Quantum Entanglements, Part 1 | Lecture 3

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Quantum Entanglements, Part 1 | Lecture 3
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Lecture 3 of Leonard Susskind's course concentrating on Quantum Entanglements (Part 1, Fall 2006). Recorded October 9, 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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same page with respect somebody elementary mathematics complex numbers just 1 important Devon concept villagers were wrote very quickly extremely quickly a complex number as a real quiet and imaginary part and on two-dimensional surface the two-dimensional surface horizontal axis represents the real part functions as a function as the real part of the number and the imaginary part of the numbers plot on the vertical axis and 8 . this plane of imaginary number Our the X and the height is why than the complex number which is experts are white are we could also represent another way we could represent better in terms of a distance which we can call are an angle that we can call later and now just from elementary trigonometry the distance X is Our Times the cosine 0 experts aren't cosigned trader and why is art times time signed later so I 8 times foreign signed data earlier KBR outs of it factory are out disease is R & cosigned plus I signed the combination cosigned fate of course I signed later called Ito today I played he showed how I not all that definition Ito I later call sign beta plus ii signed later the reason we call it equally I favor is because it satisfies certain rules of exponential rules of exponential is that when you multiply exponential so you would end the expo each of the 3 times each of the 5 is easy to be a 3 in 5 days you could say that using elementary trigonometry you have till angles they and you right heatedly I've tried purchase cosigned fighting force I signed 5 and multiply them you get something with a real and imaginary part be real party is cosigned stay the course by sign faded and signed by and handlers my society as well as 1 of the multiplier used to what you
get is closed I think that would get what's before you get cosigned cause I find my assigned signed by right call sign of they got passed by that's about as a high school
trigonometry formula call sign of the son of 2 angles cosigned cosine minus sign side then plus the imaginary
part imaginary part is later signed plus cosigned by vibrant signed Bader and that's
just signed our data plus buying and by definition Peter realized they there is cause I'm plus I signed in this just cos I'm this is just eagerly I think the plus 5 so from now that trigonometry we discover but this combination paid precise has all the properties of the exponential of later now all of trigonometry anything you ever want to remember about trigonometry could is all stored in this formula for example if you could remember what the call sign of the sum of 2 angles it is all you need to do is multiply eatery I it the I find this form and you'll discover the cosine made up of 5 years because physical side by side all of trigonometry I don't know why they think you teach trigonometry in high school I said then the end of it they call the Stanford don't know this formula OK love another factor with so we could check a two-way is we just use a formula fed a from multiplying exponential eat EIA data by all this equally I. Coast made a precise I remember what the complex conjugate everything it the complex conjugate is just the same number except would be imaginary part changing sign so it's the reflected number In the lower half playing the numbers in the upper half plane minutes reflected in the law have plane or vise versa bomb the complex conjugate eatery I data is cosine minus signed data but that's the same thing he will remind us all a favor that's the same thing as he efforts cosigned later Miners I signed and as you remember back this summer that minus the signing of the is the same thing as a sign of minus the new realize this is just minor side so the complex conjugate Vitaly I. later you just get by changing the art the signing of the or here what happens if multiply these to you know what changes on the way it is in the top half the show is over recalled that multiply eagerly I favor times it from minus I figure you get 1 so the class of numbers which are a class of complex numbers which are of the form Eda the RIA data on numbers which have a special property that when the multiply by the complex conjugate the result 1 and was another To say that the only other way to say it is of X where plus Y squared is equal to 1 the class of numbers for which X squared plus Y squared rumor number times its complex conjugate is just the sum of the squares of the real imaginary part and so these numbers which have a 1 the I Baillet specialized of ours are numbers for which X squared plus Y squared is acquittal 1 0 hours then numbers on the unit circle circle of radius 1 the simply called numbers unitary numbers numbers long before me the I've made a call them unitary numbers and the numbers that have property that they lined the unit circle and 8 times a complex conjugate is exactly what every complex number a unitary number a real numbers a real positive number and the real positive number is just a radius our or the distance to that point on the complex place like that that's so we we need to know that tourist complex numbers there is nothing else because think of offhand Richard Noah's Ark the other word for a number which is performed eatery I is pure phase angle in here is called the phase angle the phase and the radius is called the modulus against my size so number which has no armed front door or 1 of a number of units circle is called a pure phase but just a phase so
swell the I swallow the truck which wrong when I refer to a pure phase and that means a number close unthinkable source are OK now all I am possibility of
quantum mechanics who are now they go through the basic postulates upon mechanics or Kwan mechanics is is a calculus the calculating probabilities off after a while you get to make the Wishon for maybe you get some pictures of what's going on but think of it as a calculus of words are calculational procedure for calculating probabilities probabilities of what probabilities for different measurements different values of measurements the things you measure recalled observables you prepare systems in certain states not in California on the yacht for certain configurations that called states and other states Our labeled by or described by that there In a complex vector space we call them how far 80 I guess with for you could then subtract them Neukom multiply the numbers and they form complex vector spaces worn No. 2 0 Art we normalize them in that particular simple case that we studied which was the two-level system we could describe this out times upstate for example must times downstate where alpha and beta are complex numbers on the coefficients here such bad after storm out and they star data on the probability Is that if you with some measure Opel down you would measure up with probability alpha star Alpha down with on data don't just to make the probability altogether eagled the ones we require that the coefficients are such that the the sum of the squares of some of the articles of the square the square of just 1 but as
