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The Theoretical Minimum  Lecture 2
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Title  The Theoretical Minimum  Lecture 2 
Title of Series  Lecture Collection  The Theoretical Minimum: Quantum Mechanics 
Part Number  2 
Number of Parts  10 
Author 
Susskind, Leonard

License 
CC Attribution 3.0 Germany: You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
DOI  10.5446/15014 
Publisher  Stanford University 
Release Date  2012 
Language  English 
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Subject Area  Physics 
Abstract  January 16, 2012  In this course, world renowned physicist, Leonard Susskind, dives into the fundamentals of classical mechanics and quantum physics. He discovers the link between the two branches of physics and ultimately shows how quantum mechanics grew out of the classical structure. In this lecture, he discusses some of the basic logic in quantum mechanics and then moves into some more mathematical concepts. 
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a Stanford University came
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figures review very quickly how quickly we describe a simple system consisting of 1 cubic them with seemed like terminology a little bit promised that calling it Cubans being don't worry Bela has to do with spinning at the moment of course more important is that it's a little point there I use the term point there I began to use a word pointer to refer to vectors and 3 dimensional space while I was writing and I realized that so awfully awkward now unfamiliar terminology and I decided to replace it with another terms with the standard term 3 sectors sectors in threedimensional space I may sometimes used the term point there but I think are more likely to speak of me back there is a 3 sector means of vector in real more general concept or are simply say that there more likely I'm speaking about the abstract notion of a vector space and the elements of a vector space dust to warn you keep the language clear now we talk about this Q. bit which has now become a spin it has some of the character of being a little 3 veterans based there is some sense rather in which it points in the direction and that we can confirm by expand our those said before once we start thinking about quantum mechanics we really do will have to think about systems and that connection with apparatus apparatus that court would have apparatus time was sure apparatuses apparatus North Shore Europe of the plural about adjust apparatus OK to separate we do have to think about the connection with their apparatuses that measure them and we don't get away with just ignoring it what what's the difference 1st what you what the basic differences classical physics and quantum corn research in classical physics measurements can be arbitrarily gentle arbitrarily gentle means you can find out anything you want about system in principle by interacting with it with something that changes anything about it by hand on arbitrarily small America for example you want to locate the location of a particle classical physics readability shine some light on you shine some light on it and you make an image of course you have to use light which is small enough wavelengths there to resolve the ire of the abject this is true but there's no limit Her 0 week or light signal could be at a given wavelengths you could make light signal as weak as you like so it carries as little energy as you like a little momentum as you like let interactive the thing you're interested in formerly image and then after that image has been formed it can be magnified magnified in the sense of making it bigger but what's the right word for amplified it can be amplified so the experiment that Sofcom Camille arbitrarily gentle and still find out anything you might want to know about the system is the result of that an experiment can always be done which has no effect would ever on future experiments on subsequent experiments for example right afterward that classical physics quantum physics is not like that quantum physics is such that experiments inevitably change systems any experiment the measure something will change something else it's almost as tho the experiment To measure Corning was such that every time I looked at the Corning the calling jumped up in the air and end a randomly flipped itself will that hasn't happened and we can measure the state of the Corning gently without changing it on mechanics not so clear after world is clear square that cannot be done OK but let's talk a little bit before we move on to more about vector spaces in operators in mathematics of a series of articles about the logic of it and what way logic in quantum mechanics is different than logic classical physics our let's go back door Spain an apparatus misery all draw a little Harold whether truly corresponds to anything that looks at the classical arrow or not is irrelevant we have to draw on a blackboard and drummer blackboard will be classical the pink and pens or classical plot but the and apparatus the apparatus is a window that the as a number that comes out of that number appears in the window tells the result of the experiment and also happens that have a direction in it which tells you which slot which side is up how oriented factor might have a whole set of 3 taxis fees are you had already and that but that's some important just up axes is what's important In 20 apparatus it is pointing up the only in fact no matter what direction the apparatuses pointing in when it interacts with the big that produces on number and the window and that number those either plus 1 or Murray this 1 no matter what direction the apparatus of pointing in that no matter what has happened to them before him when apparatuses pointing up opt will say that what has measured is the Z component of space but put some taxis in here Vt up next to right and White House blackboard only apparatus is pointing up will say saying that it produces an express sentimental number which measures component of this bid now that particularly interesting to think of it as having a directionality only states component of the spinel imagining that the object he has a sense of directionality to alkalic customers has direction and directionality too late well we can't prepare despair in the state of what we think is pointing up by simply running it interacts with the apparatus of the apparatus of Peters with a plus 1 sign it was how the spinners up that we do it again with the same apparatus pointing the same direction as saying Spain we'll get up again in up again again match since quantum mechanics allows for reproducible experiments if you miss the same thing several times a role with the same apparatus you'll get the same answer over an office things are not that weird and it's also important because they have to have some concept the reproducibility of experimental results in order to make sense of things so this is a good thing when you measure something and you measure again and again and again you have your a comfortable put over the counter what I did say that experiments unnecessarily so on gentle when you measure anything you will affect everything into you effect something and this case when you measure something you don't affect vexing figure for measured again the same thing again you get the same it's OK now you could say are now that I have detected that the spinners up let me turn the up of the of the apparatus completely over and played down if this wealth Mama measuring the temperature I don't think we would expect that to measure might miss the temperature of we turned over right now about in car decent monograph so they are further say that mother temperature does not have a sense of orientation is certainly not newsstands and and that this is a true spend we can apparatus over initially apparatus told us that the spin was plus 1 after we've turned over and the experiment began it will register minus 1 so seemed to indicate that there is a sense of direction our but of course is not the same thing is true no matter what direction we hold the apparatus and if we hold the apparatus in the horizontal direction will come along and detect a spin the we know nothing about we have not yet made a measurement on it maybe somebody else's made a measurement farm it maybe some
