The Theoretical Minimum | Lecture 3

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The Theoretical Minimum | Lecture 3
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January 23, 2012 - In this course, world renowned physicist, Leonard Susskind, dives into the fundamentals of classical mechanics and quantum physics. He discovers the link between the two branches of physics and ultimately shows how quantum mechanics grew out of the classical structure. In this lecture, he works through some of the mathematics behind vectors and operators as used in physics.
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amount or mathematical interlude with begin with mathematical interlude again about what you're algebra about the vector spaces but the idea operators of fully do I want the let it be fully get operators I was a few more things about vectors a few more bits about the mathematics of vectors most of these bits have to do not so much with deep mathematics as good notation good notations it is worth an awful lot when you could just manipulate symbols and ways let us on a prearranged and the do it easily and comfortably was enormous benefit map with doing abstract mathematics certainly but also the abstractions a mathematical abstractions of physics we abstractions talk about were largely due to Dirac Paul Dirac who really was the 1 who saw the way acquire can fits together so much talk about that we have a basis states space of states which will come back show is possessed a linear vector spaces we decided to promote applies state numbers of new states back and laugh vectors abstract vectors maybe idea of fraud may be march universities the space stage just abstract go multiply them by numbers of our case complex numbers that you can and them and there are 2 kinds of vectors of bra vectors and a cat vectors roughly speaking related by complex conjugation off a vector space has a dimensionality dimensionality is the maximum number of orthogonal vectors it could confined find in that space and wants you know the dimension of the spaces you could look for a basis of vectors the basis of vectors is a mutually orthogonal collection of normalized the terminology was sold normal based was the only that the role final each other normal nor normal means normalized normalized simply means that the vectors argued at length a collection of them was maximal toward can't find anyone because no more actions left at the point of basis and the number of them is equal to the year of the amateur hour suspects give is such a base you can make you can't write any vector in space as some kind of some of those bases vectors so we can write for any vector aII metastable the what I I years they for the bases vectors of course a bases is not a unique just as there are many steps on mutually orthogonal vectors in three-dimensional space there are many steps are mutually orthogonal vectors in Marion abstract vector space we pick 1 we pick a set that set a new label FORT we can write any vector as a Osama how fast and I exactly those foods which which were referred to a moment ago officer a set of complex numbers as sort of complex coefficient crimes basis back yard basis vectors rub off ahead I'm labeling I is kept representation but now all we can do it this is we can take the in a proper I want to try to calculate these are in terms on some quantities of swallowing the vectors of Charles right so all we do is we take a look from the both sides of the equation with the bases vector J. can't vector J. the left side we get enough product of vector aII which is think we're trying to express that's equal a song or a powerful I have a product of the vector JAT with that guy but they did prompt other vector J. with my eyes he is either 1 0 0 hits 1 if I enjoy the same vector 0 otherwise why because they were final defies J and then normalized namely viewed might does so this bracket over here this product the brass cast is just a chronic symbol Delta JKE I don't JI 0 0 1 and when you do this on it just our 1 and only 1 contribution the contributions which I is equal to J. on the right-hand side you just get Alfred J. son songs collapsed 1 Turner that's now equal J. a lot so early learned that we learned that the coefficient here In the expansion of any vector is just in a product on Jason vector with target vector with the vector which
I'm described after the fact I wrote Jay there're 0 events it could also be off I was matter which fire which J. C. Iife coefficient is in a product now I could use that to rewrite the sum up here I now know with the supplies are these guys over here so rewrite star not all out I am going to start without 5 I and then write office abide by officer by next to it or here in the office of Lord of the that's that's all our nothing nothing month commuters stage not wanna right alphasub some so I write off ii end but some magic formula that 18 that there can be rewritten in terms it's coefficients or terms of it In a product of the bases factors times the bases vectors themselves somber why it's kind pretty formula and it comes back Over and over again was says whenever you see this summation of white times are high in that form the suffer away is just a equals a but it's a expression for a vector in terms of its component I laughed so 1 simple fact the same thing is true for brought there's exactly the same thing Bravo using you would write the following forms from brought back there is also a song or ii of the in product of a bar brawl with we bases factors behind the basis Bravo basis rather actors just complex conjured if you like a basic basis kept track of these equations withdraw however find out that they were enormously useful enormously useful Our and powerful even know are very very simple are now icon of the notion of linear operators states in quantum mechanics are vectors 1st a little bit about how that works in the case of the stables observables observables need the things that we measure things always measure objects that we measure quantities that we measure the observables are related who lead operators in this space I saw was a linear operator and to take us the rest of the evening forms in detail what has to do with observables but that I've you have a moment which is doing a mathematical interlude now what is the notion of a linear operator Vallejo operator is process if you Iike which you apply vector somebody or rather I can remember who might John Wheeler like to describe these things as machine you would say a linear operator is a machine with poor little port into 1 report you put an input vector and then you turn the wheel and outcomes comes in output vector of hinted safely which acts on input vector given out of vector star of Webster starts call In a linear operator the nearest will you a sophomore just talking about operator at a machine egos 1 outcomes of Ilorin and we'll talk about what it means for the human I took it is some operator and that means that giving when Yemen acts on it simply gives a rector unique given any it gives a unique beat not being the Commerce may not be Converse officer may not be true it may not be about that given a B on this side voters only 1 unique AT which is rise but it is true that In apply to any vector Eddie Jews a unique reflection of a which is a process and acting on a noun or as a means of and linear operator well 1st of all Kelly act on steep climbs a will times a vector birthright what's right and can act on Zepa and JAG times is itself vector because you're allowed to multiply that by complex numbers Zeiser complex number the rule about Lily operators is if they apply it on constant kinds elected they just give back the constant times they would have given foreign a class or just a bruising EU complex number the complex number can be brought through and an acting on twice a gives twice
