for every vector really want that vectors standing for right equal signs of probably shouldn't write equals mathematicians would have a stroke if wrote equals a is represented by a column vector the column has been treaties and it's just a table of numbers but a table of complex numbers table complex numbers supposing you want to multiply that column vector here but I can't stand the rule of a simple multiply every entry by that constant Societe times vary it is represented by a C 1 c to see we just multiply all the components by the same number that's multiplication had a had 2 vectors together but suppose we have vectors and a and B a is represented by the column was entries are a warning 283 is represented by a B 1 B to be free but these are the basin Beazer just numbers Norway and we just get a 3rd column vector whose entries are anyone plus B 1 8 2 Class B 2 0 8 3 Class B 3 so calm vectors an example of a vector space power and they are the basic example that we will use all over and over again in our business said mathematicians would have a stroke of I called the columns vectors would you would say columns represent sectors and you recall these components or set of components vector Okayama that's the idea of an abstract vectors may very strange thing is that the states of a quantum system particularly quantum bid all incidentally you not restricted to read this is of particular cases of three-dimensional vector space we could have a two-dimensional vector space that's actually be more interesting from Roma in a two-dimensional vector space just as entries at the next simplest thing if you like the B numbers the next simple as vector spaces a two-dimensional vector space and is a three-dimensional vector space and four-dimensional space and so forth and they all make sense OK that's how you get used to it sooner or later start thinking in terms of abstract vector spaces if you pursue this subject books there's no later learned quantum mechanics honestly incorrectly without going through this mathematics swords it's absolutely now Jack if we were talking about the the simplest vector space just complex numbers then every complex number has its complex conjugate that means that there is a 2nd vector space which is the based Of the complex conjugate numbers that concept of complex conjugation is an important concept and it also exists for other vector spaces will you have to do basically is right the complex conjugate of the entries but the rule when you're writing the complex conjugate vector I want to write complex conjugate of the sector in 80 you draw up the other way this way at stands for the complex conjugate a evecta furthermore about you represent date might be rolled back the components of the role back other complex conjugate 81 star 8 to stores so what see a row vector always think of it as the complex conjugate of the corresponding to the corresponding Congress if I tell you lose Our a role that that AT which is in correspondence with a column that a always remember that was a complex conjugation operation this is the basic idea of a complex vector space written out in components everybody happy with that anybody who was not usually that that that's right Bibi notation does not put star Over here but do remember that the Stars is implicit in turning the year of the symbol backward remind you this symbol was quoted can now the symbols color brawl this symbol of Quebec a bladder right now and that I think that 1 of them was a brawl 1 of them was a cat it doesn't matter if it's completely symmetric between 2 to remember that the complex conduit complex conjugate was original meant so that Mets the notion of a complex vector space and finally must find a way but there are a lot of pieces together you can multiply 2 vectors the product called we discusses before I realize I'm getting senile North talked about as before but we need we really need to do a right now the concept of the inner product of 2 vectors be it a product of 2 vectors the easiest way for me to describe it is in terms of components I don't want to spend a lot of time giving you the most abstract definition our remaining mathematicians annually and plug you use just just just as clever month by half Frank ahead on me he actually mathematicians also use components but down anyway by the next concept is the concept of a product of 2 vectors but it's always sea In a proper to actors and the concept is really the product of a vector with the complex conjugate another vector into very much like our multiplying a complex number by the complex now complex numbers but complex conjugate of another number could be the same number could be another number so we do the role of Will I'm going to tell you in terms of components is the role of I want to be in a proper between the Vector B and fact that AT threaten like that it's got Brock kept its a bracket that's where the terms Brockett came from half a bright a bracket as a broader have a vector is a cat again I forget which is which is cat right Yasser cat as was a Brockett get the rule of this you take it to a take 2 vectors the B vector it is a role vector there I'd be 1 star B to starter that's the representation of the bee that there and the representation of 81 82 we do not start the inner product between them this product is gotten by multiplying the 1st component of B with the 1st component of a E B