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agent on available to artists who loves secret person has done something that's been but we don't care we don't know about it we can't without detective we make a measurement what will measure again we will measure plus 1 or minus 1 and if would begin to think of this please as having a sense of direction to it we might say that will measuring then 0 is X component of the free world previously who measuring is a component the list given names the city which is a component of the spin along the vertical axis that at the moment yet I'm not asking you to think the notion of component is same as the notion of component in classical physics which is the name for the moment the singers Eamonn Everett's measures plus or minus 1 the same thing happens when we turned the apparatus on and we measure the same thing procedure we measure something that we can now costs in Texas and X is always plus or minus 1 and the same thing if we can be apparatus around so facing trial in fact no matter what their action or if the apparatus it always give this past minus 1 this would be a little bit strange model of a bit strange if this object really wore a classical vector who vector perhaps a vector of unit Lane if a vector unit Lane and the were really measuring the component of Spain along its own taxes there is no way that it would just give plus a minus 1 components of vectors even the vector is required to have exactly unit Lane is a vector with exactly the way a pointer 3 vector of exactly unit length that we turn it in those other things with the x component of Lizzie component convene any number of between minus 1 1 so this is a funny vectors vector at all but still opposing Davison from orientation the results of experiments do seem to change when you rotate detectors in particular as you begin by and things spin along the zaxis indeed up direction cause up that means singers is equal to plus 1 and then rotate the apparatus detected again you won't necessarily get plus 1 sack you may get plus 1 or you may get minus 1 with equal probability you could do the experiment over and over and over again half the times you would get cigna exit was plus 1 half the time we get Mexican miners warned even tho you knew thing with Z was exactly plus 1 in each case so I'm sums funny but what is true is that average value of sigma X would we want would be Zero if the initial starting point was that was up along the zaxis same funny at Y axis of initial value of the spin was prepared to be up along the zaxis and you rotate the apparatus a white orientation again you'll have equal probability the average value of signal y will be 0 even go beyond that and say Let's start with the apparatus not vertical let's start with the opera the apparatus along the x direction and use it to prepare the so that is known to be pointing horizontal along the x direction lurid Cigna Exs eagled the plus 1 and then the Warriors Sea experiment always will get random but equal probabilities for up in there for our plus and minus 1 you go the threat now cause I'm quoting what would b the results of experiments on space I've never done to experiment the figure could of birdies and no doubt that if I have the skill to do it what happened what all of them did the mayor of a town ex wife all we owe it let said a cast that lets us that they last what start with both the went before we yellow may come back for a 2nd and you're right we want allowed experiments measurements and choir mechanics distinct ways we could do lot of experiments are really matter which which we do we could imagine having a large number of identical spins prepared to be giving away however they were paired that we do experiments on a large number of them all we can do expiry Peter experiment from estranged the large extent it doesn't matter which were talking about how tell you when it matters but at the moment or what talking about what you said start the apparatus horizontal status will it and detect get rid you get William plus former minus 1 friend you could then take the spin out of there but give it to your friend who are does some magic with Spain's you don't know what it is you don't know what the hell he does with it he flipped around Jos it puts pocket takes his back to you and tells you do it again again you'll get fast 1 or minus 1 maybe that he did nothing in which case the 2nd form of get the same answer you have 1st part or they have turned over which it is to get the Office of Civil War have you get it would plus 1 or my miss toward the if you would do a repeated experiment knowing that nothing else was done to the space in between you get the same answer if you do the same and that the focus off Taylor general let's suppose that we already and axis of the year detector any direction so let's make a notation for direction the unit the unit 3 vectors unit vectors in threedimensional space were always indicated by putting a little hat on top of the symbol so this is a unit vector along a particular access we start without detector along the axis it is not horizontal my vertical mucking around 0 2 but just cattycornered in some some complicated way it our we start with the speed with whatsoever work with this previously having been only in of prepared along with different taxes and next how we do that while I should've gone this slower we start with the detector along an axis through experiment and discover that Longley this thing comes up plus 1 then we rotate the apparatus In the end direction and we do other experiment on the same spirit it will either come up plus 1 0 minus 1 along be annexes never in between but B average value of the spin along the Amex's here will not be 1 who will not be minus warned what will it be it will be the cosine of the angle between him and in other words the component of idiom of only in highway right that a strike that from a formula for about a symbol averages in quantum mechanics Our parents angular brackets like this was called the From of Corbett kept the whole thing makes a bra which is British so bracket OK the average value of Cigna am not that it was originally are already in the along the indirection measure Longley and direction is the cosine of the angle between and which can also be written 18 . the ordinary
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dot product of vectors also just that the component of in a long and spin was already Anthony indirection you measure along the end actions and the answer is plus or minus 1 of my family average its work classical physicist might have expected if you measure the component of the spin which is already in 1 direction along some other direction is exactly the component of em all along I still them many many times and collect averages it seems in fact that average quantities behave like vectors or River for the individual experiments on very resistant to giving any answer other than plus or minus 1 OK so this now I know how to use the Morag truck was Yemeni our Machado the mother apparatus that was better but it will you you said that there be a spilled the annexes there these measures only and that and you get a random and a random bias bias in such a way that the the average about that example would mean let us In the name of the same direction and you'd be rotated was in but I'm in that case 1 right I would just be indication that you the same answers and was up and then experiment to get up and up and up over Oregon OK orders being added staged we have no no no friend Margalo among lobster say the expected value is not the expected value if I gave you a probability distribution of like this what would be the expected value that you have the expected the least expected value would be the 1 right and that of the average value would be the war right expected that expected values and average values are very different in general the average value average if you average quantity that you get the standard sense of averaging in standard average it's also called the expectation value but again that's a misnomer not it's not the value that you expect necessarily Zhao said the last the measures we what we know that With the place again after that it is not well they are different experiments to Duke and do what on spin the results of those experiments it can be used or particular experiment UUU due can be used to label What is you measures at a quarter of Troy it said that Fed death he but has said that she had doubt that it is not geometric mean obviously to begin with but we did discover that they turning thing although we always get the opposite side on the house some difference between the apparatus of the apparatus that asylum like a Mama measuring temperature some notion there that is different go through B upanddown never tell side we find out that sums and really does behaved average for a rare for their approach but it is voters With changes last year where the Earth may have that at best but that will be Fadlallah I gave this guy but is loaded told fact the facts are that you 1st prepared by putting your apparatus along 1 direction and the measure we're apparatus along the other direction average value of the result will be in doubt but every single event will be a plus 1 arm left to set a date anyway give headed a rest after this just of words gathered it said that might still be right but the number of a result can only be plus or minus war and the averages plus 1 in every event was plus 1 OK let's talk about logical a bit of a problem of quantum mechanics is a logic the logic of the of quantum systems is different than the logic of classical systems Bob brains are classical systems in each capable of thinking in the quantum logic nevertheless we can get a glimmering of quantum logic is like by thinking about these experiments on we gonna talk about the calculus of Proposition or the nature of the state of the system the space of states of the system let's start with estates assistance basis states of the system of thing classically from moment Bessette said of possible states they are heads tails of 6 5 4 3 2 1 