another way to rewrite the same equation is in a product of 80 Real estate with itself is just equal to 1 that is equal to alpha staffer plus star beta so this is the abstract way of writing that the public probability ends up 1 so states state vectors normalized called normalized normalized means the length of the vector in some senses 1 an altogether the of the cover breweries 1 that's the basic pasta with the state's our this is an example of the two-dimensional space of states two-dimensional doesn't mean the world as two-dimensional it just means that there are 2 independent possibilities up-and-down heads and tails we discussed how you might measure whether a spin was offered bound by putting it into magnetic field and seeing it emits a full-time are not sold given half an arbitrary state of electron we can measure of explained that it gives off a photon than saves I guess and it doesn't give off a time and we saved up and the relative probabilities of probabilities photo are just call efficiency others probabilities of course do not completely determine the complex numbers here or not well if I multiply any complex number by a pure state so it doesn't change its magnitude if I take a complex number and I'm multiplied by E. B. I say that faded any angle I get new number was magnitude whose Star Alpha is exactly the same as the original Starting number and so knowing the probabilities it is not enough to completely determine what these coefficients are more information these coefficients just the probability find that without additional information but for the moment Mrs. and of course it is more than 2 possibilities up down and sideways or up down and all the chocolate chip away at another 1 Gamma chocolate chip however but Tom those awful normal yeah right report they represented them a particular basis basis of up-and-down we represented them let's just of we represent them up was just 1 0 down 0 1 a to each other normalized both of them sums of squares of each individual 1 1 of the Cold War tho normal vectors viral normal if we be if we have a higher dimensional vector space with more possibilities reject them we might have 3 vectors altogether 0 0 1 they are still orthogonal and normalized 0 should go here up-and-down in these of not he and we're gonna have to remember the difference between these vectors these abstract sectors which represents state and vectors in real space cassettes telecom operate at pointers He's pointers vectors label them slightly differently on right now what about the quantities that you measure they'll called observables the things that you measure and by Asuncion ever New York may be more than 1 thing you could measure about system and you could always real numbers the results of measurement are always a collection of real numbers about you may combine them together for example a book complex numbers but the results of real measurements your pointers on dialog whatever our real numbers and so observables are things which when you measure them among other things you get real numbers as a result they are represented by matrices or operator strictly speaking they represented by linear operators which you can take just being matrices for our purpose a special class of matrices permission matrices permission H E R M I T R N permission for her might or might pose a mathematician permission matrices and the the fine him much 1st defined a couple of concept about matrices 1st of all there's the concept of the transport of a matrix the transponder matrix drive bad moments and off bad Thomson's no reason why Acevedo by Philips Circuit B 3 3 5 Is M 1 1 minute am too 2 and 3 3 and then the off diagonal comments let's just put 1 of them already here here and 1 and he is and 2 1 and South but once other
Robert the trains foes of matrix is just your act matrix did change rows and columns all it means is you reflected about the diagonal you just imagines written on a piece of paper and U-turn over the piece of paper so that anyone tool rep replaces 2 1 and so forth the way right back the trends borrowers the trends His just written by putting tee up here and it's just the bag no elements are unchanged but be off diagonal elements and 1 tool becomes 1 and M 1 2 2 0 0 adjusting to likewise over himself for OK let's call the trends was we can write that the former transpose matrix IJV IG element of it is just the JI elements of the original matrix Interchange Online Jay rows and columns is called transposed could the next concert is her mission conjured about permission conjured as a kind of complex conjugation complex conjugation matrices permission conjured name embossed tore operation was the 1st is trends bows and then the complex conjugate misrepresented by 82 bagger 2 represented by that I don't the represented by 1st of all Trans posing but then complex conjugate eating everything all the elements that bag moments as as the off we write by saying that beat me she conjugate dagger our matrix is to interchange rows and columns and then you complex conjugate so let's tour example it is an example of a flop was Hawaii 7 minus of 6 plus 4 ii 19 miners To but let's see what's the dagger be permission conduit of that matrix well bad elements they don't move when you transpose strands posing a bag limit stays the same place but we have the complex conjugate we get former miners Hawaii now the element that here firstly have to trends was born former as to why we have to try and polls which means we put we think the change rows and columns and then complex conjugate serve and we get bored plus 2 life here but this 1 complex conjugate in this 1 complex conjugate goes down he and 6 minus writing and that no matrix elements word is but it gets complex conjugate seven-plus right so that see that's the idea of Our permission conjugation flipped the rows and columns and then laughter and then complex conjugate a her mission matrix will 1st look talk about symmetric matrices symmetric matrices let's just go back to the idea of trans boat was without conjugation Trans a symmetric matrix is 1 that is equal through its own transport another words when you transpose if you get back the same in order for that to be the case all that to is that would be elements the reflected elements on the same as each other so I would say that ended 2 1 has to be equal to M 1 to to warn that's the idea of
a symmetric matrix and it just is exactly what it says summary symmetric with respect of flipping it to reflecting about the main diagonal a symmetric matrix satisfies my Jake equals and Jr at a symmetric matrix reversible concert mission matrix is a little more complicated work says mission matrix is a matrix which is equal put all our missions conjured of lumber make products and your machine conjugated it is equal to the original matrix it's called permission it's just called membership at the definition of a mission 6 bucks right mission matrix like by formula for it yes yes yes there's a former right here MIJ is MIJ complex conjugate must Syria these were incorrectly and and and Jr can't grab what it says 1st of war Is that the diagonal elements a real for example the says if you interchange I jail under bag Normandy just get sandbag so as M 1 1 is equal to M 1 1 star that is I'm 1 1 Israel's so soul real numbers on the back novels single number just real real real Rio meaning any real number in any 1 of these places and the money off diagonal elements over here but suppose that this Softbank Parliament over here is the complex numbers then the reflected off bad Norman has to be the complex conjugate then that's called her mission so 2 0 0 7 0 1 plus ii and 1 minus side is remission real elements on the diagonal and complex conjugate 20 reflect this is a kind of reality property of thing being its own complex conjugate from number is says the 3 0 from matrix the concept of reality is not that the matrix elements a role Rio not that each matrix elements separately is it on complex conjugate but each matrix element is used these complex conjugate of the reflected will find out with the Seabay shortly