what would've give on a an acting on 3 times a 3 times what end-user made an acting on or IT firms and complex number I just use ii terms whatever and with you money at the 1st rule about Harry and the 2nd rule about linearity is In practice on some of collectors a posture bt then would you get it is just some of what the machines bailout for aII and with machines but our forbade with machines that South when you put in a Class B physicist some of what our am can't quash army be Nevada but that's the notion of a linear operators no water it than America are firmly recess of full over rules about when operators but she will you be more concrete about firms everyone to make your work less suppose that and try as a equals be argues leaders as well be so import from now b is out for me and you just letters it wasn't anywhere want and fans 80 equals be I'm interested in me component of BT along the axis I know that I mean these objects over here after they are the components of the vector aII along the various bases wrecked the the direction officer other components of vectors and I did the components of the output vector I would like know concrete way work article parents won't be so defined our are just project module direction or 80 and I want donors and both sides of the equation project or 2 projectors figure in a product of a vector this by definition is beta or I another as if I were to expand B E In the same way expanded maybe I would use the coefficients are so we learned Obama power this is that I how milder here will cut the with this
to rewrite things in terms of components or rewrite everything in terms of components of the actually become a row arithmetical operations could do do well here as well we use this fret that whenever you see about that OK you could substitute for it Osama but that right in here where is substitute Purdum ship should review of the Commission variables not high in this case I have used the external consummation variable here so it's solid Jake as an hour automation variable J a fresh stuck in AT a somewhat of a component part G-8 vector a J ramallah NYTimes jetty for the proper rats just Alpha J. so we now have a formula completely in terms of components on and J. R. get this but a number of products are numbers and J is a vector and you could take it to product lines J. is a vector and vectors kept vectors have promptly basis and findings jury pay for cars Alfred J. that's equaled bad idea so always have to do is give this object and what is a subject for you again you have some operators abstract entity you have a collection of bases vectors applying the abstract operator a vector Jay and take it to the proper called me and I J S son number it may be a complex number but a the number of another words it's possible to characterize linear operators by what called their matrix elements of core the matrix element of the linear operator I am missed and Alpha J. some knowledge a married with a raft summer rigidity but back in someone Jerry is equal debate if some ideas you the collection of G-8 somebody a hand you a victory in the form of its components Ukiah then start sweeping through it and calculating what the components of the output are in terms of the import MIJ is a matrix has 2 industries to square matrix the dimensionality of the spaces In them Emma isn't and we and in the body and make square square matrix R-Fla can be regarded as a column I'll suddenly regard as a column and is also or column Carlos because it was a cat let there be was a cab vector I saw another later right strange saying this is 1 way to write and a equals B is 1 way arrived abstract His another way derided as a concrete songs and the 3rd ways to recognize this is just a multiplication of a matrix time column rector your right the matrix it by displaying all of its components and mourn 1 and 1 2 and 1 for 8 . 5 and 2 1 and 2 2 0 about not as just a array that exhibits the matrix elements will rob where the call with the matrix elements of and times a column that a powerful 1 now 2 . 5 hour framed and what is the rule of this this of course this across the results in an output which is they don't want to talk about and I watch the rule its implied by this equation over here it's very simple if you want the 1st entry over here you take the 1st role and you take it in a product with a column he and theater William Colby of product and 1 1 Alpha 1 Plus and warned 2 Alpha to and won 3 of 4 free right down the bottom exactly this formula and 1 tool sorry M 1 1 powerful 1 I'm going to Africa to and so forth is exactly the standard rules for multiplying a matrix by column lots of knowing rough 3rd way to represent the action of a matrix on a vector this is less abstract its concrete if you know the matrix element you exactly what to do with them hilly thing about it is it does depend piculet choice on the basis of the specifics of a matrix elements here and the contractor will depend on you choice of want you pick hours a basis if it but you could chosen a different basis which Casey's components would have been different matrix elements would been different afterall Amaker songs are themselves related to a basis vectors and stole a Hilo something he gained something by going through a major representation would you look is these geometric idea of a relationship between 2 vectors that in the pain of the basis for hours with you gaze is a computational actual mechanical algorithm for producing or for the machine itself here is a machine misuse a matrix and what it does is a grind throw and 1 1 powerful 1 blob of arrival because the the next role buzz of begin goes annex world knows a thing again and just fits our there fireworks this is of course also true for ordinary electors in three-dimensional space not specific we tend not to use them as much as 3 dimensional ordinary vectors Hi they're quite are Christian ensure plural and corn chips were among gay head is based well article that would be a precise way whether it can make a difference let away you see through a problem certainly a wrong choice of basis can leave you completely confused I think what you really meant camera and different choices of bases lead different physical answers had better not be doing something wrong yards often the case that convenient choice of basis will make your computations especially simple are a bad choice of bases might make them horribly complicated Astro will in the the answers should not result from express OK so now we have we have the idea on a linear operator operating on several different um mergers are young is is that it is within he can also known alone or numb with operators act on kept vectors to give as I complex