for the particle outlying would have an infinite number possible states but the state's formed a set of some type a proposition about the state of the system is basically a subset it's a subset of states for which the proposition a stroke or pink soul after the my proposition is the dicey shows and even face that's a proposition and it's true would have these states of the dying and so it corresponds to a subset tool for sex and thus be contradictory statements and not statement not the guy you show even face but obviously equivalent to the statement the dice chosen on face and sorrow in classical logic may not of a statement the not means the contradiction of the statement of this is the statement it's true for subset of states in here but not statement is a statement that a busy complement or collections are things which are not in look at us let's go toward rasping propositions spinners up along the zaxis what's the knot of that amount that is if we measure the spin along the caxis we get something else other than 1 the only other possibilities minus 1 source it equal minus 1 along the zaxis is not a statement of the spin up along the zaxis so logic go logic makes some sense but we begin flying very peculiar differences when we start talking about the operation of combining propositions through the use of end and all our hand and more are different in quantum mechanics on classical mechanics who I want only that out it's interesting and so rarely explained that the that I'm sort of like tour to explain yacht at work in the class of their that both stated that head
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just after the that rolled off more of yet OK you're kidding at something important which were going to get you on that claim that in some sense the number of states of this Spain the finite it's just up or down if we measure along the zaxis on the other hand they also looks like we can't kill star apparatus in any direction and prepare the state in any oriented along direction so it looks like there's an infinite ambiguities in the state of the system this is what this the whole point of raw of trying to understand the states of funds in the connection between the continuously different direction the electron could point and the fact that any time we measure along a given access only Upward Bound for comfort but a good question but will come OK let's self let's talk about the proper calculus propositions if I give you 2 propositions AT and be like informed tho combinations of them 8 for and a and B Lester secrecy what that means in terms of our sex every after Proposition hours Davis 4 being there's AT and B means the set of states which the long both a entered b so it's the intersection of former set a premise that call logically intersection are set theoretically intersection is basically an operation this point is In a and B of Iraq straightforward lead and Eddie given in tool propositions you all always the media Arenburg another way the right where it is this is a set of points in here for which they Proposition a N beat is true AT and true here and strewn orders are what about what about these 2 subsets a and B a and B is still a proposition but it's always falls there is no 1 there is no state for which a and B is true I so they are ace and that you could put together in a reference NJ over here is away 600 pounds and my friend but that there are combinations for which a and B is always obviously it particularly if you take a and not it's always falls back but still the definition of a and B Is the intersection and it's true or false state to the intersection of all is a little more complicated Worley said 2 versions of all our there is the exclusive or an inclusive or in English language we usually we use the word or we usually need be exclusive or only say that a moments Joyce and I'm kaffir around Catholic Ireland there really mean that I can't be box that can be both so the usual or is exclusive or the 1 you normally think about the logic the term war inclusive it means 80 it's true what is true it's true of B is true and it's true of both of them are truly inclusive or are so I can correctly say I am a physicist and I am a father that's true but I can also say I am a physicist or I am a father who was also true why because both of them are true the inclusive all inclusive all our corresponds to the union of 2 steps if I have an and I have be inclusive or is Everything in both sets or either set anything that most it's anything in either said or in both sets the exclusivity or would correspond to things in 1 said but not the other said the exclusive or includes are there will always with whenever I speak of the of or I always mean the inclusive war OK that's carried an order Gross could you say about them Barbara March but Yellow River we're going to mount use that concept to try to formulate a set of and or a statement about the band and then asked how would you verify how would you verify would verify them and doesn't mean anything to verify OK so let's take the 2 propositions a and B The be same Lizzie is equal to plus 1 plus 1 and Cigna exorcist AT & as they sit next physical plus 1 both propositions we would know how to check you want but we want to check a with those of the 1st a we would turn on apparatus vertically upward and measures space rejected what's it like to find out be wastrel falsehood turned the apparatus 5 classical logic if Iraq 2 propositions you could form a combination a or B US star with a or B cells take your baby and let's think about taking its true value is it sure is that not whitesonly will design experiment to try all the let's go back a step that the imagining ghost in the works has Our previously unknown types unknown as originally oriented the spin upward along the zaxis he knows we don't as far as we know we know nothing but there are the ghost in the works of oriented the spin vertically upward just there doesn't doesn't matter but we could start with suspenders upward and let's check with a or B stroke I would here but begin by measuring the spin along the zaxis we put our detected our apparatus in the vertical direction we measure where we get the we get past 1 rear plus 1 because the ghosts in the works as a has made a point to begin where we don't have to go any further 8 would be strong a or B includes the possibility that both a and B a true requires to be true is that 1 of them is true and we just that discover that a is true self a or B is definitely true but now but still the experiment another way and in the end order which would not matter if we were doing classical physics and the reason it doesn't matter what we do classical physics is we do an experiment that has no necessary affect the outcome of a leader express experimenters gentle enough so let's build the expanded in the opposite order instead of 1st finding out whether AT is true but 1st ask if he is true and they then will come back and ask give a use true white secretly the spinners pointing up in the easy direction the ghost knows that but we don't and so we come along knowing nothing about this and say Why should I am measure what measure a 1st but 1st measure West 1st measure the x component of the state where we get we get it plus 1 Alliance 1 and we get plus 1 0 minus 1 with a 50 percent probability sows a 50 percent probability that these plus 1 but B is not plus 1 it doesn't mean that they the ADL Halaby is wrong and we still have a chance that when we measure aII reports warned OK but the problem is that in measuring B the 1st time around we already and would prepare a B along the xaxis were ever we get where this or this way we
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have prepared that's been along the xaxis what will the next experiment give only measure along the zaxis 50 percent probability of being up 50 percent probability of that means we have a 25 percent probability of finding both the statement X was able to minus 1 end sitting with Z is equal to minus 1 we are a 50 percent probability in the 1st experiment that X was not equal to 1 and then the next experiment we have another 50 percent probability on top of that the Cigna Z was about equal to warn so what we have learned that probability of onequarter that both a and B will fall when we did experiment in 1 or more we don't the experiment and the other order it is certain that or B is right why because he is right so who was showing classical physics that the idea of a or B symmetrical between and beat the idea of the video or statement a or B E a warm B food and BU or a stroke and we find that in quantum mechanics the concept of a or B at best can only be defined in a way that depends on water in which you determine a or B that 2 propositions it would be specified in the quarter chemicals busier give aid is part of past a certain expert Gary and which sets the I would but I would also say that any attempt to think of a logic of experiments will be different than quiet carrots while higher words AT&T's measured a certain we being a are about said the ghost and woodworkers made this and to be pointing up along the zaxis that we all know that but it's true that if we singers 1st under those circumstances we will surely