why this is a good definition and I can't feel these are your homework to prove that the Harry very simple but just there don't have names because that was simple elementary Putin's 1st award take 2 vectors can be take me it is product of 18 would be November when you formed the role that the beat you use the complex conjugate he show star be star 1 B start 2 0 complex conjugate 81 82 the 1st pyramid is but a few did change but will put be A. here in the bees here we're dead this is correct not complex conjugate the way I think of it is that the role of actors like a complex conjugate certain column record and if you interchange if you interchange Our the order of these you're taking the complex conjugate B in the complex conjugate AT&T multiply complex conjugate just come out of the 1st the 1st element on the check is the inner product to be with a is a complex conjugate of via the product of a would be that's had it's extremely elementary anybody seen the is familiar with it next statement that was less obvious take any matrix permission or not from share man Charles Turner are of this is an operation that I haven't performed yet from the defined point of view right now it very simple your other matrix M a new multiplied by that gives you a move that this is a new record and you take that no sector and you take the inner product would be you could work out what this in terms of matrix elements of Annie entries they don't be very definite expression 1st multiply the matrix by a multiplied a by the matrix you be left all of the new vector did you could call a prime and you take its in a product of baby this is related to not equal alright equal but there were no changes in the change everything 80 per diem left that's kind of a complex conjugate of 80 per beyond all right and put permission conjugate of them everybody has been complex conjugated and all rows and columns of when you and beer into changing column wrecked overall that and your complex conjugate saying I am renewal figures from conjugate you interchange rows and columns and you from a new complex conjugate this is the number and His quantities and number of complex number Gerald M. gives you a vector in a product of a vector with another vector is a number simple number it's not make Texas now that there's just a number this equation is not right but I do it complex conjugate I've complex conjugate in a have complex conjugated be enough complex conjugated am after the whole thing trumpets conjugated so that's a a simple fact about
about dumb our complex conjugation of matrices sectors but let's suppose that our matrix is permission let's suppose of the matrix in question Is permission if it is mission commemorating I can erase this hour conjugation sign equal who its own permission conjugate definition of emission of sulfur emissions matrix on from now on let's think about her mission matrices for emission matrices mission matrices b misses call matrix element score the matrix element of M B a matrix element of em that is the same as the ADB matrix on the very complex conjugate not 1 less thing that will make a point for you if a and B I happened to be the same AT and B happened to be the same vector so outlets would b a will be equal a word 80 per worker says it says matrix element of em evaluated between sandwiched between 2 states which are the same as saying state 80 on both sides is equal to which are complex conjugate was at but a Yemeni that 3 0 of Iraq mission matrix this was a fear that I was contested cruel but just proved it if you have a root of permission they tricks van we should give the subject name it's called the the expectation value why it's the expectation value will become more with that as well as do with anybody's expectations is something else it just hasn't made it's that expectation value of and in the state a I remember just keep in mind the way we're going to be describing states His through the way we describe observables makers so you could think of this basically as average value of when the state of the system is a yacht but no Persian Barack now I'm still early learned we learned that were having heard mission matrix its expectation value is revealed well we haven't gone through the details yet but that sounds like a good thing if we have AT and observable which is represented in which when you measure it always gives you a real number then obviously it's average value should be real and that's what we find referee permission matrix its expectation value a a given state Israel that's what makes the her mission matrices special and awards make what makes them permission matrices natural candidates have something to do with observable quantities because their expectation values in any state does not have many many states any 1 of them was satisfied book any questions up from now you where Blair it of fun mail the the bed there but that anything you really measure is always a number now you can you could decide to say I will describe points locations of particles on the surface of the earth by giving a complex number I'll give you an address in Manhattan thereby giving you a radio and had his way out on the grid and give you a complex number our about you don't have to do that you could represent any observable quantity as a collection of real numbers you can always represent the results of a series of experiments by a series of real numbers right and I real numbers are core gerbils with they below the complex we get together and complex combinations just because of the technical historical definition we don't call our part collection of real thing patient will lead to think about my number flinty shop for Miller Lite Light 15 seats but may not desirable but Sunderland on date and most of them are I think that I think is a better I think the question if you're really worry is why do we get why do we have to use complex numbers quite that what you really want use it that would driven complex this will become clearer but has to do with time and reversibility and we're not there yet we're not ready for a week could of thought about doing quantum mechanics were all all-wheel numbers I'm never having gotten compote as we missed some very important things you will see that you will find out about them but let's let's write down the postulates examined the consequences and then I will tell you what we go wrong if you restricted use of real numbers it would have to do in particular Armed with b evolution of state vectors we have not talked about all is how things change with time really has to do with how things change with time that aware that I come from hold on hold on to the question a Newcomb back again I I realize it's a different question and U.S. but I think is the right question why what complex numbers or is it just a mathematical formalism described Paris a real numbers in certain ways use OK what wire we driven to use them in quantum mechanics that has to do with time so it will come through it on that we still going mathematics we cannot do this subject about mathematics Hey I can't fake it
