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The Theoretical Minimum | Lecture 1

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Automatisierte Medienanalyse

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Stecher University prior to
last quarter leaf studied classical mechanics and classical mechanics it is fairly intuitive the basic ideas about things that you couldn't visualize picture in your head motion of particles motion of objects it gets a little harder when you start thinking our 10 particles all at the same time our but nevertheless the whole thing is based on arm concepts that are drawn from pretty much the everyday world of Gaza gets more and more abstract as you go along the crazy Frank were obsessed with elegance kept making it more and more and more mathematically elegant grungy unions plus brackets blah blah blah turned out to be tremendously important are the abstract are not all that difficult to understand and the basic concept is the concept of beating well before that the concept of a space of state what is a state of the system adding before that the concept of a system closed isolated system that doesn't interact with anything and whose motion and whose evolution whose history you wish to protect classical mechanics is predicted to determine a stick and the 1st concept as I said after the concept of a system is the concept of the state of the system our systems can have a collection of different states we talked about some very simple examples the 1st day day of the quarter last time I just described you the most elementary simple systems consisting of nothing more for example the Corning heads and entails the only states of the system our heads and tails they form mathematical set the stage 2 of them for entails 2 points 1 space to states there weakened draw them as dot the form stands heads this 1 stands for details and no it's easy to picture near head from there we talked about the equations of motion equations of motion or laws Our evolution of the simple very simple system our couple points heads tails we formulated some very very simple possible laws of motion the simplest being nothing happened if you start with heads of state heads heads heads heads heads to start with tales state tales tales tales tales that was a very simple motion more complicated law motion if you begin with heads you go fails To begin with tales to go ahead and oscillate back-and-forth a basic idea is the idea of updated if you know everything there is to know about a system of instant of time and classical mechanics everything there is to know essentially means everything that's needed in order to predict the future if you know everything that needs to be known classical mechanics consist of a set of equations which tells you how updates the state given its value at any given time had a goal from 1 time for the next hour bad basic idea for example while I don't think described was again we describe it late last time part that's extremely intuitive in going from these very simple discrete examples tho world of continuous motion indeed we have to abstract would have to replace the idea of a small number of states a small find number of configurations by an infinite number of configurations for example the motion of a particle along the line we have the label the states by the points along a line as well as their velocities and it does get more complicated but nevertheless classical physics consists of equations which tell you how update to state and in a given time and completely determine stick can exist easy to visualize at least by comparison easy to visualize because it makes use of concepts from the ordinary world which your brain is rather hard wired to be able to understand not always give a little sermon at this point end and the theme began probably of her that doesn't times of about the way 1 has to think about physics beyond classical physics for you moved past classical physics I don't think he can be said enough so rarely said that I think I will say it again and that's that wants physics moves test the realm of parameters that has to do with ordinary experience objects which are so small that they're far beyond what ordinary physics is about all velocities which has so fast that far beyond ordinary velocities when we moved out of the range of parameters that were familiar with from what their experience we inevitably run into things that we can visualized nobody could visualize the motion of an electron it's just not wired for it nobody could visualize four-dimensional space for four-dimensional space-time our little long 10 or 11 it's not a waste physics is done I often some of the stuff that goes back and forth between some people if suspect people in this class about trying to explain themselves to each other was about to ask questions of each other about a certain abstract physical questions and I I know is very much that allow the mistakes allowed the confusion happens be costs of trying to visualize things using the old old-fashioned or standard into intuitions that your hard wired for you may think that physicists good physicist are especially good and visualizing these things and not I can visualize 5 6 7 dimensions any better than you can but I know how to use abstract mathematics described it is no substitute there really is no substitute for the process abstract thinking and using mathematics to describe the things which are beyond your ability that directly visualized the example I always give and women would do it again tonight I just wanted to really focus on it and realize that you're not going to Wednesday in quantum mechanics but trying to visualize this summer of funny form of classical mechanics won't work you always get a wrong are but just the idea of visualizing space Of
dimensions different and 3 dimensions in particular I always used to say OK how many people here could be visualized 5 dimension and everybody would say now we can't of course the few fake there's a few of clowns in the class with a I can do it I can do it but because they know they could you can visualize 5 dimensions Our but don't begin to realize also probably heard this before that it's equally hard to directly visualized two-dimensional so I would say the class OK how many people Malcolm visualize 2 dimensions are Matamoros everybody will raise their hands and I would say no you're wrong close your eyes try visualize 2 dimensions they would OK I see see two-dimensional work could face what they're saying is not two-dimensional curves space what they see as the two-dimensional surface and they did in 3 dimensions even try one-dimensional space one-dimensional abstract one-dimensional space is just fine you cannot see that line in your head without seeing it as a line drawn on plane and the plane you can see without seeing it as an freedom of what's going on why can't I tell you this but prisoners of your own neural architecture you know all architecture was built for 3 dimensions I hire you get around that well you don't get around that by train yourself to see 4 or 5 6 inches you get around it by training is softer think abstractly about have 3 dimensional space is a point X Y and Z than four-dimensional space Is X Y Z W and you learned to manipulate the symbols that's not to say that after a while you don't start the new kind of intuition of things you do but intuition is not the process of physical visualization of the same kind as when you visualize AT way you as a wave in the ocean are still a warn you about that and I will tell you quantum mechanics is about as abstract anything can be our well perhaps perhaps things are even getting a little more abstract now forget quantum mechanics is about things that you're evolutionarily developed the global structure is not prepared to deal with directly how you deal with it abstract mathematics again men