1 star 81 Cosby two-star 82 1st come 1st Raul with 1st started 1st entry year with 1st entry a 2nd treaty is 3 supposing for example are very ready that sets the basic concept of the inner product it's just like a multiplying 2 numbers together except we get a turn to each of the each entry 1 entry into it for example if we take in a product of a vector with itself let's take any with they being In a product of a vector with South as a form 81 start 81 plus 8 two-star 82 Lotus 1 thing about this a number of times its complex conjugate it is 1st of all positive and 2nd of all real well could hardly be a positive the word real it's a positive numbers so the sums of the in a product of a complex vector with itself His always a real positive number it can be thought of as a kind of societies of vectors actually the square of the size of the wreck if you like to think of it as a square of the size of deck of the link of the vector In measures if measures the magnitude of that so this operation of farm in a proper it is absolutely essential as will see the interpretation and its complex of sectors like this which represent the state of a quantum they adjust their digress stop doing mathematics from minutes and talk about the quantum bid again the quantum bits could be pointing straight up we could invent a symbol for the state for the configuration of the electron pointing up let's call it a rickety that we could call up all we could call a vector all we can intrasquad plus I think I'll just caught plus to indicate that pointing upward or we can have an electron which is pointing down nine-member when we measure the electron we Eva find the Upward Bound nothing in between and that we can identify with me that there might just an abstract notation this stands for the state of electron pointing up the stands for the state electron pointing now known classical physics we would never under any circumstances think of any of these 2 vectors or multiplying my numbers who just say this means electron up this means electron down in quantum physics the general state of an electronic or electron spin is a vector in a vector space which we could write AT plus plus plus and minus and minus another word two-dimensional vector space we could also represented by a plus a miners and he is the rule of the road this summer so there some coefficients that we can a had them together where and he is the meaning of those coefficients released a partial meaning of those coefficients 8 plus star 8 plus is the probability that we find electron off remember it's positive it's positive Rio a start-up firms is positive and its real estate magnate food of the your square of the magnitude of the coefficient of it up configuration and then there's the configuration the probability of finding electron down a minus times a star from defined that's the interpretation of a part of the interpretation just to give you some orientation where we're going the quantum state of that electron which 1 we measure it we find the probabilistic distributions either up or down and nothing in between that's represented by saying electrons pointing in some funny angle is given by a complex vector complex vector space which can either be represented abstractly like this committee Rep. concrete Lee as a column and these squalor or magnitude of the square of these entries two-year are simply the probabilities defined upper down presumably probabilities at 1 1 of the rules about quantum mechanics is that the factories that represent the true we have a vector space of some kind and abstract vector space but sectors which actually represent the physical states of the system are the word is normalized normalized means that the sum of the probabilities His Warner so a plus star plus plus a minus start in it should be set equal to 1 this is like considering the only vectors of unit Lane for that doesn't mean that in the vector space so there aren't other sectors of other magnet to they are but the physical states of a quantum system have to be normalized which is the same thing is sitting in some of the probabilities toward normalized vectors in the vector space represent the state of quantum system that's a very abstract concept why ideally did driven to such very abstract concepts like Elie visualize is the same way we visualize classical physics well because we don't have the wiring and we have to rewire ourselves we have to rewire ourselves to learn the think about the quantum states of the system being an abstract vector space once you get this idea of a abstract vector space and how it fits together with the states of the system flying you could understand all of quantum mechanics so any questions about that's the entries coefficients or equivalently the components of the vector when you square them or better yet when you multiply them by their own complex conjugate give you the probabilities for the 2 possibilities that represents a quantum state if a star a year and thank save this way in a product of any with itself that's given by a study plus star 8 plus plus a minor starting minus that should be warned so June quantum states should have a problem with themselves which is equal to 1 and that's just the statement that that's just a statement that the sums of the probabilities of probably of blossom miners and they're just indicate that 1 of them corresponds to spin off the other cars and we could call may want to occurred during a different name I could call a up and a down or I could call them want we don't have to don't they are the coefficients Of the kept costs and mind OK for backers step What about Bequette class how should we represent that they want In some basis let's wait for the moment I don't discuss the the ambiguity basis vectors will come back before moment we have a basis serve British talk this sector will be represented by 1 0 better be represented by 8 81 in the upper place 0 remote place or about my cat might kept is represented by a 0 and 1 this is just some arbitrary our