I can play a lot of stuff about quantum mechanics and give you also to mystical experiences about entanglement and when you walk out of you you will know more about entanglement and you know when you walk in but you have this fuzzy feeling about it there was a wonderful mysterious but your it would sell out so if we run a really do it honestly are forced to do what are the better I really do it right in the symbol and the simplest possible context and I'm giving you the simplest possible context discussed quantum mechanics basically two-level system and they will do 2 two-level systems in there will talk about the entanglement of 2 two-level systems and we understand that in some detail endeavor and you know what the words really mean the next concept and basically the last mathematical concepts that will lead the today and possibly for quite a while his concept of eigenvalues and eigenvectors particularly eigenvalues and eigenvectors of permission operators so what is and I get value and I well I give you any matrix are operator matrix when operated there may the if that's a In general may not b certain sectors that 1 I applying the matrix through them don't change the vector except for multiplied by a number another words that may or may not be and typically there will be In fact in the case of permission matrices they always are certain that there we take a matrix it will exist certain sectors which have the property not every sector it will exist certain sectors of our star associated with that matrix which are called that eigenvectors which have the property the 2 just multiplied by a number the number depends on the vector I will give you an very very simple example right now supposing we have a diagonal matrix In general who will be called lying in value and eigenvectors its complex but if is our mission then it follows the plan is real but proves that but suppose we have our mission matrix we have found and eigenvectors and I give you let's see what we learn by taking me in a product of this equation with AT itself have and 80 80 and multiplying the equation armor left for a land that is a number it just comes on a outside national back nuts about 5 and a half minutes remember that we proved that any religion matrix 80 m a its expectation values Rio we also proved that in a product of a with itself Israel so we have an equation that a real number is equal to a real number times alike in value well let me is the Eigen value was real or another way to right it is provided by AAA and Horatio 2 real numbers is certainly Rio so the 1st statement is if we don't succeed and find a eigenvectors an eigenvalue view of our permission matrix Eigen value will be Rio that tell you the secret the secret is that and is an observable In the values of it that you could measure when you do an experiment arts ID value means the result of an experiment to measure there will be 1 of the Eigen values Of the Matrix and that's the significance of eigenvalues possible measurable values of these observable that's represented by and we didn't some examples from while yes the land the Eigen value on the values that you measure the eigenvectors that's associated with that particular Eigen value that eigenvectors represents a state with their when you measure the quantity and with certainty you get value and so the idea now is in the eigenvectors go together there certain states for any given observable certain states and certain are certain states which when you measure that particular observable you will with probability 1 get a particular answers and that answers the Eigen value so let's just dove example I need easiest examples involved diagonal matrices is a diagonal matrix it's just there year of some letters M 1 1 0 dollars just to buy 2 was good enough for us and to tool as a diagonal matrix what are its eigenvectors remember vectors which would you multiply them by the matrix you just get the original back-to-back forms a number understudy against the 2 eigenvectors of the matrix 1 of them has a 1 in the opera place let's check what this is matrix do we 1 multiplies suspected well we take the role multiplied by the column that puts and upper place here am 1 1 Crabapple Court Rice 0 times 1 plus and took 0 2 0 but this nectar is just M 1 1 times 1 0 there's a what's happened the Matrix forms of that it gives you number times the original vectors and other lawyers in particular doesn't change the just using back the original vector kind and numerical number in this case in the medical number happens the B M 1 1 so vector 1 0 0 is in eigenvectors of this matrix with ID in value and 1 1 there's another eigenvectors and the other eigenvectors used 4 0
here no 1 here but still we get and the top place we get M 1 1 0 0 plus 0 times 1 0 and then we give 0 ton 0 plus and to 2 times 1 and 2 2 and that's just aimed to 2 0 0 1 so I found another eigenvectors matrix and vector equals number upon the same sector and this time the ideal value was too Cho so the toll that there's 0 1 and 1 0 on both the eigenvectors with different ideas values and 1 1 2 to see where this is going for example if I'm interested in art then the spin of an electronic I might describe it by a matrix with his where they won an a minus 1 then Eigen values would just be 1 minus 1 the Eigen states would be 1 0 correct equation sorry state would electron up is an eigenvectors Our a certain observable which is represented by a matrix any other eigenvectors 0 1 and that is eagle from minus 0 1 minus comes from minus over here so the tool the 2 states of electrons up and down our eigenvectors of the spin operators spin operator which is usually called 3 2 definitions simmer 3 years the best they're eigenvectors with 2 different values in you see the pattern that the value that you measure plus 1 0 minus this 1 is the ideal value the eigenvectors is a state in which the Eigen value is desired measured quantity a question it then inside bizarre normalized rome young we normalized than they give you another example which is last of the West tribuloides inequities there were 3 by 3 and buy in major cities the same pattern would be there that if the diagonal matrices of diagonal than the diagonal entries are the Eigen values and the eigenvectors adjust vectors with zeros everywhere except in 1 place 1 so you could check that out for yourself experiment around the back of our matrices we give you want other example the other example involves a diagonal matrix these matrices incidentally will play an important role in of talking about 4 from finding me just arbitrary matrices intermission it has real diagonal elements 0 was a real number and off bad elements a complex conjugate to each other 1 is its own complex can't you also happens to be symmetric are both both permission and symmetric end now let's see if we can find the eigenvectors event yacht rather dented do of the algebra on show you what they are and then we will see that they are I the toll eigenvectors this case OK now what's wrong with this eigenvectors Mr. Big not normalized 1 square plus 1 square used to but is a statement of a vector was an eigenvectors it doesn't matter what normalization double Triplett a anything else still lying that the you want normalize if you would write 1 over square root of 2 0 1 0 0 0 square it doesn't really matter our checking whether it's eigenvectors but let's it just a worth remind ourselves always normalized at what is this matrix times is back I dig 0 0 1 and we multiplied by these 2 elements where we get we get 0 all-time high command 0 0 times world Mosquero to post 1 warmer squared 1 over squares and likewise in the bottom we got back exactly the same vector so this is an eigenvectors on this operators with Agon value plus 1 what about 0 1 1 2 0 0 I find another eigenvectors yes I can't reason I could find this could signal the answer show the but the aides and is not hard to find as defined just 1 illustrate the ideal eigenvectors another time value would go about finding bloody your illustrated the other possibility is 1 hour square to mind 1 over square of another words instead of having the same element here and here we have opposite let's check out is that Xerox times this 0 1 times this is my name C minus 1 or squared of 2 and then here we have won over square too 0 0 0 war over supporters through and that's the same thing as might the vector that we started with we have a minus sign near the opposite sign here it is among its original back death so if found another eigenvectors and this time Eigen value it is my guess what referring eigenvectors when I get value plus 1 and eigenvalue minus 1 I defy you to find any of the argument that there on wrong on and that's there are no more Eigen values or eigenvectors right got noise multiply and eigenvectors by members of the eigenvectors but no more no more of which are not just commerical multiples of the same White and now that approved a fundamental theorem of quantum mechanics very