relativity it's also true of relativity it's worse inquiry mechanics OK let let's begin with quantum mechanics at very very simplest most primitive question you condemn Maersk what is the state why I think we can as the most primitive question is what is a system nite on the agony answered it it's sort of undefinable you start with the idea of the system are very think I will try To give it a definition but in particular closed system a closed system is 1 which at least temporarily is not interaction with anything else it's isolated is not interacting with anything else and so it can be described by itself or want to entities with nothing else involved in classical mechanics or lets it's it's even money the a question of classical mechanics it's a question of classical logic the logical hold logic of quantum mechanics is different than the logic of classical mechanics blinded of classical mechanics again begins of course for the system which trial described but the next step is the basis of state the collection of possible states of the system if the system as a discrete system like calling or die trying 6 5 6 faced by it can have 6 states of course a real by real genuine die in three-dimensional space of course can be in all sorts of orientations it is infinitely many states for what I mean by a dying now was I needed abstract mathematical died which who states just consist of 1 of 6 numbers should be space states is a set of 1 2 3 4 5 6 This is the spaces state of a die This is a space of slates of calling Nov again an abstract Corning which only has 2 states had details be heads details Of the abstract space of point particle moving along the line was everybody or what's the abstract space of a particle moving along a line well I remind you that the terms things space base consists of the set of possible positions of the particles and the possible moment that possible velocities and sold for particle moving line possible states of the system from plane and the plane has a position sometimes called Q a momentum sometimes called P and every point on that plane as a possible state of the point particle moving along a line but set in this case it continuously infinite set off 2 dimensions in this case it just consist of 2 points in this case it consists of 6 points and we can now start building some logical concept for classical physics for example concepts such as poor and what do they mean arms or an errand our concepts which apply collections of states let's consider a proposition proposition such and such proposition could be that the die is located has a value which is even is even value that's a subset of the space states factor consists of 3 of them from my mistake in 3 on 1 3 in 5 yes and another proposition would be the die is or I'm not argued were possibility is that another not overlapping set of configuration but similar case let's take 2 propositions 1 of which again is the die is all right the other proposition is that it is less than or equal to 3 hours see is 1 3 5 used tool for sex and less than or equal to 3 is this said Over here so now they're overlapping sets and we can ask for concepts like end in order for example
only proposition that it is the guy is both Westerner equal to 3 and 5 it is the intersection of these 2 subsets 1 and 3 or and Western equal to 3 so the concept of an end boats in this subset end in this subset the mathematical our concept is the intersection of 2 set is intersection of 2 sets of steps which satisfies both things and that's the concept of given took up positions each proposition is a collection of states because of the end is intersection when about the concept of are all are becomes a concept of the Union whatever want estates here ii there is less than 0 equal to 3 or 5 Misty inclusive or incidentally inclusive or means a commute both that's not the intersection of the 2 sets and the union of the 2 sets the union meaning everything in both sets everything in here it is I don't lessen equal to 3 or is on conclusively with both being go out so the inclusive or translates into the Union and and concept and is the intersection of sex that's the logic of classical reasoning to a large extent on is a concept 2 sets are completely non overlapping this 1 and that 1 that's simply means the end statement this fall's there are states in there which both plays trove serves on the whole logic that is connected with the space of state a set and logical set theory when at the door of the logic of set theory but uh it's important for me to tell you right now that the logic of quantum mechanics is different is not based on the idea that the state of the system it is a sad state of the system something entirely different and only in certain limits only in certain limits which systems behave classically does whatever space states acquire mechanics were reduced is does it approximately reduced the concept of a mathematical set factors the study what is it's a vector space will talk about that time but before we go before we start to move on to talk about the abstract mathematics of vector spaces and spaces of states quantum system let's not talk about experiments and described the simplest possible systems inquiry Canucks it's the calling of quantum mechanics the system with 2 states heads and tails To if the completely erase a blackboard and start all over again so we have some system when Emily measure that system images something rather about that system namely the analog of whether it's heads of tails we I get headset tails let's give that a mathematical notation I'm not gonna quote of Corning I'm going to call it a cue-bid of cube it is a system which can have 2 states and 2 stands for Quantum be analog for classical physics is called the CBS classical bait and this simple the idea Corning and details will see that a cue-bid is a much more interesting love but complicated for sure but a much richer idea even tho it only has some states corporate sold the Cuba the kill it what measures McGovern measuring a moment is either heads or tails of what's described it another way let's give it a degree of freedom of mathematical degree of freedom and Paulette mathematical degree of freedom Sigma Sigma as a traditional name for 80 for the degree of freedom describing cube it and say Michael I had take on the value 1 or can take on the value minus 1 it's 1 where the Cubans heads and my 1 Cuba's tales and no content related love and heads and tails but allows us to stop being able to write equations for something we could also describe this another way by a picture I'm going to describe singly equals 1 by an arrow pointing up and SYDNEY Eagles minus 1 by an arrow pointing bear no implication hear of anything geometric for the moment just notation rotation 3 different notations has a pales signal equals 1 or SYDNEY Eagles minds and arrow pointing up an arrow pointing down OK now let's imagine experiment an experiment involves more than just a system involves a 2nd system and the 2nd system is called the apparatus thing carcinoma mechanics we never bother thinking too much about the details of unless we have be experimental physicists and really get going the laboratory in doing it the plan and we so don't worry very worry very much about it at least in the deep sense we think the system can be described completely without ever worrying about how you detected you measure it or anything else but in quantum mechanics 1 cannot get away ignoring the concept of AT apparatus apparatus so let's introduce an apparatus and I'll tell you what armed with going to make a mathematically abstract apparatus I'm not totally works the black box it's a black box with a call at 8 if apparatus as a boxer has a little window is a window it also may really is a box of a car owners with cotton union mail which says this side up this side up you want to stand up right is a little window where windowing those green and numbers appear on screen candor has some sort of a detector money and the fall wire that but there are sent to the sensors the Cupid and senses whether cue-bid plus 1 or minus 1 don't you start when your and you don't know what stated sent the experiment is to determine what state it's sell the kill begins in not only by the