correspondence the old regime if I multiply this by a plus the numerical constant day plus than we have multiplied by we are after multiply 0 but a plus 0 times anything is wrong if we multiplied 18 miners becomes a minus here and if we had them together 8 plus firms a plus kept plus a minus From the minus care we just Eddie's together here we get getting them gives us a plus a might take that into the question was so yes we said certainly plus and minus notation instead of 1 to we're I used 1 until they just labels but labels 1 which labeled the label 2 entries I could've used up and down like a used Joan Peter makeover His anything I want to label them here amusing and might took possibilities for the orientation of the electric of electron or for of the quantum date and labeling him plus in my gut OK so now we have AT the basic concept of a vector space in a product what do you say to our year my user care as 1 of your kid is not my warned terms a poster No no no no but I know that that but old bidders must yes yes you're asking you're asking whether where this object makes yes you're Logitech any vector and multiplied by any complex number in particular a complex numbers minus 1 for inside but about this it is not is not the same as this as a fist what make that clear mind this year does not mean that minus the prospect is just a label normal then a 3rd than they are met 8 2 0 0 means twice a war of life as head then they I want yes the threat that threatened life of the mother of all this care here as follows 1 last time let performed with the plan that's right both of these at absolutely absolutely with account of that very good very good very good very good will with urea pointing out something that I was about to come true the moment to you right now but look at multiplied by Indiana complex number of magnitude 1 a better that's correct that's correct shell physically we would not distinguish the state from rum From the state alright equals because of mathematical vector space and not equal but they represent the same physics however however when I them a plus and minus a miners are not the same that is a different state that is a different state for the minus side but will come to that will complement something of that talk about but not yet or do they just just the idea but the components of these abstract vectors when squared correspond the probability is that it it is best to it brought get bonfire associates plus but so are did they said they kept their boss theater can't cheers Ross brought this case she well except that right a but check were out in a drop in this case this is because want real memory are OK so here's a here's NATO exercise what see in a product of plus vector with itself will 1 because it's just more applying 1 0 get better 1 0 if a one-time forms 1 of his war for about this if also it a while where a barracks via the product of plots ears show all just check that Ross who we ought to quickly with just check the classes are brought is 1 0 remind care 0 0 1 multiplier together we get one-time 0 0 0 times 1 which is altogether 0 but where about my this class also world what's award refer to vectors who's in 0 what are forethought they're called orthogonal factors so that was don't states Of the electronic 1 up when there are tool orthogonal sectors in a linear vectors base upstate linear that's the basic set Our and coefficients appear there their swear magnitudes are the probabilities the 2 configuration the probability for opt for about half as you can see there are more possible physical we just up and now there'll always be a combination it's the linear combinations of them correspond to electron for having been prepared in different directions so in other words an arbitrary combinations like this with a plus and minus when this is not 1 0 0 0 and 1 what when you consider all possible complex numbers for the different directions that you might have prepared but still although it is up and down and the probabilities for up and down so AT state for example we give you an example of that theory vector of what's right 1 in both both slots but some out was that had only 1 cell probability Tokyo 1 square forms where we have abided by square till it the now we probabilities is payoff Mayo awards were to wear that yet you're right by fireworks the data wrote the column but that's a state with
equal probability they would probably have the formula for up now the source was probably have defined electron there are more states just opened their arms and that this 1 corresponds to what you would create if you create electron pointing in the horizontal direction of with the magnetic field in the horizontal direction you froze electron goatsucker break if you froze electron place for the magnetic field in the horizontal direction and in the horizontal plane 1 of the possibilities would be this stadium which were have probability of being up half a probability of being after probability of rock and have probability being down actually corresponds to some configuration with Electra arm is lying somewhere in the year in the horizontal so we have I have sectors around with all kinds of complex numbers out of which we can build them will see that we can do this we can build these these spaces Our electron arms which some sense of any election but whenever we measured always get Upward Bound with probabilities the governed by physical fish but circuit breaker of the 30 seconds more than 1 person is probably a little bit confused about 2 notions of vectors 1 is having to do with directionality an ordinary space and 1 having to do with this abstract concept and the notion of components of vectors labeling upstate and downstate questions what's the connection between the notion of directionality in space and these vectors here now about what I say 1st of all is they are not simply related they are related but