fundamental about permission matrices are going too fast and yell out after OK this 1 city and not yet out of casing 3 it is a definition warm warm 1 minus 1 0 0 0 and has I get value plus 1 minor Swan said no 1 is equal to this 1 1 0 0 0 Soviet to interesting things love about what to say 1 interesting about both of the squalor of each 1 of these matrices is everybody now have a square matrix a multiplier matrix by itself the squalor of each of these matrices is the same it's just plus 1 the square of this matrix you just square the elements 1 squared minus 1 square of book plus 1 square signal 1 with squares or 1 1 0 0 and multiplied by a 1 1 0 0 I the 1st element in the upper left-hand corner busy in a product this times this 1 the 1 gives us a 1 0 times 1 plus 1 0 0 1 NYTimes 1 0 0 2 0 not sorry 1 time 0 0 times 1 0 and 1 times 1 0 times yours 1 of us said no 1 squared and said 3 square on both equaled the 1 that's kind of interesting fact noticed that's also true of the Eigen value Of the matrices they will both plus and minus 1 ever square over things 1 it stands to reason that their ability the plaster minus 1 well from matrices particular matrices like this but matrices general the square of the Matrix Is the unit matrix Mrs. matrix is called is viewing matrix called I'm just EU matrix were the ones on the diagonal signatory squared cigna 3 square is the same as signal 1 square people toward the 3rd smoked independent of the the fact that the square of each of its I'll be Eigen values as equal what it means but the measured value that you could measure are such that the square was equal to 1 that means that you can only measure plus 1 or minus 1 is 1 more Latexo right down Sigma sequel to minors I I will come back to the city matrices Republican tonight but we found the eigenvectors of signal Sigma 3 we have not yet found the eigenvectors a similar and I will tell you in due time what their significance you could if who is prepared to say that because what its saw was the rest of the year basically we look that that well but it would be 0 1 1 0 nature of how those images that I noticed what there work but you're but we won't be too cold that were so that analysts say is also the unit that will be reflected in the of of the right to use him as a leader of 1 of through but was where working 3rd that's is sending 1 of best in each year that year reflecting but said the numbers that are also were not some of which will be used this 1 square also yeah I'm back to where it is also my form or they all property all that's a letter also proposes that will be in touch 2 are also passed by what was because the argue that but there resort are also W while that may prove to you now the following interesting of post and this theory it significant highly significant surpassing you'll have an observable has more than 1 eigenvectors with general well I the number of eigenvectors there has usually equal to the dimensionality of the Matrix it is mentioned earlier matrix and 3 by 3 matrices permission they will typically have 3 Eigen values and 3 eigenvectors fall by 4 0 4 eigenvectors of 4 argue about use but he is I have any 8 permission matrix M has tool different ideas about 2 different died in arrears that means the prison eigenvectors that 80 with Eigen value Alam vary and there's an eigenvectors beef With a different idea value B B then if when they can land the br differs what it says is that any NBA will be off to each other before I probably miss it would mean what it means is that if there is any of observable distinct quantity that you could measure which will distinguish between 2 different well 2 different distinctly different possible measurable answers that you could get for some observable the Nets equivalent to saying that be eigenvectors remember the eigenvectors other states In which you definitely measure land that the eigenvectors states where if you make a measurement of and you will get land Reed is the state in which if you measure M you will definitely get Lambert beat these which correspond to distinctly different measurable quantities it's distinctly different measurable values of the same quantity they are orthogonal that's that's these the basic idea is orthogonal states are distinguishable measurably distinct and you could distinguish between them by measuring appropriate our our observables for example will we're ready have some examples are 2 examples of we 1 of them was 1 0 and 0 1 these 2 0 eigenvectors of Sigma 3 notice that their orthogonal because they don't share any many elements this plan
is this forces fund 0 the orthogonality is thing that tells you that is the unique experiment you could distinguish between you distinguish between them by measuring the quantity which about which is represented by permission operated has 20 words but but you get the idea and they drove deal Matt have this is with 2 different there's been a AT&T must be orthogonal which we do this easily Labrador just at the joke goes few definition we 1st of all multiplied this equation the 1st equation on the left by they use so will rewrite it then b M. Eddie people's land that AT crimes b a I'm multiplied both sides might be by the vector B now good those sooner the thing here by both sides by so we get a M B equals wearing the being pardoned but now I would like to switch our story of the text always confused make that Ritter 2 equations now let's take our this acquired from he'd heard that brought conjugate 1 of these equations want complex conjugate of 1 of these equations which was Richard Secord this AM a complex conjugate his work areas remember since this remission we don't have the complex conjugate if that's equal Toland the B star plans be 80 0 OK now are thrilled to do necks I yes there's now yes just announced complex conjugate everything at the summit which of but I forget the conjugate something we do it we we got sorry this country a direct gap that before we do that collects complex done here is said if I interchange Bugsy which of them 1 arm was reversed by change in B at the same time that permission consular them which I don't have to do because intermission and take complex conjugate I get become accomplished view of the right side not just public complex conjugate the whole equation the main just throw away the star forward now I have 2 equations that get rid of the middle 1 here 1 of them says BAM a equals land 8 times BAT the other ones as B M a equals when the B & B it will not surprise you have the conclusion is tool for possible conclusion 1 conclusion is that land equals land the beach but Todd already and talking about to lighten values which are known to be different sold by assumptions when they landed B saying when they the br not the the same is only 1 way this could be true river upset B 8 0 0 number words let's subtract these 2 equation the subtracted 2 equations on a left-hand side we give 0 on the right-hand side we get a land that minus slammed the times be paid if a product is equal to 0 that means 1 or the other factor is equal to 0 product of 2 things can only be 0 if 1 or the other factors equal to 0 so 1 possibility is of Monday's acquittal and the big