unknown state a but I'm not sure we don't know whether it's 1 minus 1 over the experiment the experiment has that I'm not going to tell you how they have detector worked as some important for us it's only important detectors do exist river we will talk later about the process of detection but not for now so has a sensor over here but center over to the killed it and was a dope on number lights up in here you know plus 1 4 1st of all it starts before we do the detection be apparatus is in the neutral state the blanks state no number they're all sorts is quite a blank configuration of the apparatus we take the apparatus over close enough to the cube it to interact with it and that's a good faithful apparatus it will then registered number here and that number will you be plus 1 or minus 1 plus 1 or minus 1 depending on which way whether was Heads of tales that's the actor measurement now you may not be there a look at the apparatus of secondary be apparatus he kept did end recorded was gone on here that something that in classical mechanics of course we do the same thing there's nothing particularly quantum mechanical America's Bullock can never to think about it there in formulating the basic principles of quantum mechanics and classical mechanics OK so that's an experiment that measures the state Of measures where the signals plus 1 or minus 1 bowler can also think of it as another way we can think of it not so much as a measurement but as a preparation for preparation of the system is the idea of Parliament we in fact dual measure Sigma to be plus 1 supposing that the outcome of the experiment and we can erase the 1 that wasn't and start over and then say let's let's back off reset the apparatus reset it means started over in the blanks state and then read do the experiment again let's say we do it fast enough so that it was much time for anything else happened what happens what happens is that these detector records again and it so good detector and the system hasn't had a chance to change remarked it will simply recorded the same number again it will confirm the previous experiment this is employed the experiments can be confirmed again no difference between classical and quantum mechanics in this respect which is emphasizing the car inquiry can experiments can be confirmed we can do it again again and again and again will continue to get the same answer that we got the 1st place so consent and a little bit differently the 1st experiment instead of saying that it determined weather it was Heads of tails we can say it prepared it in the state which was in this case heads after the 1st detection now known to be here is prepared in the state heads we could do the experiment over and over and over again I always give the same answer my fill of a device that measures is also a device that prepares the system in a given state got the same thing would be true Oh added device determined in the 1st round the signal was has warned that it would continue to do her through detect a surge of Corsican be bad detectors not all detectives or will make the detectors work the way you want them to work in which case it just was the bird detectors didn't do what you want to do possibilities you your actually has to do with the head and tail this is 2 possibilities if you that of familiarity from with his entails OK now the something you wouldn't do an experiment 1st so that I can start with a preparation the preparation determined but the spin was horse Spivack or spin and giving away it it is a spin cube it is in the state plus 1 right so the cube it and the other notation is an arrow pointing up we've prepared we now know we did little times to check but once we check it we now know that this killed is up and if we continue to do it we will continue to give the same answer value the summit funny after we've determined or prepare a Cuba will now take the detector and turn it over 8 madam this side up a wrist shot under detector Over and now we're going to probe be Cuba again I wanna learn in this case we discovered we discover that instead of getting plus 1 we get minus 1 by turning the detector over we get the opposite was telling us well as is telling us about the nature of this system was telling us all of that of course then if we if we think are upside down detector and continued to do the experiment over and over will continue to get minus 1 because what happens affair fester we do with 35 times we determined that we do we turned to detect the backup right we get past 1 well whale learning is that would ever this system years this particular has a sense of direction our toured the former drew it as Bob verses then owned there really was a cents In widths orientation being distinguished returning turning the better detector upside down on interchange Oppens then well that's what normally up and then turned over look at upside down and it changes the signing of a Cuban that syndication that has some directionality inspects spatial directionality associated with none now all cubit still have special directionality associated with them but by doing this experiment we discovered that whatever the Cuban is it really does have a sensible orientation and optimists and damage it in space so that a piece of interesting information we might begin to suspect that some we might begin to suspect that since of has a directional audience its that where's pointing up it's different there was
pointing down we can make yes so that's that's the basic idea of cue bid which also happens to have a sense of direction however if Cyprus pain and body hectares or measure it the measures about or measures it now but noted something else all we might is we might believe is all right but we've given that the detector has an orientation it and does have an orientation to it is a right-side up alongside up in fact in the detector we might even think that is something which is itself an orientation something built into the detector which has an orientation and end noble we think we might think that this detector is he kept thinking the value of the component of a vector along the direction of the detector itself so when the detectors right-side up and Texas saying he it's detecting the component of some vector In the past our action when you turn the detector over its measuring the component of the same object except with respect to an axis which is flat so you might think this thing is a vectors Cuba so now we do something they are let's swept Sabri also let's turn to detect the roadside Webster and the detector on the side we have initially determined that the cube it was off we definitely made sure it up we've done a thousand times would be right-side-up up detector we know for sure Opitz not back and now we detect take the sideways detector aII side up over here when it's internal arrow pointing in this direction over here and there what should we expect ordinarily ordinarily would say Well if this thing is going to measure now that the vertical component of some vector but the horizontal component of some vector we know at that it was pointing upward by the initial preparations for what should you get here well what's the component of this vector along the horizontal axis deer out so the answer is we should get 0 if this were classical physics and his work classical little vector that's exactly what we we get so we go into the experimental Robert and we get mad 0 but you don't 1 or minus 1 Indian which 1 we get 1 0 minus 1 1 2 minus 1 associated the be symmetrically