find it a Uribe's drove but remained under the same circumstances the same degree of lack of proper president of knowledge with measure B 1st we find out that there's a 25 percent chance that he'll be this does come past statement here is for questioning he operated a now going to be the for size well not only that but you have to specify which order your determination right so that's a itself an element of something new that or statement does not unified 2 statements and a symmetrical aura symmetrical way quake both but used up their there but that doesn't matter just 1 it was a matter quick there were to be just which it said that this time will get a Yahoo when applicable which I find right what saying for that 1st there should not cyber said and that then shows was on that but do were there when the resolution was not good where they met for the 1st time back but southwest work workforce with elevation made Mr. moved read by an or regard wiry or as violating the at vision it give it that but that brought here that was all revealed that the next thing which I was going to review but I'm not going to because it took so long vector of the revised euro quickly vector spaces are set of elements you could a elements this is not something you would do with sects you could and 2 sets of Kostunica bigger said but our has no natural single element which is the sum of 2 elements policy or will be elements Our set theory namely the basic individual objects on marketing to command and make no elements of the space of states a quantum system is linear vector space and that means given any 2 states you could and then some sense and make another state referring to to explore our this logic is peculiar logic this peculiar distortion of logic turns out to be the logic of vector spaces a logic which is embedded in the structure vector spaces in a very neat and tidy way and that's why we ultimately why vector spaces around 40 OK trough but remember vectors of things do in a the the shuttle vectors and be them Ukiah multiplied by complex numbers every vector has a dual that which is kind of its complex conjugate recall these objects can't vectors because they are the 2nd half were kept them the cat broad sectors which are in onetoone correspondence which are for every 10 victory a brother there about the status is that of a complex conjugate so show dual vector spaces and you could take a product of vectors a brawl that deliver can solve the broad vector indicate that it could be combined together it gives a complex number being in a product is a complex number and it satisfies letters equal to the complex conjugate complex conjugate are the opposite order so when you take a complex vector complex vector space take me problem if you interchange with 2 vectors then that's equivalent
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complex conjugation that's pretty much what you need to know that allows
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you to define how giving a given the idea of a in a product that allows you to define the notion of the length of the sector we usually don't think about the length of the vector we usually think of a square of the vector and the square of the length of a vector of say vectors a busy in a product of its all conjugate duel we usually say Stacey the product itself but that's a certified abuse of terminology and is always read the reason it's always Rio is because if we change ANA and AU we just get back to the same thing but this now that thing in a product of a only complex conjugate on complex conjugate it's a real it's also a positive it's easy approved and it can be thought of as a square of the League of vector it's analogous to closely analogous to the dot product of a free vector with itself if any Indy with free vectors did we would distinguish the Braman then cat but we were right that a . about product between 2 vectors of our use of its allure also arrived at 8 . aII was the school of the lane of a be in a product of vector with itself could be thought of as a square of link to the vector and finally there is the notion of orthogonality 2 victories Our orthogonal when in a product 0 How with with it was final same definition prefer about product when the . truck between 2 Vector 0 both vectors a fine of tho the definitions that we need for a vector space the other definition is a definition of the dimensionality of the vector space on the dimensions of the vector space in wake the sign that is by asking What's the maximum number of orthogonal vectors confined in the vector space for all vectors 2 3 4 dimensions this is obvious in tow dimensions I could pick pickaback did any make EU vector discover shrinking its length and thoughts of you wait and I look around and see if I could find another orthogonal vector tour confined to 1 . us where 1 point that way but that's not that's important distinction we find 1 on directions 1 of the vector apart from apart from numerical multiple apart from numerical multiple there is 1 other vector which is orthogonal to rate and at 2 neutral only a maximum number of mutual orthogonal vectors it you could find this to be a redirection didn't find the 3rd 1 but no more defined a 4th 1 have directions sold threedimensional space has 3 mutually orthogonal vectors fourdimensional space more so for us and the same thing is true of complex vector space we describe a version of complex vector spaces which adjust column vectors are for 1 hour for 2 out 3 . up and wolves which will roll vectors which will made up Out of the complex conjugate variables so once we have a vector we start working for orthogonal vectors to it sectors which are perpendicular in sentence find other want them oppressor London look for 4th warned however many you find you eventually run out of directions that the dimensionality base for the special case of which there are only interest here it's a twodimensional vector space you could only find tool mutually final vectors you could find many painters of mutual apart vectors and Anderson real spaces these as these as these 2 . 2 men and their actions but once you find to it and there won't be another OK so we have give mutual Fargo vectors and now we have to state what is the connection between a vector space and the space of possible states of a quantum system pool the quantum system that reveal what it is still the very primitive and simple singles banned until we get a handle on that we want a moved more complicated things that jurors have debut as will push humidity Harold because of worries about what synonymous terms perpendicular final both me wrangle you confuse by something and I don't where Yom Yom where your your breath perpendicular orthogonal right angles they mean all there but that a firm that is not confused right with nearly independent is different from of Fargo Fargo is linearly independent but linearly independent is less strong these 2 sectors like that there linearly independent but the nullified bars orthogonal game so we will come back the states of a quantum system is a pastdue it formed a vector space former complex vector space in fact the space on the simple system we've been discussing forming twodimensional vector space forthright bounce back as we know some states we know for example that there is a state that can be prepared in which the spinners pointing upward along zaxis we characterize the state by the experiment that we be together in that state that experiment Have the apparatus vertically upward and created a state of the system which either could've been Namrup but let's wiki label at stated number ways we can call up we go down call plus we our way up and down of the same plus and minus we could draw them with arrows up and down or other ways could we are we could arrive as singers E equals plus 1 stint equals minus 1 they all mean the same thing up down plusminus figures
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vehicles plus 1 Eagles minus 1 all the same thing I'm so there are 2 states that we know
