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# The Theoretical Minimum | Lecture 1

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#### Automatisierte Medienanalyse

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Stecher University prior to
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
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
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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