located relative to the axis of the detector so that I got 1 but still this experiment whole bunch of cars let's go back let's go back Gilson nor cubit so the kid away were finish with it gets no cue-bid it make sure that it's pointing up and do the same experiment again well we might get plus 1 again but again again we might get minus 1 we don't maybe time is exactly the same experiment with a cue-bid known to be pointing up and we measure what we would have liked to think was the horizontal component of it we always get plus 1 of my randomly randomly but in such a way to do a great many Torre the average value 0 as many zeal as many plus ones as minus 1 to O'Grady mate times and you find every single 1 of them is either plus 1 my this 1 but on average they absorb 0 the way would kill by then but you re oriented if you reorient oriented Major was up again that it would matter whether was a new Cuba not as if could question right there the article back with question we started with a cue-bid known to be out we made a detection we got plus 1 the implication of that is cube it it is lying in the horizontal direction somehow somehow we detected 8 plus 1 plus 1 relative to know I'm not be original axes but no actress supposing we now take that same cubit don't modified but below the same experimental over and over again we will always get the same plus 1 with once that Cuba has registered plus 1 now we don't disturb it go the experiment again and again we will continue to get plus 1 of those confirmed who confirmed the experiment if we do if that you go Chen 180 but where's the others time because the but may be reused I no longer have definite about these the answer is random but only the 1st time once I do it once I do it and I determine whether the horizontal component is plus 1 0 minus 1 that is determined never do the same horizontal experiment over and over again I will continue to get the same well then I could come back and say OK and now having done that have been pretty sure that the horizontal component cover is in the plus direction I can come back and turn my detector gear this will happen I will find the random result after Prime's class at the Times miners begin once it's determined I'd do it all over and over again it stays the same or less there were the 1 AD Lowell this saga art erupt the judges they'll carry so that therefore having half the having gone this experiment and found out that I get up plus
1 I do it alright alright although a few times I get a plus 1 each time let's say then I take the detector and turned around by 180 degrees again but this 1 for Rabin's only deterrent but by 90 degrees random but what random means a particular kind of random arrests equal probability for up to now right equal probability of 4 plus 4 plus 1 0 minus 1 and that means a what to do the same experiment the whole thing over and over perhaps with a whole bunch of different Cuba all I have a whole reservoir of tube it's only a lot of them don't even know what direction there Marion where are they doing take them 1 time I subjected them to the 1st experiment if I get up I keep them if I get down from a trench at the end of the day got myself is a whole bunch of Cuba don't be pointing now I take 1 of them and subjected to the sideways express major quality minus 1 up I recorded when I got throw away go to the next 1 of its saying experiment they may get plus 1 minus 1 recorded what I get this whole collection large collection of them and do the sideways experiment for all of them what will be average result if if you are still allow have as many plus warrants as minus warns the silly average will be 0 let's try that in this case year by saying that the average value we need a symbol for the average value but after this experiment here will discover that they average value of acts horizontal component of Riyadh cue-bid here is 0 classically if we would have created a little factor in me up directions and we measured the horizontal component we simply get 0 In the real world the real world of quantum mechanics we always get plus or minus 1 but in such a way that average 0 I don't want be she would go true young process correct your lives as yes young ever there's Eggers you saw a Young Werther that settlers only that the Law most most of the cue bid that nothing happens to it so remembers renewed when you detected new detector begins instantly after that hasn't changed right that's correct now later on we get a consider possibilities where during the intervening time between measurements some interesting thing may happen for the moment was opposing nothing interesting happens between the sea and a lot of the classical Corning was a law of physics was just nothing happened don't talk about collapsing only if the repair but no let's talk about that yeah let's talk about young operational things that we might do to discover the laws on the character they put some words on such as part of a wave packet that the thing of funny average sends the horizontal component of something new 0 within each individual instant you only get plus 1 0 minus 1 that's going further let's always handling the same thing will be strolled it instead of turning the detector this way we deterrent returned it this way so that the internal vector was pointing out would still be true said OK now must go of different experiment same set up to begin with but instead of rotating B detected by 90 degrees but updated by 45 degrees fact any angle of up to 45 degrees but rotated by arbitrary angle so that the internal kept their orientation is now characterized by some angle feeder relative to a vertical salt and rotated by angle theta and of the same experiment now we take the 1st tube it subjected now be to be detector oriented of angle theta will we get plus 1 or minus 1 2 again where we get plus 1 or minus 1 we get who again plus or minus 1 never anything in between but in this case we average value R V experimental value of what comes out here is voters euro it's equal to 0 would be expected equal to co-sign a thinker cosine of the angle exactly like the x component of a vector would be or exactly like other yes exactly alike component of this vector along the detector axis a component of the vector along the dead they have a detector access would be smaller by amount cosine played a classically quantum chemical we simply get past 1 minus 1 plus 1 plus 1 minus 1 plus plus plus whatever Our in such a way as to an average the cosine of the angle as the detector gets more and more upright cosigned gets closer and closer to 1 and that just means it gets closer and closer the value of go of the total the answer yes to detect or
perfectly up right now and then you might get a minus sign mostly plus signs as soon as the detector gets perfectly right side up all plus 1 what about when the detectors almost upside man they mostly minus ones but here and there a few plus 1 and so it seems B average values of sigma are behaving as if said no it was the component of a vector this is something you know really different areas these huge lost you Sudan someone like him if only because she started with yes yes the same 1 and do it over and over again with the same detector you just get the same answer over Norway you just confirmed the previous measurements of they want Gilbert the br because you've made it the you know the top and subjected to this angularly distorted annually rotated experiment the either get past 1 minus 1 but to do it the 2nd time you just have the same the 1st time your learn anything new is if that were the case it would be no confirmed that the incident you get the 1st time was in any sense the right cancer Nova waiting to confirm the incident right checking it here and our way to check it is just cyclone I look at look at you when I see you up right now I close my eyes again and look at you again I've confirmed