not simply related for example Armed many components as a vector have an ordinary sector the 3 ornaments and then and they real numbers right x y all real numbers these factors have till components in the complex numbers that means comedy comedy real numbers as it take to describe them for them out the same but there must be some connection to warm up the the collection just yet that but we'll can't do it next concept which we're ready went into a little less time is the concept of a matrix ahead in the video on this is from the fact that they wrongly sure possible interest of way when you do this experiment on electron you either get tripped up or down and nothing in between that tells you that there are to possibilities and tells you that you should be dealing with a tool dimensional respects all two-dimensional complex vector spaces are are mathematically they throw them all the same two-dimensional complex vector spaces of 2 dimensional comfort place is not there but for 1 of them mathematically sold so you count up the the number of possibilities somebody as before I had a spin 1 particles of spent half article which is when electronic how many components would be there would be 3 months in experimental question how many possibilities are I'm really stateside family family stink possibilities are there and the case of the electron spin a case of the spin the electron there are 2 possibilities and two-dimensional vector space in all the same novel the no distinction between different two-dimensional vector spaces other than the fact that that tho we could consider real or complex vector space OK let's wet a move on the concept of the abstract concept of a linear operator or the concrete concept of a matrix let me tell you where were gone we talked about states but we haven't talked about things that you actually measure the things that you measure of it you can measure called the observables observables of system are the things that you can measure and get answers for our position of electron observable but we're not doing anything is complicated that that our if we for the label the up and down the state as plus and minus we couldn't observable which would have 2 possible values it can be plus plus 1 0 minus 1 would be an observable we could measure if electrons up we will sign the the number plus 1 downloads oversight of the number minus 1 also called observable an inhaler can measure that has a numerical value numerical value means a real number anything that we can measure that when that when measurement is recorded that gives rise to a number of numerical number is called an observer so the set in the case of electron spin and if we measure b um if we measured be ordered up pour down signed up the number plus 1 and down the number minus 1 and in that way have observable which whose numerical value when we measure it is you the plus 1 minus 1 on when they talk about a day before talking about observables in quantum mechanics with talk about observables from classical mechanics the current classical period but and classical these states of the system Our just a set of states which he represented by point I want to 3 4 5 6 these could be the SEC's possibilities of throwing abide by a single by May 1 2 3 4 5 6 so that we could a summary on some numbers to each 1 of these points we could call we could labels from 1 to 3 4 5 6 we throw the died we look at it that's the measurement withdraw the died and we look at it and we get in answer answers either 1 2 3 4 5 or 6 back measurement is the measurement observable Erickson observable which has fixed possible answers there are other observables that we could concoct for example we could concoct an observable which is 0 everywhere except that 1 value 1 0 0 0 0 0 0 0 half this means that reflect the dying or we get what we assigned number 1 the observable forget any other number we assign the value 0 observable so that Romania observables but you can make and basically they correspond 84 functions Of these point any functions of these points assigned any functions for the points function point assigned any numbers you like these points real numbers and then when you flip the died and you see what comes out Tuesday the value of the observable is whatever the numerical value of that
observable laws when you for many many observables if you could think about our interests In this simple system but there are other trivial religious correspond to our tools the basic question of whether the fight is 1 2 3 4 5 or 6 on it's interesting just not because guesses interesting to define the observable as a function which state were talking about f sub and then he represents which of these states with talking about and was a function which could be any number for each of these states that's a call the observable when you flip the dying you look at what you get in your sign number half of a particular state sub 1 of 2 of the 6 that's call the observable and that's a desirable overkill concept and classical situation here but let's continue now let us suppose it for 1 reason or another in classical physics we haven't been very careful doing classical physics measurements and so forth and we don't know exactly what state the system is always know with some probabilities off for example we may have a loaded Diallo did die which is an unfair died like the kind of 30 gamblers used and you flip it and the plans on the table there may be a probability distribution for different values of the 1 2 3 4 5 or 6 let's label that peace in also that probability the a priority probability let's say that when you flip the die you get beat and state then why is he average if you flip the dying many many times and each time you measure f you look at many many times you measure