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# The Theoretical Minimum | Lecture 3

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#### Automatisierte Medienanalyse

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Laos Stecher University all right will
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
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
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
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
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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