that the 1st time you are operatic set young from China's enormous warmer the always makes you either get plus 1 minus 1 and then you do the same expanded again on the same 1 over and over you just to perform but you have a sequence of them and you do the experiment with a sequence of them while having to prepare the same way that do radio statistical distribution finally we still have a lot of what I need Jr but brought back the but that there have the Air Force said in a final game of United ahead of cheating on not Nova saying they he said but we're and now but lost it is easy to talk about a 3rd of the gives us a goes only want to beg you did this Is that the attackers were security milestones audience sitting upside down some of the city right-side-up Rivera randomly man is restored 50 % is standing on the right side up remark close my eyes with another student 50 percent drop the right if organizers and then tells you the Tatum look the same always the same way How we may find distributors were there I was looking at difference but better as just look at the same stage who had controls the whole thing is that we're that we can't keep track and remember which of these things that were looking at them the big we think of it as the actors theory about how you appear that give it the 1 you observe that 1 period the knowledge that there is still room in my opinion Was could change if you believe that the a theory of what I want you got that schedule station that but he should be want depends on how they were prepared somebody might have giving you a sequence of Schubert's which happened to be up down up down up down up down you have got them from someplace you go from a young from kibbutz store where they were and what they did no good quality control but of all of these 2 which will prepare the same way end
identical way and they send them through here they will continue to be identical on OK so what this is telling us is something funny about the notion of the stink of the system is not as clear-cut and not as simple as a state of either a quality which is in the hands of tells war move vector which has components in different directions but for example when you measure the component along an axis perpendicular the direction of the electorate 0 of something the something intrinsically different our and the different both traced the big difference between the notion of a stake quantum mechanics an estate classical mechanics you head there but that is what I want you to produce some some might then that each would you bring up that missed becomes might as well as the original the did it to look everyone analysts say that it is minus 1 here the new separators another user will be my last time that's because it can be further up the operators of the but it'd miners were turned loose they new attorney for artists then you get a statistical distribution I think when you turn the apparatus once you have we Cuba that comes out of this apparatus let's say it was plus 1 let's say came out plus 1 and now you take that you better over here and attorney apparatus back up again the new living in a random passer-by the relatively few flu that this can turn it around the bank feel and up the random the Mucha cubic yards yet now depends on the angle of the angle not true largest small then mostly you get classes right mostly state that what would be captain it is not their boss you truck there were also among its a like you can the world ever bought all the your absolutely right that at some point we have the comeback and state the think proper there is a system and has states and understand the combined system as a quantum system composed of till quantum system but let's let's not do that yet puts divide the world detectors and systems and then come back later and say Look a detector really is a system and we have to be able to describe it quantum mechanically also as a system and on the stand the kind of thing as interplay between tow quantum systems but that would take about that that will take some some steps before we get there would he played this way young workers and 1 8 1 my guess is not right New year ideal guessing that the whole store is much more complex but still on the started more complex but not inconsistent with what I said so don't try to get too far ahead but I guess too far ahead stake simple version of the 1st and will eventually come to a much more rich understanding of was correct sectors of could just be deterred continuous for most look the much of the system please allow yes right right if you like you may think of yes for now we may think of the detector as thing which can be oriented nearly direction how measure it it behaves like a classical system eventually will have to reconcile that with the idea that depicted themselves or quantum systems hour and a half to respect the same laws as the systems of cells OK now what I lead to Lolita that ponder now there going to be seperate from this for YRO and discuss mathematics group discussed the mathematics of the state space space states of a quantum system there said it's not a set is not a set the same sense the the all of them configurations of advice is this is a set of 6 configurations it's more complicated tribal leads the mathematics Schweitzer mathematics abstract mathematics and the mathematics this is not the mathematics of set theory it's the mathematics of vector spaces now let's just discussed award vector for a moment you're familiar of course with the idea of a vector of that there is an arrow pointing in the direction ordinary space and that's 1 concept of a vector but there's a more abstract notion of a mathematical vector space and mathematical vector spaces killing composed many kinds of mathematical objects which are not necessarily point there's along the b axis Spain
I generally only get to this point I generally make on and nonconventional means that nobody else that I know of users the notion of a would never vector in ordinary space and of finger-pointing in ordinary space I call a pointer to distinguish it From be abstract mathematical concepts of electors who would talk now about the abstract mathematical concept the rector of don't think of that as pointing three-dimensional space ordinary three-dimensional space will discuss in what basic points I talk about vector spaces of vector space is a collection of mathematical objects to fight numbers are a collection of mathematical objects and numbers of the real number access for example is a vector space 2 special case of vector space a one-dimensional vector space complex numbers of two-dimensional vector spaces that we're going to be talking about vector spaces for the moment unknown dimensionality religious right down on mathematical notation for an object in the vector space a vector space is not just a need for a collection of mathematical objects completely abstract acres a vector and that's the location for an abstract vector it's dear acts notation consist of vertical line 8 cook at 118 and a letter in between and that stands for the vector a wherever rectory what could you do with vectors well you could add them wasn't that Obama described argue that given any collectors a and B Ukiah former song and this summer's another vector let's call out of the vector sector say so a vector space consists of a collection of objects which could be a if not a bad case is a collection of vectors for vector spaces a collection revert Bruce the same distinction between number and numbers numbers a all numbers number is a specific number of so they said numbers are a special case of this you can hear them you could do something else with numbers you can multiply them by other numbers well you will eventually will talk about multiplying sectors but not yet what we can do is multiplying sectors by numbers ordinary numbers or complex numbers and we're going to be thinking about complex numbers I hope everybody is comfortable with complex numbers if not you have quickly get up to speed on complex numbers but would not really don't need to know too much you need to know that a complex number is a combination of a real number and imaginary number and have multiplied and complex numbers and you need the notion of complex conjugate the many people here are familiar with the concept of complex conjugate most people I most arid areas on the going to go it now but basically for every complex numbers it was another complex number With its complex conjured mapping between numbers and their complex conjugate sets of notion keep in mind their comeback but for the moment ordinary vectors of battle the pointers pointers space ordinary pointers Ukiah multiplied by ordinary numbers positive or negative a point there which is certainly in that direction to