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about that weekend access but putting these would prepare by turning on apparatus upward along the zaxis let's give the names of these states of vectors in a vector space according the hypothesis and they can be called let's say up and down if it is true that the vector spaces as simple as just a twodimensional vector space and that means it any factor in that they could be written as some linear combination of up and down another every that in a twodimensional space can be written In terms Our 2 basis vectors up and down enough laughter right any state whatever that I about the role of the state which is made by tilting are apparatus along the xaxis we did something different it's not the same thing it's not because other means up along the zaxis is not bad because down down along the the zaxis now we know different because we know the results of experiments on these 2 different kinds states will in general be different so we can call states attend the left and right up and down was right and left Of the 2 other states we could draw them this way if we like they also corresponds to signal x equals 1 and Mexico's minus 1 that we could label and if we liked label them right and left finally we know that if we would turn our apparatus that these apparatus was pointing along the Y axis with access to more we can access to most states bigwig call those in and out into the black out of a blackboard so there is in and out into black matter Blackwood and they correspond to more wiII equals plus or minus 1 Wycombe label them in and how all the logic that I've described to you and all the experimental results as we will see proNazi and immediately it took some time understand all of this is also summarized by a few statements but the 1st of which Is that the space of all states of that simple system is twodimensional has only 2 independent states which means that these other things can be written in terms of up and down OK let's go let's go to in general linear combinations of up and down and that's what they mean let's take a victory I guess we can call an 8 AD which is over forming coefficient of complex coefficient forewarned find out plus another complex and another a Scottish call How far how far up offered downed times down the office I'll complex numbers and complex vector spaces you multiply the could be real numbers but in general they will be complex numbers let us suppose we knew what that meant what kind of status 1 where it came from the beast it's not up not down from was between next postulates next postulated is that if we want to measure up and narrative through its somebody prepared us an electron Rory spin in this state and we measured up and down we would find probabilities were up and down and those probabilities would be probability up is square of the magnitude of this coefficient probability of downed sequel Balfour stored down offered then another word basically the square of the coefficient of the square of the components are 80 in directions up and that outfit their essentially the components of a general vector along with 2 acts eased up and down there is something that I failed to tell you it's also them think posture but Potosi see they fit together in neat package there probably fuel pap consulates I'm saying how that we could we do sit down before posture for axioms but rather than try tax ties it would just the rest of the room or informally what's a relation between up and down start but yes we Aviva negative each other now they're final now that's posture the basic postulate is when things on measurably different went to states are such that you could cleanly tell the difference between them by an experiment here are orthogonal orthogonality literally means they are sufficiently different that you can with a single experiment showed the or maybe a few expands tell the difference between them uniquely a player corresponds to terms sex you might say that to set orthogonal if they don't share any element is now standard terminology but you might say it anyway there are 5 of those Jeremy elements that means a distinctly different there's no chance of experimental confuse the 2 of them experimental with nice experiment to distinguish them will not lead to any ambiguity the same is true of up and down the distinctly different and down a few measure up you measure up again and again and again and be version of that in quantum mechanics Olivie implication of that quantum mechanics for the vectors that things a chorus pontiff physically on the big was different situations far richer so up and down or far richer What if we have a combination of and was some coefficient then there's a probability to be up and a probability to be down and the probabilities a given by Alpha pop Alpha start Alpha down again that's a postulated that will see a moment with that corresponds to wealth what's the relation between the probability for up in the probability that down the answer is free and then we get 1 and therefore offer Star Alpha plus played Starbase is equal to 1 any reasonable state of the system which has total probability 1 for all the different things that can happen should satisfy the some of the probabilities Of the distinct probabilities should ever awarded so that says that's how fourstar after plus biggest never made a profit and plus after balanced star Alford band that I used baby before they should boring but that's a equivalent the statement that be enough product of AT with itself is equal to 1 what we think about but process Bush
1:03:51
ruled that out star after some upanddown is equal to the inner product of with itself does follow from the postulates it does postulates and the assumption that you will be hot perpendicularly unit vectors Ewan perpendicularly unit vectors make that assumption FOR GROWTH whichever the unit vectors denied unit vectors you could change them until they are you and then you could prove that in a product of a vector with itself is just some Of the squares conjugates turned itself could threaten the through think proved but would not say that says that a physical state corresponds to a vector and the vector space of unit Lane a unit vector unit standing for the fact that the total probability is 1 of 5 states not only equal vectors In a vector space but they unit vectors for the other terms unit vector is normalized vectors normalized vectors normalize is another word for so or physical states states award you could do that electron spin or anything else correspond to unit vector Front given any sort Jude Becker the probabilities associated with our office staff up after stars back so that a bit insulation the winner will happen to these other vectors right and left what happened the right and left right and left seem like different states but according past right and left must be linear combinations of up and down that's not too surprising because as you prepare right or left and measure offered then you get plus 1 0 minus 1 a solo its performance 1 was not surprising that whatever writer left is it somehow a combination of now than that a statement was varied In the properties of vectors so let's look at the state blast most of stay right where we know about it but we know there must be a combination of thought up plus Alfred down we can also write that in the form of a column vector how far up and down No. 1 No. 2 supposing electron is created pointing along the right direction the Xaxis where probabilities for it to be up and down and a half the Haft family is that our fop star of mistake an hour for star that now must equal half but against the bigger guests the simplest guests would be to put Guo square root of 2 0 and 1 over square to How far up is squares to itself up as 1 of square toward toughest our offer 1 half of the also warned him now we could probably could put some would put some of the things when ambiguities we could minus sign him all week Apollo complex coefficient them from here without changing the magnitude of that will come back that will come back to what that means this will conduct a without ambiguity me but let's take this as a working hypothesis for what the state right here that leaves us with the state left you know but left well right and left are physically distinguishable by a single experiment you can orient furor apparatus along the x direction measured Sigma X and determined unambiguously whether it corresponds arrive laugh what does that mean about the vectors right and left it means they're orthogonal of Fargo and physically measurably distinct 3 things so that tells us that left right should be equal to 0 incidentally we have still that up now is equal to 0 upanddown orthogonal by symmetry basically there's nothing different about the Xaxis zaxis suffered 2 vectors up and down Fire stands the reason the 2 vectors left and right must be of part of is right is 0 vector that's orthogonal to it we aren't there be is a vector which is part what is at 1 0 square too hot minus 1 over square to invalid ballots ambiguous whether you put the minus sign here but here he the way will work I will find out that is no physical difference between those 2 possibilities for an hour later that there's no physics in the overall signs are over most a vector so here's 1 which is a fire or how we checked this is a fog well let's let's write role of strike Ruud his column vector describing right let's right down the road that not parliament column column vector describe right is 1 0 was quoted as to what was quoted as well as the role of that describing left if we want to be in a proper we want to combine the row vector with a column them so the role vector prescribing Skyring here is 1 all the square root of 2 minus 1 over square but and to take the inner product between now and you take the product of a 1st coefficient terms a 1st coefficient of the product of the 2nd 1 times a 2nd stand the rule but taking me in a product between a convector and a role that do you match the 1st component of the 1st component multiplied setting component of the 2nd component multiplied and go right down through the years those who components in the mouth OK so what's the answer 1 half what was critical what was Querida letterhead minus wireless could lead to Times Square to visit a product is 0 so we found a 2nd back there which corresponds to right and left a candidate if you were right to left and what's more you notice that this sector also has the property that is but the probability for it to be up and down is equaled or half for each 1 white because the square of this is 1 hand and a square of the magnitude of this 1 was 1 so we have a nice symmetrical relationship between left to right an upanddown and family is kind of interesting supposing with merit . 2 it's kind of interesting