because I regard right approach on this temple is kind of a linear operator operators just multiplication complex number is multiply it by complex number and get the review of actors red basically the same vector but forces Warsaw something that is a linear operator you might ask whether complex conjugation is a linear operator I take I take the components of a vector and replace them by their complex conjugate the answer was no it is not R and you can check the definitions multiplying the component select their own multiplying complex conjugate is not what it does do is it takes a vector and it's a different kind of machine complex conjugation it's a machine which takes a bra vector to a cat vector convicted brought back massive elapsed strictly its called entire linear operator our linear I mean it's just opposite the linear it's just a word while the resent toward resumes called win years because process of comfort conjugation is from deep way related impact particles were not ready yet gone young white Miller Stocker let's talk about can't let us I brought vectors so fondly the flying a linear operators can act only on can't vectors but every linear operator can also be given a definition we needed definition of course we have defined here we definition but there was also a definition on any given linear operator acting on a brother there for privacy of returned her concoct a definition and I will stick with definition of what might be the rules for the actions of that same linear operator when it acts on brought back 1st of all the standard notation very very good to keep track of notation and to use the same notation borrowing needs but it keeps he was an order when your eye from a brawl that put the corresponding operator on the right side it gives something they have something and take it in a Prague was poor a bright around the indicate that mn acting on a sofa In a beauty and now let's take it in a product with another vector BT well if I removed the bracket here I would have a symbol which were require there's a M B and I really wouldn't know all supposed to do what is actually em on beef and then take the inner product with a EU or act and on a CD and then take proper will be the U.S. arrears With a standard definition of how we William operators actor through left on Bravo actors it doesn't matter in other words we defined the action on the part of the left such that Ross you that we could always do such that it doesn't matter but but but which way we do it a M V is the same as AT & B 5 that you could remove the brackets OK let's see that we can do lot to say let's start with the definition this way stalin definitions where my from star definitions way and now I'm going to use black trek up on top where you write a back by inserting this is a for his operation he has called inserting a complete set of states complete simply means that the bases its called inserting a complete set of data did nothing talk a wrote a buttery wrote a terms of its components so let's do that over on be over here the same thing over here we come right this is Eagle B. J. J. some knowledge not underwrite salacious I are prepared and about yet some racial eye injuries right so this is saying that was baby but a layer of bracket around it and back and I'll do exactly the same thing on the cat threat remember cat can also be expanded in the same way so here we are a mighty where we see here we see our 1st on fire he warned that the Fiat when the start fierce Abra vectors for complex country office of sorry and Amla I J RIT in a product of iron and fans jetty and then there is beta J. four-story half-year Sesana injury sigh and we're on the left and jail on the right Marjorie J. Baden Bari Liuka see from the way that we it doesn't matter if I 1st summer over GATT and think of him as acting on and on they can affect the beta or I saw what I think of them as 1st acting role left on how you give the same answer just a bubble sort of orange you answer if you defined you know you can would you can and in such a way that doesn't matter which which order or you put a bracket at ease your life you don't have to remember whether you meant an act on me and they knew it taking a product with aII or MXR new take you probably be saying OK but there is a sudden burst of power should not consider another question supposing the linear operator M a gives Mr. every using a and B all over the place in different ways there was just 2 vectors here is the input vector Beasley output that so let's suppose that the machine and when you stick adient away Our bid be we could ask What's the corresponding art brought back the relationship is there a linear operator which has a property that when acts on and that course and yet ma Web scored script it is therefore for every M which acts on 80 give me is there a corresponding script EMI which acts on a corresponding Braun vector forgive a corresponding Robert De over here well yes there
is a mother figure out how a figure out what it is soon enough but it is not the original and 1 example would beat what it was just multiplication by a complex number Norwood script BMB multiplication by the complex conjugate number salt water that image is not necessarily the the same script there it would be the same as script PM it was can't who was multiplication by a real number but multiplication by a complex number then in general generally will not give us a relationship we would normally just for a complex conjugate signed the throughout the operator of a action might need notation that is Standard Mr. the operator which were acting from a laugh it is a reflection if you like the of the opera about operated him acting variety is not called him it was called in to those that were reduced a teammate Jose operated version of complex conjugation the tour operator version of complex conjugation or matrix version of complex conjugation called her mission conjugation will get into words while Hawk aliases abstract but let's see if we can get for concrete notions am abstract operator is equally for concrete matrix of concrete matrix that unduly and the question is in terms of that concrete matrix is very concrete matrix for and the answer is yes I refined if we find it just by preceding dead ahead I would like to know what With BTR the matrix elements to which would it e Joss this is what I would call the ICJ matrix our of and dagger and coordinator is yet I have a suspicion the something do a complex hundreds but this is its matrix on concrete set of numbers are can we find out what they are right the autographed stepped back from led supposed In acts on the bases vectors give prime I Prime is just the result of sticking the bases vector into the machine IBM machines and this is what comes out works a process result now let's take J and piety that J. my prime won't donned a stake in a product with JT here's the Matrix our rent on matrix and as justice in a product that was a trivial operation I didn't do anything new anything received if this issue for a walk them by definition definition and dagger acting for the left quarter's and dagger acting left field on all right to ordered us cordial here and that your Vax from a left years the beach so I thought bagger Das so the equation over so what does and dagger acting Oregon it gives the