trouble is a stroke possibility but I explicitly said let us consider 2 different died in value to different values Of the same our observable if a different than when they the slender B is not equal to 0 and the only possibility is that be and NATO or orthogonal to each other in a product of B with a equal to 0 so what follows permission later seized 5 distinct Agon values I can immediately conclude food these are orthogonal let's check it out for the eigenvectors well Sigma tools where the eigenvectors stigma to give Iraq right here now no I'm sorry single ones where Mayor let's check it out for certain award notice that entries here are real and we don't have to we don't have to complex conjugate anything all the entries Rio the in the product of 2 vectors all of whose entries a Rio always have to do is take a product of the 1st 2 entries sorry part this entry with this century and they had to with the product of this of this 1 so we get 1 0 square two-time squared too a half and then won over square to minus 1 over squared to minus a have half of minus a have 0 this vector is orthogonal to effect so there is an example of the orthogonality of eigenvectors with different ideas values permission makers as I said the significance of this is deep says that whenever there is an observation that you could do no uniquely distinguish between 2 possible states to assist them Our 2 possible values of observable be states associated with them off Fargo other you could say they'll wave to states orthogonal it means that there exists some measurement you could do which well uniquely tell you which 1 of the tools is a realized in your at your In your system that's that's 1 of the fundamental you of quantum mechanics again buds take a little break the eigenvectors of Sigma too are so that you can check them yourself with 2 0 eigenvectors of signature rules with but it does the 2 eigenvectors of signature Our 1 stick over square is to normalize from and the other 1 is 1 minors I want square 1 of them has eigenvalue plus 1 or the other has I value my 1 so you could check that yourself through exercise please do it and you see the signal 1 sing with 2 wins in 3 all similar to each other and that they have the same Eigen values plus 1 minus 1 they all square India 1 they have another interest in property compose a moment I think but let me tell you what their physical means they are observable quantities and with respect and thinking about these of electron spin that tho before 1st time was 2nd time electron spin classically classical physics is like a magnet that as an polymer south of a pointer points along an axis
if we were talking about and has would call the magnetic moment in that magnetic moment is a high I hesitate to use the word vector the kind of vector points and ordinary space is to call it a pointer the magnetic moment or became caught speeding sustained apart from numerical cut speed the magnetic moment saying it back it's a it's at the point space and as such it has components has components in the vertical axis for example was back as a component which is about he the vertical direction and as a component of the horizontal direction over here and a component of a year earlier the horizontal direction the most components characterized b . there they are measurable quantities for a magnet you could decide to measure the x component of the nite of the year of the magnetic moment you could decide to measure the white component but you could decide the presidency component you might think classical you'd be right but you could measure all 3 of them simultaneously you could do it experiment measure all 3 of the components of the magnetic moment simultaneously and that way figure out exactly what where the magnetic moment is pointing but say that question whether you could measure all of them simultaneously for an electron or not but you can into now but you can measure he 81 of the x component the Y component of zinc component honey do it suppose I wanted to measure the x component Exodus Way I put in a big magnetic field and I check whether or not picture photons yes it emits a photon Bend it had what 1 component or the other and the only possible answers on minus 1 plus 1 you CFO can't you don't see a so the components of the electron spin have only 2 possible values each of the components the x component but why component Lindsay component and what's small we receive admitted that the component any old direction if you measure it will only have 2 possible value that's how little hard to think about but by midday right now what signaled 1 sigma to 1 3 are they represent the observable values of the components of the electron spin along with 3 axes of space the 3 axes of ordinary space all show you how that works and how we can construct the component along any direction moment but not that they don't have very similar properties sane Eigen values so if you measure the possible values that you could get in an express train the signal 1 you get 1 minus 1 in free you get 1 in minus 1 the Sigma till you get 1 in minus 1 but all you could ever get when you actually measure receive the same is true along any axis but before we do let me tell you last I think it's Celeste postulate of quantum mechanics and nears fears mirrors it's probability interpretation suppose you prepare a system in a state of particular state let's see let's call at State beat somebody gave you an electron or whatever happened to be in a particular quantum state now there is somehow observable coed M. some observable that has an eigenvalue land that which means that if you measure em you could get the value Lambert we may not get the value may give some other land but measure and you get 1 of its use and be associated eigenvectors news editor commas between understand Emison observable meaning peseta mission operate a Land that AD is 1 of the possible values that you can measure and the associated eigenvectors land we just any old state and estate whatever not particularly of Oregon State of them could be remarks this just any old arbitrary state we could ask the question if this system is prepared and beat and we measure M what's the probability of being and so will be Day when they are possible values you a measure him a thing you measuring and 80 Is the eigenvectors and with Eigen value and the answer is that the probability the is given by the expression in a product of 8 would be times its complex conjugate were where our fans its complex conjugate very easy later write complex conjugate is just right it can be a which is just the same as multiplying lives complex country remember that when you multiply and number by its complex conjugate you always get a real positive number if a and B are normal are used to easy here to prove that this is less than 1 less recalled award a br unit length than the inner product between them as a magnitude less than 1 so this number is always less than a regal to 1 and it is a probability it's the probability again our thoughts state what is the probability of you start with an arbitrary state aid system prepared an arbitrary state and you measure the measurable quantities this is a probability that you get land that 8 start but I said the With they outside receive again across a river research on prepare the state beat arbitrarily preparing state baby them the probability that you measure land they will be In the product between be in 80 square complex country a real positive number given by the square lawyer I caught square but I mean times complex conjugate Our a with that's a problem of notices so symmetrical Maine beat is symmetric an a and but but for the moment think of it asymmetrically prepare the system in a U.S. prepare the system would be a new measure and ask what the probability for land that's the that's a basic pass to acquire mechanics are always assuming that all of your vectors are normalized always assuming that you're vectors on normalized state b is normalized and the eigenvectors a normalized where is the powers of that is work if Beaver real vectors of real factors the 2 of two-dimensional vector space it would be a cosigner mangle their World Cup lest they still have an angle is that there's a brewery numerical southerners also deserves as it ways for users of what is true is that these adjust ordinary