1:12:32
I reckon right and left in terms of up and down I could of course right up and down in terms of right and left what do we get this week and these 2 equations the cancel or just get up will get told school squared to it 12 of the square of 2 times up Eagles right plus left and then divide square to angle fine that down as you gotta right minus left all the square says is a complete reciprocal relationship between them off if we began our experiment With our apparatus along the x direction we might have started we might have sort of left and right and come to the conclusion that up and down of combinations of left and right OK that leaves you now worth about hour is there a 3rd pair of vectors in an hour which standing in the same relationship to Orion less and less right and left and upanddown resist into each other 0 3rd combination which has which is symmetrically the fine regardless sort of we have been down and there a final to each other but we have right to left to right which is equal to 1 over square to become writer has up fast Darrell and the left same thing a square 2 times opt for minors don't we also have a right to left equal to 0 more although we have something I'm going to write but I'll say it anyway you start right and say What's the probability for up and then on its onehalf bestowed left and what the probability for upanddown it's 1 half what happens if you start with up and down with the probability for right left was also 1 half we know that's 1 half from the physics of the situation but it's also true that when I re right up and down in terms of right and left the coefficients are also won over square was too so the nicest symmetrically related to each other but is us with the last possibility of West perpendicular directions in our case which China guessed that I haven't up and was gonna have plus a damn but is good isn't coefficients here we don't know what those coefficients are you know about coefficient the complex and general Indiana where complex what clear square has the the magnitude of this square has to be onehalf we also know that the officials here have to be such that in an hour perpendicular to each other we also that if we rewrite In the an old In terms of left and right the coefficients should also have modulus won him just symmetry does that's enough to determine and completely among them throw it that's a homework assignment at a nice homework assignment was to what the requirements are requirements on in now equals 0 that but In How would win a written in terms of up or down should have coefficients was magnitude is a half square 2 is a half and also when written in terms of left and right the same corresponding coefficients year should have magnitude plus it asked to determine the coefficient from the terms of up and down with a small ambiguity which is not interstate not later with ambiguity has no physics and 1st was your overall sign of the vectors that has no physics will will find out are where the solution the solution is 1 year and ii here was Querida we have 2 per 1 0 Mosquero too 1 over square too and how what is the same thing Midas over square to them a strident and column and row vectors there's Irish further on it but Rabbi alike Georgia asking you had a writer was 1 times I say we go right isn't a vectors slid striker various column vectors up will address widow right 1 0 the up with no with no Darren component down His Joe 0 1 1 and the Dow Place and 0 in a place where about right that's equal to 1 over square 2 0 1 0 0 square too left that's equal to rule 1 0 square to minus 1 over square now let's put in in is 1 squared of 2 0 I over square of 2 and right now is equal time I won over squared of to minors I over squared it too 1st question is this orthogonal to this Our reject we take this time the complex conduit of this was this the complex conjugate of this role for calculating in a product we could construct the role that there is no need to about the but but now we don't remember the role of actors health complex conjugate entries quiet but easy enough to do it just to get a product in a product is some of the products of the entries plans a complex conjugate writer 1 of these has to be complex conjugate was product of this with complex conduit 1 done plus others is onehalf of this 1 this 1 your 1 half and how about this time the complex conjugate of this might want my minus Iowa to watch the comfort and you get of vial square my side of the square story after multiply might have squaredtoe turned her mother mind side of the square to which Midas icons eyes minus 1 minus ahead we've 0
1:21:18
3 at these 2 are to each other these magnitude of the coefficients here the square the magnitude of the coefficients are each equal half which proved the journey state over the Alps stayed the probability of being Upward Bound is 1 half how about the probability of being left or right 0 mighty figured that out definite definition of squared minus 1 by times eyes overseas warned by is now or sorry this of this idea is not in fact that this OK ally and our right and the left up to say the way it is and down and now go back out all the pages and change it I actually it could achieve them more easily just change site to capitalize with the that would be a better bet form that form Syria has left a very good question the answer is yes but will come yes that's a very very good questions on what is it that year that is telling us of our right here in court system of the lefthand chords and I don't wanna go they yet Will Contel it will come to it when we explore the holy matrices from the moment we're just doing was a guessing away bomb going through giant defined sectors which described by various state part the system OK O this in my proposal if you like my best guess at the moment for what the state's up found those a base that's debut stock in 10 of those were right and left arm in out now would you could do we saw that you could really right up and down In terms of left and right ushered out there and that means you could write I O in terms of left and right you could check what the coefficients are I hesitate to do it because I'm sure bombilate never hand I'm tempted to do it because social did a right they want a shot at the start workout old OK up is left plus right down his left friend with a square removed her so let's Celia figure out what his 1st such is doing in India's 1 or the square 2 friends up upwards up I said what left plus it is left class right left force right but With over square with another 1 was quick to gives me a 1 half the raft or 2 plus right over to rule that's scary and then class ii always quarters tool cut was down but aboard downwards but left miners right or give me another forgive me but left right for another squares To my a left
1:25:59
over another square to the whole thing looks like he said that there are bad said that it is not known but I'll let us their hair and this could just be 1 hey the squid 2 couldn't and that's Connolly back right OK so early get we get a coefficient times will last until we are all I see is equal to 1 half minors 1 might yes I over 2 is 1 my ears over to turns left plus there's 1 plus I owe to tons right it Georgia Oh said here we are faced with a heart love include doing this 1 up his left plus right down his left miners right at the same time go OK now what's the magnitude of the 2 coefficients here then a complex conjugate let's check battle see square to win this thing you see these queries To win you might have expected sums Querida tour nevertheless right maintained that if you take this kind that's complex congenerical to college to maintain take this turns its complex conjugate to get 1 half but jacket and
1:28:19