multiply by 2 Mrs. twice as long the same direction multiplied by minus 1 it just becomes in the same direction except the opposite view the opposite direction so yes you from multiply vectors by numbers and in a complex vector space we're now going to talk about complex vector spaces you could multiply vectors by complex number New Trier picture this in your head as any kind of pointer and some direction just except we're now talking about a mathematical abstractions where a vector of any given that there can be multiplied by any complex number on the use let's see what shall I used for complex number my notes I used see but I don't want Z disease standard notation experts I equal position so you can take any complex not sorry vector multiplied by any complex numbers and it is still sounds like a treasured thoughtfully the area's stopped trying to think about drawings vectors if there is a vector complex numbers you could multiply them and it gives you some no vector of scores of political at a prime so Liuka multiply vectors by numbers complex numbers and that's about it that there that's all the really is to a vector space any set of objects for which you could do is a vector space gave some examples of vector spaces ladies and these accomplish a in an hour and bees I'm just real numbers then you can and then to make new real numbers but you cannot multiply them by complex numbers is still expect to get real numbers for Yuka multiply them real numbers and get over real numbers so the real numbers a real vector space about the complex numbers of the complex numbers Aiea a complex vector space yes you can he had them and get no condo complex numbers too complex numbers you could head and any complex number you could multiply by complex number and get back a complex number so the complex numbers are the simplest example of a complex vector space they we give you another example of a complex vector spaces was incredibly complicated but still pretty easy to write down take any functions of a variable but quotes side of X X may be an ordinary real variable a functional functional 1 that's a complex function has a real and imaginary part complex functional life could you and 2 functions in yet another function certain right functions just functions of things you could head To get other functions can you multiply a complex function by a complex number and get a complex function yet again a complex function multiplied by a complex number you get another complex function soak so far functions former vector space that complicated idea I don't wanna get that complicated yet so it's a race that from the blackboard give you another vector space another complex vector space let's just make some symbols a bracket and putting entreaties to entries and arrived at and the toll entries are themselves complex numbers 1 an to Alpha equals a complex number both are forewarned offered to allow for 1 of the token be anything the collection of such symbols is also a complex vector space here is the role supposing you have 2 of them is 1 and he is 1 they 1 made it anyway had them what does that mean definition the definition Of the sound of a tool is a cold columns the sum of 2 columns like that is just that and the entries powerful 1 was made a 1 hour photo Post a of giving abstract symbols which the entries of filled in with complex numbers any 2 of them Alpha and Beta we can and then get a 3rd 1 column vectors like Pearsall columns of 2 entries like there's satisfy the 1st rule and the secular role is that if you want to take a column and multiplied by a complex number all you do is multiplied each entry by the complex number multiply each entry separately and then you construct something which is Zeke so just payers of complex number of workers formed a vector space a very abstract concept but not very hard How about triples of them sure there would be wave of column like that now would you don't want to do is in column of total column refer you don't have a role for I so columns of different form different vector spaces completely different abstract vector spaces a question about the meaning of the Act and this is not ordered but it is abstract if if yellow pages also reported that could be 0 vector 0 vector is e unique vector which would you edit any other vector gives back saying the same thing the same vector from guidance like writing it down but the closure closure of close under the cause of the vision I thing right core multiplication dealers another that if the please was a major yacht This is a usually the term matrices in this class in any case will be reserved for square matrices square matrices art collections of our numbers which forms square raises many roses columns but the general notion of a matrix can have different numbers of rows and columns and the vector would be a matrix with only 1 common in this case to rose so yes this would be a special case of a matrix but many contexts when reserves turned matrix for a square matrix and I will use it that way I will use the term matrix to mean a square matrix our so I'd I didn't want to use that terminology OK our radio following saturated with abstract mathematics this is this should be the should be straightforward for you OK now we introduce on mother abstraction another abstraction basically it's the abstraction of a can't given complex vector space the complex conjugate of the vector space complex conjugate the vector space His so like love this place of complex conjugate vectors just like you can have a number a complex conjugate a complex numbers there's a complex conjugate to go with it so in that sense the complex numbers system ally has a a tool of version which is the complex conjugate of the original numbers
so do the complex conjugate vector spaces and roughly speaking the complex conjugate vector space so that the dual vector space usually called dual is roughly speaking the complex conjugate of the original vectors will make them more precise as we go along but what it is it is a separate vector space elements are in one-to-one correspondence with the elements of the original states so if there is a vector a In the urinals space this way main areas a victory in the dual the duo vector space as the a complex conjugate has written this way and just written as a backwards Durack symbol will learn the call these things brought vectors in these things had vectors but no specific advective as Robert there but for the moment religious symbols Harlem an example of the vector space is just the space of complex numbers then the door vector spaces just space of complex conjugate of those numbers is a one-to-one correspondence between vectors in a complex country let down 1st some posture with about the complex conjugate vector space and then has set out by seeing the forgive find a concrete representation of the notion will write down some abstract postulates and still did here we said he a concrete representation of the notions that we wrote down over here