what is the average value of F after many many many identical experiment could be answered and with no of such a B average value of F assuming of course that some of the peace events is 1 another or some other probabilities of what the average value of that is the sum of all the possibilities of the probability for the in power at the end of the configuration times the value of the function hands new configuration so would you do if you add up all the configurations when did according to their probability wake them according to the probability for example if all the problem a former probabilities are equal I'm then you just get up at the end of the role equal incidentally did not 1 of the 1 and because that while the number 1 over 6 in this case but this is the end average over let's let's take an example extort according to the simple Corning up about had details but it is the probability fans will head plus is the probability their heads which is B plus there's a probability for tales urges P if we have heads we assign the number plus 1 if we have tails we assigned number minus 1 what is the probability for that observable which is you plus 1 minus 1 of its people asked the probability of plus or minus the probability for my figures every time you get a minus you give it a value minus 1 every time broader returned to get tell you give us some minus 1 every time you get a head you give it a plus 1 the average is Pepe last month as minors the difference between the probability of head and tail is the the average of the heads of the pills so this basic formula he yeah which is just the most elementary formula of probability of of the notion of an average that you average all possibilities way it with the value of the quantity be observable whose average taking a new way it according to the probability of that particular configuration and that's the average value of our experiment which consists of many many repeated experiments a that expected values sometimes called expected that and call the expectation values sometimes call the average value our there and that's all that be interested in will be interested in me average values of things that we can measure the average values of observables now I what I haven't told you is how you represent observables quantum mechanics and representation of observables in quantum mechanics is more intricate and complicated than just saying we assigned to each state numerical value that we called the observable value in that state somewhat complicated concept and more tricky concept the notion of observable or measurable measurable quantity and its related not related by well related related to the concept of 80 linear operator or a matrix a proper procedures were the same thing again the mathematicians will objected saying matrix is a linear operate out of a deal not anybody OBJECT my saying matrix is a linear operator the the 1st version of the show what did that get rich yet last war there is a euro if it is a probability that we will say the a meeting of the virus this vote yes yes yes right that's why people don't like calling it expected value because if his lead with it is a value which is very definitely not expected so I everybody used to call it the expectations about your now that's a move that will we go and say we expected value that my friend Marie go months Courtney B. expected value you will at 8 is as you point out that about an hour the area about the size of the Air assuming some of these days but they all just how that is back then always had battle is that it's not clear that U.S. gets out gives you have a problem with it there is a payoff Jr. yes 0 0 vote of IAEA not that is value but definitions the average rock so that's right so yeah fund glad you pointed out that people are simply miners both a half at the
average value would be 0 0 0 is not possible answers so average value is not expected you did not expect that the presence of the year is out there a lot of evidence that the date there as well as versions a lot of good data so I should let it pointed out that the there calling the head at is something about some temperature there is a lot of it is not yet healthy completely ignorant you completely ignorant as a 50 50 chance to experiment or nor were were there according a average the result the average had better be 0 Keyser can be can't be positive and can't be negative because people Brown on the other hand In unfair corn let's say three-quarters of the time it comes up heads and one-quarter of the time to come the details many averages what three-quarters minus 1 quarter which is a half have unfair coin which three-quarters of the time comes upset 1 quarter Condit tales and the averages a half again doesn't mean that you can measure ahead after just means that the the average value average values are important let me give you 1 observable what kind of observable which is special suppose it might take a special class of observables which 1 and 1 state after is 1 4 1 state over here and 0 for all the others what's the average value of it crap just a probability right just a probability so if I had observable which I invented which was won at 1 place and 0 everyplace else than it would be only 1 in the summer because ever 0 for all the others and for the 1 case where it's not 0 we just get Pete so if you know what the rules for calculating averages you would also know the rules for calculating all possible observables he would also immediately know the rules calculating probabilities latest it is if you know how to calculate the average value of any observable Ucon reconstruct for it From a probability distribution the probabilities for hours about observable probabilities probably different different from different possibilities on