1 of minors ii times 1 plus ii over 2 times tours for 1 minor terms 1 class I pray it is 2 over 4 which is equal to onehalf another word this coefficients although it doesn't contain any explicit square root of you it times its complex conjugate is equal to 1 over to the magnitude of this complex number here is 1 over the square root of 2 3 uniformity see an explicit worn over square but the magnitude of this war was worth through and so was the magnitude of the implication is that if you started in the state on ID into the blackboard instead of measuring up or down new measured left or right the probability would be ahead but so we found mutually related and that saga whichever up in Namur left and right and the number of pirate whichever but then our final vertical direction but each 1 if expanded in any of the others would have coefficients whose magnitude was 1 over the square root of 2 Service symmetrically relating to Mattson and what this means in effect it that if you start with any 1 of of them and measure the other component of you start with upper deck and measures Texas signal wiII you get probability at half the matter what you start with any 1 of these Ray combination annually 6 combinations and measure a perpendicular combination you always get probability plus now I've put in some hired some some postulate about what probabilities are without telling you about the uh I think we're coming along pretty well we have some mathematics here which is telling us that that these 3 collections of pairs of vectors on nicely symmetrically related to each other are end that well that's most important factor yes we're not it can but we don't have the Seville this is orthogonal to their own war on terror 3 basic graph that's 3 base used at angles with respect to each other that's a big question the question is could filed excellent question could only find another double let's call it that will now we're ready use little will love fat skinny guy bans Getty so we find another payer such that travels orthogonal scary and be product that's from that through actually came down or that the expansion coefficients on each 1 in turn these other kind of basis are always equaled 1 over square 2 are always 1 half the magnitude and now there is no 3rd 4th combination that's in the same relationship that's that's got sets up but it is a fact we were threedimensional world it's exactly factory River threedimensional world right it's exactly the the fact that statement that the 3 dimensions of space can be represented In twodimensional Loma twodimensional vector space terms holiday something else which is true if I give you an 80 column vector Webster column that the Alpha One Alpha to all are just the coefficients of expansion are up enough down if you like the question is how much freedom is their Indiana offers such that it doesn't change the physical nature of the state how much freedom does every combination on How further up enough advance correspond to reveal a different state theaters now 1st of all Lee total probability has stayed up toward so that's 1 constraint right Alsop star plus ourselves down star often down is equal to 1 is another sign that we have not done so yet but we will find that every physically observable quantities always associated with the state always involves Alpha times a complex conjugate will find that will find it we won't put in it will emerge as 1 of the consequences of the past Europe's but what it says is that if you multiply both of the Alfred C. by the phase factor that in order phase factors phase factors a complex number of the form he for me I favor all the unit length doesn't change the magnitude of either coefficient that if we multiply every entry by the same face factor it doesn't affect anything that means that I would just give Udo complex numbers here there ought to be done pieces of information that really don't have anything to do with the physical character of the state the 1st as the overall magnitude because overall magnitude should always be such that the sum of the squares of these at up to 1 so that these is 1 call it a constraint of your life and 80 redundancy redundancy is that if you multiply both injuries by the same phase factor here it also doesn't change state so that you could ask comedy the independent variables all right there which characterizes state of the system of the system many of them are well let's count there are 2 independent variables if we don't worry about either of these constraints
1:35:58
here many things as take to characterize tool complex numbers for real on their side but now 1 of them doesn't count because we have to satisfy the constraints that of total length of this is equal to 1 1 of them doesn't count but also don't 1 doesn't count because he overall phase of we would multiply everything by the same place factor here it also doesn't count doesn't change the nature the state so tool of those parameters are irrelevant to the physical character of the state have I leave right tool real numbers come they numbers as it take that characterized the direction of a unit vector 2 were recalled longitudinal attitude or something like that just a directional vector could be characterized by 2 angles these tools pairs of numbers are the same the fact but it takes told real parameters the describe twocomponent vector is closely really related to the fact that it takes tool real angles describe the direction of 83 vector in ordinary threedimensional space everywhere they had far more components here it would have been more complicated yes there is a deep connection between the fact very deep connections between the fact that we only find 3 independent combinations like 3 independent combinations in the fat skinny never I can't find that skinny basically they won't be there now without increasing the dimensionality of vector space that is in fact very deeply connected to 3 dimensions of space on space was 9 dimensional or something like that we would not be able to represent the directions of the euro will have in quantum mechanically by 2 components sectors we would need more component it's like just physically it would take 3 chooses but the direction and whether boss 1 of the pleasure None of us were minus 1 inherited the base yes yeah but that has failed bad we can't of the day Majorga studs the dimensional I'll go have figure it out that requesters harmony directions parameters are there to him more complicated matter get what mutter try crude I did at I get mad at players really scared when I realized the Union Square the tool but a at a 3rd shift was left is called to Peru's yes although much safer this would version of it it would be a version of it it would be more complicated and you couldn't described the spin of a particle by just a finger with 2 components like this it would take more components the theory that is similar similar enclosed tried for this is enough from our understanding that if we went the fourdimensional space we would need for component objects but grows of reform said states had a buffer he wasn't end to UN vectors you could get a nonunion actors you can but then the role for of the state of the system are divided divided by an overall factors until normally there they have the same significance as probability distributions which stance some of the 1 if could always go back and say if somehow somewhere along the line I got some answers for probabilities that add up to something greater than 1 Willowwood is divided through but total probability the fine normalized probability distribution thrown arm the intermediate steps of calculations there might well be the presence of vectors which are not normal for the end of the day the states of the system are normalized vectors normalized vectors and With no memory of what their overall mathematical terms that of questions Our gone through our staff will make it more precise the next time when we discuss observables and particularly discussed what the mathematical nature Of these sit met to both wanted these are for the moment we just specified themselves by experiment to hold your apparatus this way and you measure something called Sigma X that way you measure something cause why don't we use some symmetries namely good equivalent of different directions a space but to guide us into finding on expressions for the various physical vectors describing different orientations of the state but we haven't been very rigorous about it and you know I have made no pretense Of really a mathematical clear derivation of these things 1st seat of the pants guessing what logic and the next time will make some rigorous postulates about what these things really mean and their relationship these things and these things and what the relationship between them the vectors and the things that you measure is the states but these are the things you measure about questions what's the connection between them OK any more questions about the ahead Majamay are better the remember but there's only 1 way I know how think about choir mechanics or that they are and his sitting this the 3rd time that's your problem bullied me
1:43:42