brother the right time and dagger acts on the Bravo party it must give the Brava prime just flips the equation from bras to kept so we have over here and is my client Jerry that's the matrix on Bridget Jones over here but when I look at school focus also goalie have where matrix element of an Uzi in a proper J. Wilhite Prime the matrix element R and dagger Procter why Prime with J. what the relationship between he discloses just complex Carter if you take 2 vectors AT&T and you think of any opposite order the bride a conjure can be versus the Robbie times a k a the relationship between them is just complex conjugation this year is just a complex can't do this so what with the rioters is the matrix elements J matrix element of dagger is a complex conjugate Of the jail by matrix element of him so air final with matrix elements of warlike due to get the major elements of an actor is very simple you complex conjugate but that's not quite enough you complex conjugate and you would think that change vise injuries you interchange eye injuries from our waiter right at his very and Viagra is equal to 0 and J. O'Reilly I stock so now we are locked we know we know how to construct operators which act of left to do this same mapping begin when it went right now the Brock 18 of Robbie if the original operator through the same thing on a cat fight that's an ocean of permission conjugation core mission conjugation award her mission is named after some mathematician language from I turned her rich and half conjugation is the process Of the changing my injury and taking its complex conjugate In terms of matrices of I have a matrix so your memory and 1 1 M 1 2 0 . 5 2 1 and 2 2 the crash the matrix representing him matrix representing its conjured conjures gently interchange rows and columns that means reflecting about the Bible and then a complex conjugate Inc it did change in eye injury sovereign debt changing and warned with them 2 1 you just flip it about the back row and you complex conjugate so off I want the matrix elements her mission conjugated well I I can't say any more prestigious restorer said you interchange rows and columns and you complex conjugate changing rows and columns is the same thing as reflecting about the bag you reflect about the diagonal and then you complex conjugate the UN example supposing I have the matrix Fuller
check plus I Air Force I 7 and floor minors i and 9 9 plus I case 9 plus or what Her mission conjugal back at you 1st foot that flipping it leaves the bag no elements in place interchanges me off their dormant you flip tool 9 plus are 6 plus right 7 4 minus ii what is at crosses called flipping it like that transport called France of flapping in the the trans but then you have the complex conjugated complex conjugate just means were put my aside this becomes for plus ii warned minus 7 and 9 Marina site and that the matrix that represents the mission conjugate of the original matrix of OCC visits to see concrete which also abstracted has an abstract side Commission conjugate just allows you to turn equation from Brock Bravo report can in terms of concrete matrix elements you'll flip by injury and you complex conjugate that defines him back however is generally is matrices of thought of as complex object and the notion of complex conjugation is replaced by this Commission conjugation simultaneous flipping and her nor was a real number a real number was 1 which is complex conjugate afire right GE is equal disease star you jump up and say that is a real number was a corresponding notion for operators it's a notion of the and operator which is its only permission conjure operator which is the only religion conjured that means you interchange complex conjugate you back same things that is called permission permission is analog for matrices only analog for operators of real I storage says our mission matrix now with specialized to a case of her mission matrices just 0 out her mission to permission from matrices dizzy and are real floor numbers are mission operate or matrix is 1 which is equal to what all persons conjured In terms of matrices MRI J 0 2 NJ RIT star where Stratus Suweon constructor her mission latex and Harry ought to weaken constructive the 1st thing about her mission matrices is diagonal elements are always real take a bag of Parliament for Abagael element on injury are equal I J JI farm so I that diagonal elements are always real so her mission matrix 1st of all has real Bangalore elements 1 is evident in Norway minors 2 . 6 on the diagonal off the bag know they have the property that when you reflect and basically complex conjugate so fruit but Iowa here we would put minus Iowa here for commission matrix if approach for minors I here you would perform plus RIO here and if you have 5 0 he put 5 over here I saw her mission matrix has the property first-order Aguilar Osorio and off diagonal elements not symmetric but with a different kind of symmetry symmetry will you for reflect a complex country that called on her mission this should make has enormous importance to America permission makers have enormous importance In quantum mechanics they represent the observable quantities measurable quantities the things that you could measure we're contract number do a bit of wrath especially for the single Spain I hope tonight but before we do we have to know a little more about conditions matrices in particular we need a concept the concept of eigenvectors and I get very fast and gone quite aware that if you don't anomalies ideas have those our struggle tonight to to keep up but there the standard ideas of what textbooks on linear algebra you can't find it can quietly and burned Rooney's ideas barked well 2 3 hours OK so blacks talk about her mission was 1st talk about eigenvectors and I all we even think about these things incidentally in ordinary three-dimensional space where the vector spaces just displays of ordinary 3 vectors are there are operators they represented by matrices of evaporators operate on vectors to give our vectors and did glee of a matrix all interesting often operators role interesting 1 mean be interesting way of the complex will generally have the property where if you pull a vector into machine secular comes out will be pointing in a different direction that will be generally true generic Richard also directionality is not preserved by linear operators as a rule on the other hand maybe particular sectors typically there aren't particularly vectors art that when you apply a matrix those vectors will depend on which matrix talking about 1 operated there may be specific sectors associated with a specific operator that simply get multiplied by numbers there was a command the same direction as an example after example of operationally operation on a vector is a rotation about Max Factor be represented by a matrix arm in 3 dimensions if you rotate the vector space about some matches generally speaking on arbitrary vector warned the different direction however there is 1 particular direction for which that doesn't happen mainly along the axis of rotation of European about the axis of rotation vector comes out pointing in the same direction becomes our multiple of itself Baroness case to go equaled with self quite so that that the notion on eigenvectors of an operator they are at the vectors if there I not necessarily for example and rotations about 2 and 2 dimensions drawn or eigenvectors there no directions Comerica old directions which Camallo are affected so in general and may or may not be our car eigenvectors affair eigenvectors defined foreign way again and when he was named for the generic operator is acts on and eigenvectors butts call the eigenvectors I for I do