let's try Daryl abstract postulate about the dual vectors space so 1st of all for any vector theory is 80 Yuri one-to-one correspondence next if you take 2 vectors a new and them together if they the Doole of summer it is just the some of the duels I soviet figure sector which is made up of the sum of 2 other vectors its reflection in the dual space it is just the sum of these individual dual vectors is postulates is a postulates if you like but then that's or obscure the 3rd assumption is that if you take any vector and multiply by complex conjugate must abide by a complex number its reflection and Nadal's based on your guests cost involves the door vector OK but not multiplication by Z a multiplication by the complex conjugate of zinc V stock busy star is complex country videos it this would be true of complex numbers if you take a complex number and multiply by another complex number In the complex can't give the whole thing is a product of complex conjugate the salsa saying so we think Dole vector space is basically just complex conjugate the ability of we can make any sense out of this in terms of all on the terms of his column records here is a D yes yes a possibly as soon as people a good question guests from bad sectors and you can see that from here I should I sorry somebody asked me Are there any other rules about the definition of vector spaces indeed there are among other ones this that may be all areas that matter which you wanted to demand yes but didn't see that here at least some of 2 vectors is given by numerically heading the complex numbers here and of course numerical would be additional complex numbers doesn't matter which way you will we be but but OK so let's come back now tool for up on top and come back the concrete not an abstract representation of the complex Oh breakfast have back there this object is represent things the vector the 1 that I erased with Bader as representing the Vector B but right now I don't need the Vector B what about the object this object represents a what is the object which represents the Dole that is the daughter of a corresponding Nova over there and he answers it's sustained numbers except that complex conjugate the same pair of numbers not quite there complex conjugate powerful 1 the 2 stars but just to keep track it with talking about different vector space we write it in role form not as a column but as a role this is just a truck remember that this subject belongs to the doll that space this subject belongs to the original space in a different RIO so the Dole that the space image powerful 104 2 0 is out for 1 star offer to start that's CEO vestibule there I know you can check you could check our with this definition are postulates namely the US and 2 vectors and then take Dole whether or not to get some of duels and you can check whether would you multiply this by a complex number and then Multan and take its toll whether it is multiplied by the complex conjugate the number of yes so it just laying out a pair of complex numbers like this makes a vector space and the doll vector spaces the same pair of numbers except complex conjugate and laid out in a Rome laid out on a road just to remember that it's not the same creature doesn't live in the same space of things not the idea of Allah a complex vector space the game very abstract and very easy final concept works supper final concept but they may be enclosed 1st concert modular areas but I want people fight as hard as another hard will find a familiar but a lot of people find a familiar got those adults find this familiar apparently didn't find it hard that's also the product of 2 vectors the inner product of 2 vectors inner product of 2 vectors is analog of the dot product for pointers analog of the products reported to be in a product between 2 vectors sectors is strictly speaking the inner product between vector and vector and ah the duel of a vector get into vectors let's call them AT&T when does not form the product of 80 would be product instead 1 pigs B and constructs it In the
Dole vector space and then takes a product of the vector a with the tools to be in acts a language you take a cab sector and you multiplied by a bra vector when you do so you make but Blair kept bracket bracket the bracket of courses this thing over here and the 2 together I think this was called the captain's form of called abroad Brock kept them In a product of 2 vectors it is really a product of a vector and the rule over there now is it the same as multiplying the bra vector a with a kid Vector B question is if the same multiply the broad with declared a brought a with it be in theaters now it is not is not the same operation but factor closely related called upon the axiom numerous was still talking about axioms the action is that these 2 are related by complex contribution complex conjugated a could go from from the kids vector to abroad I complex conjugate beta go from the bra vector can't back there so this is a product of a roughly speaking times be start this is proud be star turns 80 this is the complex country of this a another plaster let when the axiom number of complex vector spaces In a product they will be the complex conjugate of in a product derby were they start but that it did not if the language of force and area product is all that is using their owners but is still called the head of special they've got usual faces and you think about our real vector spaces and real that a real numbers no difference between the number is complex conjugate sell the real vector space satisfies the same postulates except that the complex conjugation completely a trivial operation but you will if take 2 public sectors it does it doesn't fall tribe more or is this could you could invent such things that of course a proper vector with itself would not be real was OK who cares so you could invent such things turns out not to be terribly useful in the context requires mechanics sorrow your it's largely a questioner works useful here was useful wines being this concept but yes you could do you could certainly defined such a thing not commonly defined think they have a lot of use this structure colors or over all kinds of places now we we can't mechanics Serena's sister complex vector spaces are very very common things in mathematics and OK let's check now let's see if we can postulate a construction for in a product of the Dole death remember weeks when you were you complex conjugate when you take times they original vector this is supposed to be now I guess I called a B. a I awarded written down limitations would be and the notation for a column vector
for a role that the B complex conjugated I'll tell you now what began we're making up the rules which will satisfy those consulates and is very simple construction it's just beta 1 star time out for 1 was made two-star and Alpha to another words you take the 1st entry of the electorate finds the 1st entry of theory of video of record Alpha editor the 1st and the 2nd in 3 times a 2nd entry so just a generalization of multiplying made at times after but it's also a generalization of the dot product member about not products of ordinary factors you simply take the sums of the product of the components if you have to 2 pointers and labeled by