so that raises the question now what is the mathematical representation of observables and that is matrices matrices or linear operators so let's talk begins a little bit about matrices and February let's talk about observables all our lives forever matrices for the a matrix is a thing that you can multiplier vector with what act vector word it it does something Quebec an operation on a vector was not just any old operation on the a linear operation and that going to explain that because we will we will need to explain our but whatever it is call it and him from matrix the abstract notation it's a faded an object which you multiply vector by and you get a that for example 1 simple example is just multiplying by a complex numbers multiplying by a complex number is they operate it's a very simple operator multiply the vector and get my vector of the number 2 just takes every veteran doubles size by the General make sure general operators at within consider all ones which are represented by matrices matrices or free have our vector which we represent X Back column there and I'm only going to write two-dimensional column vectors coasts because but do you'll immediately deduce what to do if you have a few more interest here we can multiply matrix and matrix as a square array of numbers and 1 1 and 1 to end 2 1 and 2 and general these numbers a role complex numbers In general the all complex numbers be a and in a complex vector space can all be complex numbers and the role for matrix multiplication which we discussed last time for multiply matrix by vector is just if you want the top entry which you could think of was the copper you take top you multiply it by Victor you take in a product of a copper or with the vector which will be M 1 1 81 plus M 1 2 0 0 8 2 0 and man down here and 2 1 8 1 plus and 2 2 0 8 2 personnel vector I had to draw up pretty wide because I was getting some numbers here but sister that column vector with 1 column and its constructed by taking this role the column in this role column and those of the 2 and 3 but that's it that's all matrices are the you couple examples cup of simple examples from very ordinary vector space very ordinary vector space is just the point of the arrows that you could draw on the blackboard Yuka multiply a vector embodied in this case real number double-A-2 could triple Iftikhar multiplied by minus 1 and you can add 2 vectors so that vectors a you draw on the blackboard on a vector space when they give you and they have what components components are the x component his ex Hawaii and the vector has an ex component and a white component so we represent the specter of Let's X component of its why component and we could ride in a form X component of that their white component of course now I'm only going two-dimensional space these would be real numbers on the Black Course it wouldn't be complex numbers and this would be a tool real dimensional now let's think about some operations the 1st interesting operation which is most easy it is to stretch the vector space for example multiply every vector but to multiply very better by tool will magnify the whole of space by a factor of 2 0 and a vector we'll get stretched out twice its original Lane that saw what happened here is the matrix that represents that operation it's just the diagonal matrix To that means that entries are along the diagonal photo diagonals matrix is called principal think of our Member of the other 1 is called the unprincipled but aware
after while the good guys and the of bad at was 1 bad about God let's just calculator we get the other entry we get to times VX was 0 the upper entry just becomes To VAX the entry we get 2 times so as advertised This is the operator which simply doubles a length of every sector very is a moral almost similar similar things but supposing I just wanted to double the white on what would that do the vector space it would take every vectors and stretch out why component by a factor of 2 without stretching out its 6 components so would be a stretch in the white direction by by a factor of 2 but no stretch in the x direction how we represent that again diagonal no 1 here in the X place the X X and Y Y place factor what's jacket 1 times VXQ those VX too times BY gives us twice new wife so that's a stretching of the vector spaces in the year and the year the White direction likewise we could put the 2 0 here no 1 here that would be a stretching of the vector space which stretches about this direction 1 or 2 more works just a visa is a very easy examples so far only involved diagonal matrices did a matrix which corresponds to a rotating every sector by 90 degrees they get and rotated by 90 degrees and other words This is the operation which rotates the plane by 90 degrees of 31 is in them will check and see if we could see why it off diagonal wha 1 1 1 might miss 1 excuse me all that let's do this 1 1st let's do what this is before I do the rotation let's do this the see what the stores this 0 times VAX one-time VYE the interchanges BY in VX interchanges we wanted but see geometrically with that corresponds to a reflection it's a reflection of the vector spaces about this diagonal takes every vector and reflected about that Barak just flipped it about pentagonal that's this makers is another matrix you put a minus warned that are here what is this yeah this do you why might we want it might be makes it interchanges X and Y but then throws in a minus sign for 1 of that is an operation and that only a few rules as with our time but I'm so you what the stars and rotates the vector space by 90 degrees the go 0 3 reviewers I certify Rodale in 8 years in a row rotates Roach stereo and really is 1 of the dead were militants voting their 1st sought after a reasonable