not course I was not I give me canceled and he chloride anyway I yet act on 1 of its eigenvectors eigenvectors on things which are independent of the matrices given the nature of matrix operator it has certainly I give directions certain argue vectors and rollovers word definition is a set of vectors which just get multiplied by numbers West cough earlier I eigenvectors year the Eigen value is called land the simply multiplies now different eigenvectors of the same matrix will have different ideas Mario's there may be In general laps a generic situation if there are several eigenvectors vectors we will have different ideas however eigenvectors or not completely unambiguous if you take an eigenvectors multiplied by the number the result is still fighting back with the same regularity of multiply this equation by complex numbers I can bring the complex numbers each rule cannot find our that numerical even complex numerical multiple of eigenvectors is also an eigenvectors are so strictly speaking you should talk about Eigen direction was a set of vectors Inc. or not worrying about their wings it case this is the direction of this is definition of can I get vector quite some matrices have eigenvectors of the matrices don't have eigenvectors there's something important about her mission matrices of emission operators permission operators not only have eigenvectors they have a complete set of eigenvectors complete means that there are enough of them to expand any vector an even better they have under also normal family of eigenvectors in general a different eigenvectors allowed different values Aziz said but it will be true from matrix is her mission it has entered the dimensionality space is in there will be a mutually orthogonal eigenvectors lures eigenvectors of her mission matrices the fine are based on a letter Paul proved part of that era and it's a very easy here program would approve the other half of the opera over half Webb said that if you have 2 0 eigenvectors with different ideas values and that's a generic situation of the Eigen values are all different you have 2 0 eigenvectors with different ideas values will prove that they orthogonal whichever you are jobless approved but you won't not out of eigenvectors on to you have enough room that's not part it's not hard in any case it strong Archuleta proved that if we have still eigenvectors Amazon mission and is and a cause our mission matrix I use matrix an operator synonymously which means that E equals and OK let's suppose that Mr. missions and it has an eigenvalue and by the 1st observation is that I didn't values of her mission matrices our revealed that it comes before what I said previously if there is an ideal value is real appear how we prove real let trio well let's think this equation and flip it over into Woody brought equation of flip it all rental abroad equation my foot right I take and then I replaced I am that arts due March because there was equal to address right inside I flipped Rwanda In general complex conjugate complex Larry our OEM probe editorial every time you forever but from brought us to cats you have the complex conjugate older members earlier Barbara Madoff assumption areas remission sought to go protocol all which encourages finale of courtship melody cake In of product on both of these equations us with on the upper equation would take enough product on the left where are you aware that give us give fight might plans when the RIT but do the same thing over except now multiplier right he missed and this are obviously the same and where are ammo after my right as saying as this but this is not obviously saying not on West land that is equal Poland the star I I hear same as I hear cancels out and therefore we come to the conclusion that of a matrix is her mission a single photon emission conjugate letters I did values are equal to their own complex conjugate Real ID values from Mission matrices a real respect now let's proved that if we have 2 different Eigen values corresponding eigenvectors must be orthogonal earned them a lot of other means for physical point of view it means physically distinct that was a experiment that you could do it could distinguish the 2 of them on ambiguous that's what our father narrowing means from physical point view West approved the eigenvectors of ending with different Taganrog different bargain value physically distant distinguishable give Avenue OK so here we are would start with on equals land or I R and JUA equals land the Jade Jagger through different died in which I enjoyed 1st thing will do is turn this 1 all over and turned it into a cat equations abroad is equation with think this 1 and turn it into a broad equation that were Jerry and Bagram remember agarose or equals the real number Lowlander J. eigenvectors Eigen values of remission matrices a Rio J. now here we have our case for we we have Joy equation what you do to them so we can relate them organ donor and the relate them is to multiply this 1 by J. on the left and this 1 by ion right I will get the same expression on the left side of the equation so stood out Jerry J. Hogg Ari war left was a sign the opera more origin or
Alamo right is not necessarily certain sort require a requiring those strangers and the Minor slander J. Ponzi enough product from J. whether on must equal 0 and are just written affairs I've just written that this is equal permits and transposed title when the I'm slander jetty Lee in a product of JI equals to well if the product of 2 things I have 2 things and I slander J. Indiana profit Ji Javed product of 2 things is 0 so as unambiguously at least 1 of her 0 usually says 1 of them is 0 the be enough product may or may not be 0 but let's take a case will we know that the eigenvectors correspond to different values of the Eigen that charity eigenvalues different I'm interested in the interest of the eigenvectors corresponding to different values of the idea that you sold by assumption land the RIA is not equal the land Jerry the conclusion is that the inner product of Iowa Jerry must be equal to 0 so that means I J or or forum so we now have a fear of me I eigenvectors each of our mission matrices with different Taganrog value are orthogonal Arlette Sert just take the generic case where or the Eigen values of different wouldn't come back what happens with some of them may be the same amount important right now if all the eigenvalues have we have to be different than each other all the eigenvectors are mutually orthogonal to each other if there are enough of them and your job is to prod the proves enough of a form of based Lawrence's collection of things global injury he is a special case of a kind of basis which we labeled with small eye injury but the basis of eigenvectors basis of eigenvectors Of the mission operator mission operators will be identified with observable quantities and say