three-dimensional vectors real vectors Mary three-dimensional that the bar product of 2 vectors to unit vectors is equal to a cosigner the angle between sought cut like the cosine of the angle but not the real thing not real In general the swampy real for multiply by compote backed by the please that there is a God farm workers hurt her Exxon if beat happens to be 1 of the eigenvectors of ammo with different I get them and AT and be orthogonal so what it says is that if you prepare the system in 1 eigenvectors corresponding to a different Eigen value than the probability that you get AP is 0 perfect to prepare late in in 1 eigenvectors measure the probability you get a different Eigen value is 0 laps that's part of the year of the probabilistic interpretation he had always says ever quantities definite legal 1 thing surely article the other thing but the but expressed in terms of In a products why that yes sir as a barrel all yet we get away with too many many such things this is the simplest of things we could do and moment this is our basic quantum postulant probabilities let's say give examples what's the probability let's take the case With a 3 is equal to 1 0 perfect would prepare we prepare is equal to 1 0 which means by the way that signatories include plus 1 but now that the measures 3 Wheeler measure signal signal what measures signal 1 memory may ask What's the probability that we measure signal 1 that we get lost 1 war minus 1 case so let s what what the probabilities that if we prepare areas expand we prepare the system would spend up by by putting in a big magnetic field pointing up and then we measure signal 1 which is a component of the spin along the excess what's the probability that we get plus 1 what the probability that we give 1 couldn't guests for where work about right so where's Eddie is an eigenvectors signal 1 with Eigen value at 6 plus 1 so aII is equal to 1 over route tool 1 over route to what inner product between these quick quick quick quick were I was squared to erect the grouped with square that 1 half so a His 1 over square of and kind complex conjugate that's usually written this way and complex conjugate its absolute value square which is multiplied by complex conduit is 1 half so what this tells us is if we're preparing electron upstate and we measure the x component we have a probability 1 half forgetting plus and it will also be true that have probably 1 here forgetting minus the sea that what's the eigenvectors that's associated with minus 1 of 1 minus here the Treaty of the 1 here they of as an example and this is a characteristic example you prepare a system in this case the system is prepared with spin vertically up and measured in the horizontal direction but go the opposite way get the same answer let yachts to let's do we start beat now this time we 1st prepare lecture horizontally we measure the vertical component that could be the same in a product the horizontal this time be Mr. just them interchange be a further widows which there'll be very we've prepared the system would expand point being with the x component positive and we measured busy component which just exactly the same thing when him let still more complicated case I've told you what the eigenvectors of Sigma to are so let's as the most complicated question we can ask What's the probability that if we prepare the state Ted With signal 1 equal positive 1 that means B is equal to 1 0 square 2 1 0 square and this time we measure signature will not sit 3 but Sigma too so this time B is 1 of the 2 0 eigenvectors of segment to with stickers what 1 or squared to fight over square to start this is 18 this is AT take the eigenvectors signature so we start With this spend pointing along the x-axis and we measure it along the Y axis we start with the point along the x-axis but that eigenvectors warmer along 1 axis and now we measure it along with 2 axes OK could somebody computing in a product of these 2 I think I can begin a product of these tools we 1st of all have the complex conjugate forget complex conjugate 1 over the square a two-thirds over squared 2 is 1 half and then plus but we have the complex conjugate so that means miners are 2 or two-time grouped with 1 one-half mile this is the in the product of a would-be I'm not to be a B with labor force what's a complex conjugate of this would itself but get that we have multiplied by 1 0 with a 2 0 plus I'll I eat over to show could be imaginary parts canceled we have a one-half times a one-day afford an argument we have a minus 1 half times 1 here with my with my side of the imaginary parts canceled the real Prada 1 half 1 him was 1 quarter plus one-quarter one-half so we say if we the way the say this now is if we lineup the speed direction by a strong magnetic field and then we measure the spin in the perpendicular direction we have I have problem in any impact in the perpendicular direction we have a probability of a half them this way and have votes what so Andy polyrrhiza school the spin polarization any spin state if we started out with a strong magnetic field and freeze some direction if we remove the magnetic field and measure on
orthogonal component the spin it will have equal probability of being up and that was about the components of space and in some arbitrary direction let's discuss the components of Spain along some other access our week maybe it's a good to stop our well but what's what do with an earned value promise to do is to get lot of cases last year that nite we are talking about their of players these are some of the however I am going to be but not so right so we have to we have distinguished our world vectors from pointers right so here we have a situation where we have Willie Harris of point there which has been forced the point in some directions and we measured the point being this unity with put perpendicular direction earned respect her that what let's just say that or better yet let's look at the eigenvectors which corresponds to the point being this plus 1 the X direction and compare it with 3 eigenvectors corresponds with the point fingers In another direction and not orthogonal eigenvectors about orthogonal vectors in need in this and yet now right about 1 there 0 but what is the thing which has Zero probability of pointing out like what but now wonders pointing down the 1 this had been down in this sense is orthogonal pointing out so we have those so let's use the word perpendicular commune ordinary space and orthogonal is of our vocabulary pointers instead of vectors vectors means state pointers mean there's orthogonality means state being orthogonal being distinctly different from each other measurably different and perpendicular means perpendicular at 90 degrees in ordinary space In the pen mean something a little different independent doesn't mean low perpendicular so let's stick Woodward now our independent simply means that pointing in the same direction and a set of of the concept of independence linear independence with the butt of which is not which is different than bargain mutual bargain hours so in independent bargain our quite different our linear independent anyway OK