answering through it to be a place that we finish yes yes yes I check that a factory that well mostly will be saying Look it doesn't matter right it doesn't matter on these OK so let's let's talk about the mathematics of ignoring something the young preference reports that's true but error Of those just about mathematics of ignoring things from 5 . 0 dimensional right a circle is onedimensional has 1 parameter an angle on it but supposing I tell you that there's no significance to where you are on the circle is no significance for every all points on circle or equivalent to every other point than you might say well for all of interesting questions the circle is just a point if you could ignore away warrants there is only 1 point on the circle you on the circle workers if you're on a circle run circles were point you can say the same thing about see another example what other examples but what sectors example if you have a or was a complex number a complex number is a point on the plane but for some reason angle away you are plane doesn't matter and the only thing that's interesting is distant but now supposing you have to complex numbers and all I tell you is that when you have 2 complex numbers thing which doesn't matter is the sum of their face some of the faces some of their angles then there is still 1 angle leftover which may be important difference of the angles so far I have to complex numbers and I tell you for reasons that I haven't specified yet but the overall phase which could just be thought of as some of the phase angles of the some of the phase angles for whatever reason plays no role in the physics of the situation there is still the difference of angles that's exactly what the cases some of the angles that big of complex numbers that go into these entries here is of no physical things we can't that out for real parameters for complex numbers the overall magnitude of it doesn't count because we divide that out that removes 1 of them and then the sum of the angles doesn't count for whatever reason there was another 1 that leaves you with Tova was to real numbers and that's in correspondence with the 2 real numbers takes Poqet directions space a lot of subtlety but nothing varied difficult to mathematics is really easy mathematics of quantum mechanics is easier than the mathematics of classical physics our 2 by 2 matrices Maryland simple mathematics is easier some familiar the hallway of and selling it to our policy areas be down Carlisle said left month there is an l funnel doesn't mean perpendicular threedimensional space physically distinguishable unambiguously distinguishable left and right are not physically distinguishable uniquely from up and down because of your have nothing Oriental longer left and you measure Upward Bound you might get up those to to a last week Senator Kerry yes that's right so here you can vary independently the probability fop and now of course a halfday enough to warrant right a halfday up to 1 but ignoring the question of whether they had up to warn you have the freedom to independently varied coefficient in terms of you and the coefficient in terms of D the fact that they have to to 1 Missouri another factor young don't confuse orthogonality In the space of states with orthogonality in turn in in threedimensional space the analog idea in phase space or in the configuration space of classical system orthogonality of 2 probability distributions of Think of suppose you have space or the configuration space of some classical his based sex and I give you a probability distribution a simple probability distribution might be Zero everywhere except in some patch where the probabilities uniform OK we can no longer Our orthogonality between us 2 distinct probability distributions is that they don't overlap with another overlap Our shippers or ambiguity and that's COR saga malady hot and then owned don't overlap if you're up not bound to Banyan about now if you think about there is a situation in which orthogonality is staying in space and in the space of state the polarization of photon polarization of photons rooms with different do you want to get into it because it's more complicated but who would the better if it was a different word for perpendicularity in space in North Fargo priorities in the space of states different concept it's very much what's going on here that means only into along right columns yes can
1:51:24
I do have an experiment which can distinguish between a spin pointing up and spin pointing down you could bring your apparatus over to it detected and if it up 0 . 1 per cent down 0 . minus 1 give a minus 1 can't you use that apparatus could tell you uniquely the difference between a spend pointing along the xaxis this pinpointing along the zaxis what would you do to tell the difference what would you measure no matter what you measure it's not going to uniquely tell you the difference if you measure Cigna Z and you get plus 1 at money thing about say racks it doesn't you could have gotten signals equals plus 1 B it was pointing along busy accents and you gotta with certainty 1 all or it was pointing along the xaxis would probably have grab pointing up is no way to tell is no way to tell by a single welldefined experiment whether the ban was pointing up or on what is measured Our with 1 expanded to restore calm will WorldCom off and their and from that you cannot tell uniquely whether it was no you can tell from the ship's exactly the same as measuring and if you get opt in your measuring it does not tell you where there was initially up was initially sideways with probability ahead after that no unique single experiment Ucon due to distinguish this from this there is an experiment that you could do to distinguish that's what makes up in their own orthogonal but down unless they overlap overlap in fact the probability that if you are left but you measure down when you measure Singh was 80 half a measure of the overlap between overlap perch means technical language Miles Sobral said that it had originally they say the measure saying that can against the quick we're trying to do was confirmed the proposition Spain is up or the or the excon Z component of spinners up or the x component confirmed that that's a single proposition capped to from a singleparty state is up or the this is a list up along or along but a single proposition if the 1st detection measures the spin off along the zaxis were finished Greg Anthony and since the setup was set up before here and we know that in that case we would get ready cricket a fleet of a state not a young young proposition young trip propositions technically corresponds to her projection operators projection operators on commuters it would be proper to summer summarizes everything about it but observers are summarized part of that point your pens or were order you will leave us this time with a of cases along the caxis excess visualization it that it had urged others it is of interest well now have a young you know it is it is a visualization still keep in mind tho this interesting fact that on average the spinners if you measure many of them many many many of them the average values do behaved like but components of the little vector in particular if you take a large number of and you make sure that they're all pointing upward and then you measures some more bleak component of them With your apparatus being tilted average that you'll get 1st the oblique component would just be the projection of the pride vector onto the oblique direction exactly as if you were measuring the component of sorrow the army and the upshot is if you have a large number of them in many cases you can visualize with classical physics after all for classical physics is classical physics for large extent be approximation of systems which consist of many many parts uh a rich or behave a simultaneously the same way the problem a coalition that social occasion I suppose there is about to be more specific than that gage invariant does have to do with these things factors what does have to do it them but um it does have to do with them but I would kick into that right now this is what they it is about the efficacy of Dallas the tips think the blast Slick Rick absorb part of bitches but it's not like it was really say what value for really say what his captors but they once we measure it we can measure again and the same thing again and they will be consistent but relief we put absurd state that was that would be the same exact preparation but can't hear you can always think that a system has a property just your job to find out what formal or please visit us at
1:59:01
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Magnetic moment
Aleph
home office
Cut (gems)
Combined cycle
Stock (firearms)
Electronic component
FACTS (newspaper)
Excited state
Angle of attack
Direct current
Cartridge (firearms)
Hour
Reciprocal lattice
Raft
Noise figure
Miner
Week
Last
1:25:55
Effects unit
Schubvektorsteuerung
Measurement
Order and disorder (physics)
Relative articulation
Basis (linear algebra)
Bird vocalization
Phase (matter)
Apparent magnitude
Rootstype supercharger
Turning
Roll forming
Ship class
Constraint (mathematics)
Star
Typesetting
Halpha
Combined cycle
Electronic component
FACTS (newspaper)
Excited state
Angle of attack
Signal (electrical engineering)
Deck (ship)
Moving walkway
Measurement
Miner
1:35:58
Weapon
Matrix (printing)
Patch antenna
Roman calendar
Analog signal
Longitudinal wave
AMHerculisStern
Particle
Phase (matter)
Turning
Phase angle
Constraint (mathematics)
Tool
Magnetic moment
Mechanic
Combined cycle
DVD player
Photon
FACTS (newspaper)
Excited state
Direct current
Pufferlager
Perturbation theory
Cartridge (firearms)
Mintmade errors
Plane (tool)
Measurement
Gas turbine
Level staff
Noise figure
Transmission line
Day
Schubvektorsteuerung
Measurement
Fire apparatus
Mitsubishi A6M Zero
Apparent magnitude
Lecture/Conference
Steckverbinder
Month
Grey
Redshift
Digital electronics
Sigma
Electronic component
Car seat
Map
Classical mechanics
Finger protocol
Angle of attack
Spin (physics)
Photonics
Week
Last
1:51:19
Sensor
Schubvektorsteuerung
Electronic component
Measurement
Absorbance
Summer (George Winston album)
Order and disorder (physics)
Single (music)
Fire apparatus
FACTS (newspaper)
Pen
Finger protocol
Excited state
Ship
Rack railway
Spin (physics)
Direct current
Lecture/Conference
1Wire
Cartridge (firearms)
Signal (electrical engineering)
Spare part
Rail transport operations