that eigenvectors orthogonal has something or other to do with physical differences that are measurable unambiguously between was most states of the system OK we are now prepared to state the principles of quantum mechanics we're not mathematical information or not mathematical of the state the entire area shall we could put on a blackboard now what quantum mechanics is and what is and what the rules are or what the correspondences ours for the mathematical starter with the army or the physical meaning of these various are you won't get hit 81 understand the 1st time rapidly foreign newbie peculiar talking about above course will work out in examples will see how the ideas work so don't get disturbed therefore I throw I do now or through sounds Rubin meaningless that's OK by the time we finished with some examples it should be clear how we use my 1st of all there on measurable quantities the result of a measurement is always a real number it takes at least 2 measurements and somehow took compatible measurements to measurements both before perform simultaneously to measure a complex number the basic ingredients of basic Miserables are real numbers efforts comes out of you'd be Hector Yukon measure taught them if you can measure tour and it is possible to simultaneously measure of things and put together a rental complex number but really is just measurements of each measurement each thing you measure has a real result associated with it the physical measurable so called observables corner mechanics are observables things it you could measure the observables are identified with permission matrices for mission operators observables with my cue with ridicule from full but nor right equals a represented by which put an arrow are represented by a militia leader of course operators but for first-rate let's just say in this study this being in Ruffin was way we still identified some observables be observable so we identified with the components of the spin observables a identified or associated with our apparatus point your apparatus and some direction and measure the spin along those directions we said we called the free measured the spin along the z-axis we call the thing was we can measure it all we could turn our apparatus oxide and registered next all which turn apparatus and half from the X-axis to Y axis and measures should why we talk about our most things we didn't talk about is being permission operators have really talk about them being operators at all but we did talk about measuring these will be calm permission operators in prime time next a possible values that a given observable can take on where are they forcing Missouri whatever possible plus or minus 1 right for pain but passed on all the observable values measured an express hours observe values are the Eigen values the real because matrices are permission but but if should also the eigenvectors were about the eigenvectors the eigenvectors correspond to steady state vectors straight rector eigenvectors on the state in which the corresponding quantities are definite another word in which the measurement of them is unambiguous all over the yard through on this bed for all 3 of these observables the eigenvalues had better be plus or minus 1 OK now come to the eigenvectors the eigenvectors for example the idea that the weather given I give value is a state of the system where few measures that particular observable you will definitely get the Eigen value associated with that idea director of the eigenvectors but for Bozo are ever backwards for should eigenvectors oversight turned more West Side the physical meaning our members physical meaning state Quinn Wedge Court observer Wong
has on a ambiguous but an ambiguous result for example we talked for example are barred states oct and now Bozo states in which said Missouri had definite values when Iraq Tunisia is definitely 1 when you're bound soon was used that remind us toward a deal on a vector simazine is statistically you plus or minus 1 with a random rest stop up-and-down now must be the eigenvectors of sued Missouri left right for the state in which Cigna X was unambiguously plus warmer in this 1 so left and right must be the eigenvectors of acts were missing exit were tried figure out what next knowing it's values noise tigon values and that's eigenvectors and nausea illustrated other called in an hour and in and how what were the eigenvectors of signal white so we have 3 connections and now a 4th connection get enough a general state not an eigenvectors mountain but given general state a somebody made an electron or a system of some sort and left it in the state and we can talk about what it was the probability that we measure of Eigen values and at different values may be several different idea very this case there would be a plus 1 minus 1 but more generally thousand eigenvalues million United values whatever whatever number labeled body Media away wiII what is the probability that when you measure and you get land the RIT store as be birthing here sir West inside we put probability physical meaning probability right just as the probability a measure of body not to measure land by my items are measure land party the result of the experiment comes out a particularly Eigen value of the observable and were rare Mitch back is related not related to its equal to a Toby comes in 2 steps 1st you calculate if you have a we vector a directive which historically the state of the system you take it In a product where the eigenvectors the projections of 80 on by the component of AT and T I direction found that in general was a complex number that is not a you multiplied by its own complex conjugate To get a real positive number and that is the probability of multiply this by its complex conjugate is right it hate this in AREA are complex conduits of each other also write this the absolute value of white squares the individual components before we squared them called amplitude called probability amplitude probability amplitudes of things which you multiply by their complex conduits defined probabilities their real positive and that's 4th Schwab posture of quantum mechanics Sunol independent I use them because so because they're reasonably intuitive EDIFACT Yukon probably get rid of at least 2 of them and prove them from the other from the others we need to do that this is a good starting point are and these other basic for posture would quite attracts the other pasta work is that when you measure required you prepare the system in a state of definite value of quantity I won't bother writing down when you measure something you are in effect pushing it Ito a state of definite value of that quantity depending on what the outcome of your experiment that outcome measures some particularly our common experimenters some particular Eigen value and major pushed Systems Inc it was great to be Eigen spate of our would drive argued Warburg bearer pitched Woodward who wrote hounded he said that the ability to do to get that yes they could have used a flawed the try to will be promise probably probable probably with probability mysteries probability of measuring when voters were by and large a probability on measuring and body was the squalor of the inner product of 80 with Clyde See as if you like its projection of any onto the ice direction of a component of direction now an implication he wasn't implication here about the nature are state vectors coming from the assumption that toppled half of those lost if there is a there Lula's dot . 