time clock rotating clock duck a solid urea yup yup rotates vector by 90 degrees see what matrices correspond to is a corresponds to a transformations of a vector space stretching things rotations what complicated kinds of things another example would be as sheer this is interesting to try to work out the matrix that corresponds to a sheer emotion play sheer motion Heller orphanage assume I takes every could back well it's along boy slides stuff this way for planes so it's due to the right by an amount proportional how high you want it takes every point and shifts it the right by an amount proportional to how high you are as a matrix of described back motion the forms the plane tilts this axis as were among them right that 1 house if you find it the next time up 0 you would be answers sheer motion but in general matrices correspond to transformations special transformations null transformations are represented as matrices of a linear transformations be specific but these other ones will be interested in and we're gonna be interested in a special class of this is very abstract and for the moment you will not see why sisters definition but I gotta give you some mathematics and interspersed with some physics and we give you the mathematics today and next time I will show you what the point is the notion of a permission matrix the notion of a her mission matrix I know that the notion of her mission matrix is corresponds to the notion of a real number Our kind of concept of reality versus imaginary but it's not the doesn't mean that the entries a role revealed what it means is that if you can't take an element of the matrix MIJ that's a that's some element of a matrix to could be some big matrix over how big it is the summer's over here and then this
and JIT NJ on May Is the reflected matrix our if this is 1 tool over here and then M 2 1 would be over here M 3 5 might be over he M 5 3 would be over here so it changing rows and columns is a kind of reflection of the Matrix about the bag of mission matrix is 1 that if you reflected complex conjugate another words MIJ Siegel and JRE star this matrix was just the number was just take a case of numbers and I tell you I have a number which is equal to its own complex conjugate was at tell you about the number but to Rio but has Normanton Harry Potter now From matrix that's not what said the it doesn't say that entries are Rio but a kind of reality property and they give you some examples of matrices 1 which is permission everybody not artist over mission H E R end AP SHU when and got permittee early reached your entire life yet but yes Mrs. something over he and a good shot at us the truth is that it a review of the of the what Review saw a massive now the final reels another won't renew flip that you get the complex and it is an example that's is 1 thing it says the diagonal elements are real because M 1 1 was a status as an 1 1 is em star 1 1 if you put 2 rows and columns on the bag you give the same answer for example I equals 1 J. equals 1 man 1 1 equals M 1 1 star so 1st all is Abagael elements of Rio real numbers here 7 3 that they elements of complex hundreds of each other so he is a matrix for example that is not her mission former and to that not her mission because 2 was not the complex conjugate of 4 this is our mission war is a complex and for server matrix is real and his her mission and also symmetric but is another 1 which is permission for plus ii and for side 4 plus ii is a complex conduit for my side so this is the notion of our her mission threats and I wanna tell you that we're fine next time Morgana postulate next time that her mission matrices are the quantum a version of our observables classical version of observables functions of CEO the state point there the quantum version of observable permission operate a which means of mission matrix but more or we it like there was there that major man in a state of high prices advanced after some probability is this is not so good years later this it would of our what they were they the change of any vector which is ready quit you're asking about but but permission upward does not have the property so unitary matrix is 1 which doesn't change the line and it's it's the result love of operation electric due to the electron kick it it had to do something Our after you very quickly now well no I won't bother your because we didn't do it next time matrices or more generally linear operators are the basic quantum concept of the observable thing that you could measure and then discussed what kind of matrices correspond the Z compiled up they component of the spin what kind correspond to the x component of kind correspond to the Y component what kind of observable what kind of matrix corresponds to other components of spin along some arbitrary direction and so would see then that there's a connection between these vectors matrices and so forth and real directions of space but that the takes some time take some of the more pieces were all of this is no way that you could understand honestly what entanglement his work bells the inequalities what what the basic setup of quantum mechanics is so it more mathematically file a if I had my way I would love to be able to teach this without biking is nowhere 1 honestly work correctly you want the basic goods ingredients for the quantum mechanics we have to go through the use of mathematics wants what through permission operators vectors and so forth and the rest of kind of smooth sailing so it's relatively easy stuff but so far we've had to spend more time on mathematics and I'm out like but OK among the minor
roles the preceding program is copyrighted by Stanford University please visit us at stanford .


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