components both of them than the Da product is just the sound of the product of the components does that satisfy the rules appear Lovler rather rules vicarious must Arista rule dearest rural the role was down to flee we're like a letter last year OK what happens if you change Alpha and Beta Florian this becomes 1 star after 2 stars times Bader wanted data to if I interchange there and that's equal to alpha 1 star Beta 1 plus Alpha star and has just a complex 100 really is complex from so into changing health be really does interchange number the value of the product and its complex country thus the motion of a complex vector space and an inner product former comp works breakfast place now let's just take a couple of of special cases of EU products what about being a product of vector with itself or a vector with its image in the bill face let's take in a product of 80 with a well I know about that 1st of all that let's upsetting be equal that tells me today in a product of vector was itself is equal to its own complex conjugate I just substituted fat from B a in a product of a vector with itself was equal to its own come on complex conjugate what kind of numbers equal to its own complex country a real number so the in a product of a vector with itself is always Rio as pretty obvious just from here if I take Alpha and multiply by its image I get Alpha Star Alpha Alfred to star for to each 1 of these Israel not only is it we'll also something else positive Alpha staffer number at times John complex conjugate is always positive real and positive in a product of a vector with itself is real and positive and is considered to be the square of the Lady of the vector the square root of the inner product of vector with itself defines the length of the vector so complex vectors have real lengths real and positive lengths also a square of the beyond both square and the league itself undefined be positive that's not too or obscure every complex vector has a link which is the square root of the sum of the squares of the components but squares now mean multiplied by complex conjured next burst North important notion the notion of orthogonality of vectors notion of orthogonality of vectors is receives is a very important 1 do some examples of emotional Hawthorne between 2 vectors it is exactly the same as it is for ordinary vectors 50 in product read dot product if you like the inner product 0 3 in a product of 2 vectors 0 0 they are said to be orthogonal defines all perpendicular the case of 2 vectors are orthogonal in a product between them 0 with Celia make up some of Fargo vectors very easy although what about reading a lot about take In a product of the episode water is just a complex conjugate of 0 of complex you 0 0 0 0 0 so saying that 2 vectors are orthogonal it doesn't matter which order you put that the notion of orthogonality doesn't depend on order although the general idea than a product now we have emotional final vectors In a real nectar space honey a define the dimensionality of the real vector space forgive me and definition of the dimensionality many definitions of course per hour and a half the 2 of them lost more abstract without them around it Of course correct the maximum number of orthogonal vectors it to confined space so the vector spaces two-dimensional ordinary two-dimensional vector space ought to mutually orthogonal vectors you cannot find the 3rd sector which is orthogonal both of them so the maximum number of mutually orthogonal vectors in 3 dimensions rare I'm not 3 mutually orthogonal vectors because there are many sets of 3 mutually orthogonal vectors but given free I cannot find the 4th 1 or 3 dimensional vectors faces a maximum number of 3 mutually orthogonal vectors and the same is true of the same definition its definition is true for a complex vector space the maximum number of orthogonal vectors it confined is equal to the dimensionality of the space it does correspond to the number of entries here a number of components it is the same thing you could prove that exercise prove that the number of components of the vector is seen as the number of mutual fund vectors but let's just talk with just under any have mutually orthogonal vectors we can write sure loads and loads of them but a very simple case would be 1 0 9 1 0 0 1 0 complex numbers William real numbers a special cases complex numbers story of a complex number can be real his 1 and here's another 1 0 1 1 has a 1 the 1st place it has 0 0 in the 1st place this 1 0 you get the idea inner product is 0 times 1 plus 1 time 0 OK so the 2 vectors all us forces right in terms of column vectors 1 0 and 0 1 1 0 0 and 0 1 orthogonal vectors In a product of this 1 with the goal of this 1 0 as because 1 time 0 4 0 1 0 Pressler plenty of orthogonal vectors around but in 2 dimensions of the sister dimensions once you've found toward of them you cannot find a 3rd observers oral exercise proved that there can be no victory which is both orthogonal to this In the final that that sought to refer to know fully Yahoo 0 0 is always a fog of everything right yacht our rights or to be careful to say the maximum number of
nuns you throw off our mutual father is good point right OK that's a mathematical interlude for tonight complex vector spaces and the next time we will discuss however these spaces of upstate what means for the space of states of these simple kilobits be two-dimensional vector spaces very very different than saying that they consist of points set the space of states is a vector space and then try and make contact with the mathematics of vector space the strange things that are happening what seemed to be happening here would be cruel these zones are a funny properties from measurements us what we wanna go and physics of Experimental Physics on-the-spot 4 them With on some abstract mathematics over here and next time we want to connect them end show what that would be abstract mathematics how it represents a heart could be used to represent this unusual kind of on larger of Cuba he told me they rose to ah yes Miers rejects yes absolutely will construct in a product out of products are also to products yah be our product bears to make just 2 vectors is a maker of him of course Tommy represented the 1st heard as well we are a product of the vectors represented this way it is a matrix but will complement all classes wide use of air bases but not it the basis of functions can be thought of as column vectors buttons but with entries Our continuous so we have alfalfa is the function and an hour from the 1st entry all 4 of the 2nd century instead of that we have a continuous family of golfers after evaluated at point decks 1 Alpha evaluated point decks to Alpha evaluated point x 3 because we can't really writers literally as as was a year discrete collection like that but feel like think of these an office here to assist just a function of 2 variables as a function of an entry 1 until instead of a function of X it's a function of a variable which is either 1 or 2 3 us complex functions form complex vector spaces in a product we will discuss that if she plays that consider this will get what's going out of state auditor might want get by the O O that completely random completely random is clearly random variables you're talking about the relationship between the different replication branches twins subjected alter the same questions you think you're asking whether they are correlations between the independent on correlated variables yes the 1st time that's a big difference between classical mechanics and quantum mechanics and classical mechanics you can measure something sufficiently gently affected them out disturbed In quantum mechanics what ever you measure something you always disturb it so this is an example of how measuring something disturbs it puts it into a new state so that's a good point OK for more please visit
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Formale Metadaten

Titel The Theoretical Minimum | Lecture 1
Serientitel Lecture Collection | The Theoretical Minimum: Quantum Mechanics
Teil 1
Anzahl der Teile 10
Autor Susskind, Leonard
Lizenz CC-Namensnennung 3.0 Deutschland:
Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen.
DOI 10.5446/15011
Herausgeber Stanford University
Erscheinungsjahr 2012
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Physik
Abstract (January 9, 2012) Leonard Susskind provides an introduction to quantum mechanics.

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