4 but back we if you want the total probability and after warned then you want the sum of all these objects when you some or what might over various possible values of but you could measure you want that people want to want the song indeed allowable state aII we want the of Tracy Ortega a tape or not I the bases a lot Top equations where you have equations which tell you how deal with exactly this combination what is this somewhat on May I this just a store this says the inner product of a with itself should equal warned the toppling of a should be quota 1 must not surprising about 4 per cent this says the sums of squares of a component of AT and award for ordinary vectors of disabling was 1 of Africa's weight a equal 1 its equivalent in terms of the Alphonse we got rid every offers there also quantum some of our 1st III off i.e. equaled 1 well I capitalize and so there is 1 a combination of the officers is want real
equation are complex equipped a real equations and that means is 1 real combination of the offers which has to be constrained offers are not completely free army officer not completely free roughly speaking the sums of squares some of them plans a complex can't you have 1 of these is less of a furor independent quantities thought there is another factor which we have not gotten to yeah we will get to it on you could still see it from here 0 probabilities he probability that you ever may want to measure or that you ever may want to were off to calculate will compare with experiment is always connected with a real quantity like this the collection of such real quantities are is invariant under a certain operations the operation is multiplying the state vector buyout everybody needs multiply number a ring referees multiply by each of the they view all the components of a by the same overall favors every probabilities unchanged every probability is unchanged dot even if you rotate Macy's its unchanged so physical quantities don't respond to a change of the fees that means there are tool fewer degrees of freedom tool fewer parameters In the specification of a state then the collection of complex numbers that would define a vector and the abstract space to want phase the overall faces and the other was the magnitude of it welcome back OK that's taken us a long way are probably saturated by marks over the continue Kamarhati if they have a yard our these are always positive think the complex from Falls the devolution where now we have direct how how things change with time know where the level now on where we worry classical physics when we just said the state of the system is just a point in the 3rd among a set of that's about what we are at the top forward we not talked about helping how things change the particular how the states systems to cover more Thais back saying they had examples of measures humor that were comment I have RSS shortened added that goes on sale so we're not aware of by bus we have got the idea that was used in Riviera she adds you know what got your back uh E quickly he said that last part were overtime with new measures you push it into the state you are measured not measure something you get a result the result a statistically determined meeting to say that Romania 1 result another result but want to get that result and you know what state the system there it's in the Eigen state of the quantitative measures with the Eigen value that came up on you see under detector remember that could have apparatus measures Missouri which said that not only measures and restraint but it prepares the system in the state of definite Missouri and leaves over in the state In which said movie is definite for the next measurement by means of a leads to the United States motion was OK what you we reporters rule Aja a crossbencher Strom you're very general formulation and now we wanna try out to see how it works I want start with the 3 observables stigma X Y movie and to our concern about 1st Austin Missouri now let's start in the bassist associated with the measurement of the spin along with the access take that as appropriate places are saying Lizzie has 2 0 eigenvectors bite from the experiment so we talk about we will learn that we don't experiment soon as a has 2 0 eigenvalues and 2 eigenvectors 2 eigenvectors with co-op inbound and their associated with him was plus 1 like this 1 must be the eigenvectors of Missouri and I get values a silicon immediately say was being planned Art is equal top plus 1 kind White Pierce a measured value measure Sigma z is going to be a plus 1 hour CIT in value and is the eigenvectors because certainly upstaging was definite for downstate which are right there soon was a definite were announced they yes always minus 1 minus warned and that means the ideal value was minus warned cars down not turned down equals that would be an ideal eigenvectors eigenvectors a property owner operator acts I think it gives the same direction back are you stop here is what we know about
certain Missouri let's steelmaking compute its matrix elements and exhibited as a matrix of what I say exhibitors a matrix I have look I must pick basis so I'm going to pick the bases opt in that 1st of all 1 thing up and then are orthogonal they must be according to these principles because of singers observable corresponds to our mission matrix if different Eigen values the eigenvectors must be orthogonal for her mission matrix and so up-and-down will prove 1st will learn that up and of fodder that's it's 1 I and doesn't matter whether we right up down bound up with carotid Rapisura let's Ely computer maker Chalmers the matrix elements are just be a ready E. that we get when we consider before possibilities going to by 2 matrices for possibilities are going to be 1st of all we had in a product of his back up BDR pop trachoma drew 1 1 with was Ackerman be off To file let's not sure want world approach stored over here which calculate ought to Missouri opt fast equal Tuesday extra right forgive us off plus 1 or pass waters 1 kind up Thursday at a Procter Rob shelf prisons op as a basis vectors its normalized and this just equal to 1 of its magnitude was equipped widen should be called 1 as positive got positive to Rio and this just plain 1 so here art opt for the 1 1 place is just 1 now let's were about to be off diagonal elements off diagonal element of golf bag element that's going to be able to be in a product of downward what's at 0 over here I would get exactly the same thing if I put down over here and Opel values this equation over here but soon Rosiak from bound and then take its overlap with Optima we do 0 again and the last story is certain Missouri down or fat well you've already here she was 8 times values might service just a quote from my 1st Baron around much that minus what so this filmmaker associated with Sig Missouri but when it acts on a vector call murdered in particular were acts was column that groups associated with this is a 1 zeroing Amado price source just drive which 1 0 4 0 get 1 times 1 1 0 Parma 0 0 we just give in back-to-back saying