now let's let's count the classical pointers I'm interested in something Your Majesty now component of the pointer along an arbitrary direction along arbitrary direction in real space what direction picket direction Alec picket direction we direction by picking a set of components of a unit vector we unit vector and had a set of components in the poll political unit that SRO unit pointer a unit pointer we pick unit point there ours in real three-dimensional space 0 1 2 Bahraini 1 is the same as X 2 was the same as we foresee the article a somewhat arbitrary pointer so some arbitrary pointer this point along that direction and let's take it to be a unit vector which means it's 1 unit in it's described by a set of 3 components the sum of the squares of components have they add up to 1 but he over 3 components such course that game the unit vector is usually described by putting a have over like that in fact means a unit that this is a standard notation for unit vector the unit point them as please correct made on Monday due back a unit pointer is indicated by little hat and has a set of components the components are in X NY in zeal and 1 end to an end those other components and they basically this component of this component and the you add them up and you get a pointer pointing some arbitrary direction these other components of the pointer know what what about the component of along the direction if I we come to regret that the component of the speed of the component of the spin pointer longer the arbitrary direction here is given spin-off now looking classically thinking about classical magnetic moment classical spin that our stood advance in Hey it is component of all morning directions and that's what I proud you it's a component of 1 vector along the axis of another sorry pointer what is not end its see 1st component the x component costs in a 1 time and 1 plus segment Tool times 2 plus signatory pretty apparent and parade Will this gives us an interesting can vary In the quantum mechanics will be operator which it is a component of Spain all along a direction mainly we take the operator signal 1 multiplied by in 1 and to it they operate a stigma to times in To add to with the apparatus With 3 and multiplier brains that a candidate and his physical activity which we can cost of about areas but an operator operated a week in numbers just numerical numbers of here said muzzle matrices of multiply matrix by number we just multiply all the entries in number and then we ended up with like this then actually given 3 and the 1 and 2 and 3 weaker near the mouth of the stoic and one-time signal 1 what signal 1 signal 1 1 affair multiplied by and 1 books August that's signal 101 what about sigma took turns into said Nadal is minus ii party and now we multiply by by into sit with 3 times in 3 but to the and 3 which is 1 minus 1 multiplied by and 3 sisters and 3 and 3 0 0 now we end the mud where we get on the back of these have no diagonal elements this that forget and 3 and 3 Miners century we get
in 1 minors I 2 men and 1 plus to is a 3 3 components and 1 and 2 1 and 3 the sum of the squares should be equal to 1 because unit vector and here is the operator Carlos funds for measuring the components of Spain along the direction here so what's expanded experiment is we beg electron really put it in the magnetic field pointing along the indirection we see what we get proper down this is gay operated corresponds to that in 3 diagonal lines and 3 and 1 minor signed 2 was a summation yes termination Rio Rio and this was a complex country of theirs mission Is square equal the 1 0 0 votes are needed if any Utrecht well I add are expected it was sealed here so I would build in America show you an important property and let you check it yourself this is an important property all these matrices 1 you can check bulb arrival well derive its importance and then allow you to check if there were robbed by yourself let's take a signal 1 person with tools to more than 1 person would do and supersymmetry and women this former former lead in this form and Square where we get well works 1 in 1 forcing tool to proceed 3 and 3 times 1 and 1 for us in the 2 0 0 and 2 plus in the 3 and as mobile Begala slew of terms the 1st terms 1st Tanzer big things like signal 1 squared times in 1 squared multiply the South targets it Watson born square 1 so from the from me in 1 squared sin 1 squared terms will get in 1 square what about from the similar till in into square for about the same with and 4 3 squared what's that that's 1 outlets look signal let's look at the beam which multiplies In 1 friends and to hear the thing which is the multiplied In 1 turns into an 1 to wear Whitaker contain contains signal 1 segment but that is another term like that which is Sigma Tuesday no 1 a one-time which is signaled 1 on the left signature won the right and the other in to 1 left signal 1 on the right while I bother writing said mourns simmer tourism adorned signaled once the scoring calling them both signal once been repealed because when you multiplying matrices we'll water can be ordered accounts when you multiply makers matrices in general have different products when you multiply the different orders that technical terms that they don't commute where he or she will find themselves if you want to do this easy each square there is a rare progress at the Valley may have a to weighing already I tools these you'll and under terms similar emerges in 1 well they said what were sleazy you made This is a year X 1 what's is doing here it was that we all want that we want the square of the component of the spin in any direction to be the same as in every other direction is nothing special about a new direction as well child multiplied and multiplied that multiply I miss you go home and do yourself proved self that signaled 1 time signature is the opposite of the negative of 2 times 1 now maybe that commuters as special property they intake commute which is to say that when you change the order changes this sorry this is something to check to sit down and do a numerically once said to oppose Sigma Cosimo 1 0 0 similar to a signatory cosignatory the 3 sigma too with 0 let me if I met with 3 minutes but the 1 case 1 case for an hour the rule of law avoided Majorie numbers but doing so 1 3 1 one-time signatory What's that that's equal to 1 1 0 0 0 times 1 0 0 0 minus 1 and that's equal to achieve over 0 minus 1 now warned 0 that idea barite very big the facility of 0 8 1 0 0 0 minus 1 0 1 1 0 right in the 1st element 1 5 0 0 0 2 1 0 again but now it's go through 1 times 1 it 1 and now we go down this corner 0 10 0 minus 1 0 2 notice when your multiply them in opposite water you get exactly the same thing except from side when you Eddie's 1 symmetry in the trees and 1 them together you give 0 same Mr. for signal 1 sigma parents in reducing rate so these are 0 so I learned River learned that with this particular choice of makers to to use the squalor on any component certain along axes and Texas whatever it is 1 that we'd be Eigen values of such a matrix of plus or minus 1 so any matrix widower right but Herbert 3 the Eigen values of any matrix of this form of plot summary warned this will check next time for what it says as the possible measurable values of the bid along any axis Of course of minus 1 is misses a very weird thing Enquirer character only get plus a minus 1 measure along any axis and then take another axis you could also make it possible in this 1 for summers warned will be the probabilities which reflect whether tool in of global that's enough for this time in our probably saturated viewers OK and member of the error of the preceding program is
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