they will get a guaranteed that will get the right the eigenvectors are supposed to be mainly just stop Maryland the eigenvalues of course were minus 1 in the right way factory found want the stigma major change shit Missouri reserve cost rose above course recalled how we may refuse to next warriors Missouri let's see what we can find out about sigma acts now I'm if I use the X basis use the bases of left and right but also just find exactly the same thing I wanna stick with the up-and-down based or stay clear basis but find out what the matrix elements of XR nor was the property of stood next the property of seeing that acts Is eigenvectors are left and right so for example saying Exxon right about equals right or about Exxon West equals minus left those auto possible values that Mexican take on left them right on the eigenvectors of awards a state in which next has definite values plus 1 minus 1 and issue final belching acts but I want defined its matrix elements in the upper down based so what I really want them with so much more here the sea of weakened compute this same bases were either with our republic nitric Mary I worked so hard to get it string Missouri rococo want minus 1 0 0 0 now we watch stigma acts of 1st step was raised by racer waste from blackboard will act let's start when Warner for maker so upset 1 by 1 up said next off Norway have to know that up is a quote from left us right or the square official Memorex but we're originally defined left and right To be plus or minus down all of this quarter to with them we can solve up and down in terms of laughter right down value is left are starts right minus 4 F with the notation we have West rival was found last are used write my initial left over ridicule so if there was poor with it left here we have left it on with unlimited abuse notation and just right this as left plus right
square tourism me Our symbol both you I think you know what I mean I mean the brought back left plus broader the right over squares but that From a vertical strokes renewed confused sources right has left us right or square to say next left us right or were fresh or I can't take Abbas were resisting cigna X. are part periods which are carpentry former Gaza opened here White 1st of all was a factor of 2 like take out swerved from here were from Europe forward your have also prepares 1 have been sworn stigma experts left it give back left except for minor signed weapons were used to exit right give back right with a plus sign staunch lacks acts on areas were wines giving you pursue next or wary remember drugs it gives right mind left instead of right plus left hits the vector right and used plus 1 it a back left in minus were again these are not things to be numerically and use of vectors right minus 0 effect and how we have to do is take me in a proxy well we we'll product of right with right is what 1 in a product of right with the left is 0 apart from left to right is 0 and in product left left Midas returned firms minus the result is a canceled 1 minus 1 4 Oklahoma Blake B product Hall of of a vote about 4 K 0 first-order 0 over here but they are part matrix on how it's easy to get down down matrix renowned manager stone and you would do exactly the same thing you accepted it would start with the right minus left right minus left right minus left next venue would say warns should exercise its right of course right where it has left me to my has left next act it takes right minus left to right past life Wesleyan a product of right plus left with my right minus left 0 again right with writers 1 Westwood left is minus 1 across turns urges Iraq 0 0 a year now referred year zeros effectively life so let's try what's tryout my which are up there don't let's put down their up will be right plus left or stigma acts in the Narrows right for after Lewis J. next right plus left it makes right miners left now what's era product yeah Procter right with writers 1 year Procter might left with minus left is also plus 1 to raise then again across terms between right and left a 0 so we get to divided by tool which is worn worn and then if we work at our will get another 1 over here that's we've got your more white West just stop from and and say what is it that we have we have a pair of linear operators were a pair of matrices was 0 eigenvectors far shows eigenvectors here are often found his fellows eigenvectors Our right in West exactly what we were talking about the previous fight for me state in which the observables have our ambiguous results RRB eigenvectors will right now will last for a single white where we know about him why are you now a sale are left those laws because they don't have a fresh from Big Bear Road wiry figure just failed to cross the X it out of the village said nor why don't believe some of the algebra you are you were the for the rules are well incidentally Our Nevada grow Kerry before the this rules OK server was in an hour I cannot remember which 1 was which are start fake it overcomes are wrong comes out wrong arms I think it was Bob plus I think are right turns barrel Our what woes off my ears tight turns down there I think I left our factors square you have to start with apron chapter that in hour our orthogonal to each other if I'm took their workers in an hour can I will find a matrix where In
our eigenvectors Wilhite in value plus warned minor sworn duty late it's very easy limit you throw simply tell you what that matrix isn't it unique opportunity if you know the eigenvectors and you know the Eigen values June 1 way of doing it Mr. solve for Robin in terms of now what you solve Robin now in terms of our communal harmony accurate announce your Westwarren in the multiple at here the now I am writing white show warned all miners are in I writer will exercise you're not really needs go improve there's different things you could take this matrix and show that these are the eigenvectors are showed the eigenvectors velvety figures matrix show those eigenvectors was a maintenance of eigenvectors or 1 eye and warned minor side square of tourism matter eigenvectors are eigenvectors don't care about square of these are 3 Polly matrices that represent the components of Spain in different directions after I think I think that's enough for tonight where was going do hand my notes was exactly this question what happens if you our component of spin along an arbitrary direction can we using what we know the prop I will do this next time we'll go next part of rehab for component of Spain along arbitrary direction and we want to calculate using the principles we want to calculate what the probability is former measurement of Sigma me in various the bottom line we should be able do agree have enough information I think it's probably more than that we can get him 0 for tonight it we go off provide a a side wall above do not let you know what you know New York interesting nursing adapter compote conjured for your dinner in a product in where our you multiply Times warned me and then air not times minus sign 5 times plus I now know away that somebody like me no so quickly y with their thinking is having made the same mistake of course former pro credit card will furnish from from court for more please visit us
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