 So, when teams come, we have a new problem set, and we'll have our problem session this week. The following week, we'll be on travel, so we'll try to kind of think about what we're going to do if our problems say they're up, and what these schedule will be there. But we'll take over or we'll have another call. All right, so last time we were really talking about the heart of what we, to this day, cannot completely wrap our brains around regarding quantum theory. There at some level, we wave our hands, and it has to do with the question of dynamics. Okay, and what we know in quantum theory is that there are two seemingly very different and distinct kinds of dynamics, which is to say ways in which the state of the system should be changed as a function of time. In one scenario, which is easy and rigorous to define, we say if we have a closed quantum system, as I say here, all the degrees of freedom, physical degrees of freedom are included in the system. Whatever it is, whatever the stuff is, particles, fields, what have you, it's all, that's all there is. Then the evolution is unitarian. And that followed from saying that if the state started in a pure state, it had to remain as a pure state. And moreover, it's going to be a little bit more clear about what is rigorous and quantitative and what can lead by a pure state later in this lecture. We loosely defined a pure state as a state where we have maximum possible knowledge about the system, but we'll be more precise about that in a moment in just a little bit. But because we have a closed system and we neither gain nor lose information that a pure state stays a pure state, and therefore, and since the total probability is concerned, unitary maps preserve the norm and the norm represents the probability and there it has to be unitary. Okay, so that is clear, I think. And properties of this kind of evolution is, first of all, it's deterministic. That is to say, the state of the system at a later time is completely determined by the initial condition because this map is something that comes from the physics. I mean, we haven't really talked about physics yet. We will, I promise you. But the physics determines what this is and we know this, we know what the state is at a later time. So that's promoter kind of dynamics from classical physics, right? Newton's laws and trajectories, planets, what have you. Okay? And then we said there's some other kind of dynamic in the sense of another way in which we, the state should be changed between one time and another and that is what we call measurement, whatever that is. And being somewhat vague because it's hard to pin down what that act, what it is. But what I say is somehow if we learn the outcome of some possible alternative realization, then, for example, if we have something we're calling a meter and it points after something happens to one of the eigenvalues of some transmission observable, then the state right after the measurement is given by this rule. Okay? And so in this case, this corresponds to a projected measurement and we have there the projection operator operating on the state and then renormalized. Okay? And this was, if there was generally a degenerate eigenvalue, then there's an underlying assumption that this meter doesn't distinguish between the other eigenvalues of the mutually commuting observables. Otherwise, this would be a projector on the substates associated with those other eigenvalues as well. Okay? And that happens with some probability that probability is determined by the Born rule. Okay? That's the Born rule. So this kind of scenario is distinguished from this one in that this is not a closed-wilding system. There's something we're calling the meter that is interacting with the system and then it points after some kind of something happens and it points along this direction. And after the measurement happens, we have this rule, which for the particular case of a non-degenerate eigenvalue means that the state afterwards is equal to the eigenvector corresponding to that eigenvalue that this meter points to. Okay? And again, that happens with this probability. And this kind of evolution is distinguished from this kind of evolution into very important ways. One is that it's random. Okay? As you say, I can't predict with certainty, generally, unless the probability is one for zero, which how the state is going to evolve or how I should evolve the state depends on which tense you want to use. The paths are pretty active. So it's random. It's stochastic. The other thing is that it's irreversible. That's to say this is not a reversible map. I can't from this know what the pre-measurement state is. Whereas here I can because this is an invertible map. So if I have the state at a later time, I can just invert it and find out what the state was at the initial time. So this is time reversible. And this is irreversible. Now, that might give some bit of a clue as to the question that really one would like to understand is if this, if there is this kind of distinctly different type of evolution from this kind of evolution, we would like to know when we, what physical situations do we apply this kind of evolution and what physical situation do we apply this? Well, this we know. If we have a completely closed system, then we apply this. But as we said last time, why don't I just put all of this in a big box? In which case everything is enclosed in my box. And isn't this then unitary? Because we just said if everything is included, then the evolution is unitary. If we include all the quantum mechanical degrees of freedom that constitute the meter as well. So now there's a little, there's this piece here that we haven't really discussed. And this came up in our discussion last lecture in the notion that the one way that the Copenhagen perspective is that the meter is classical. And that there's somehow some kind of emergent phenomenon in a macroscopic world that we call the classical world. And the meter lives there. And it's, we can't actually draw this box around it because the box doesn't live in the quantum, and because the meter doesn't live in the quantum world. Now that's perhaps to me not completely satisfied because you know the meter is made up of stuff, atoms, molecules, solids, what have you. So it seems not fair to somehow bring it outside, say it's not a quantum object. But maybe there's something effective in some way that it's not completely a crazy kind of distinction to make. And that is the following, this question of irreversibility. So typically when you learn about this, probably in some way, of course if you think about it, the measurement involves some kind of irreversible record. So the measuring device records the outcome. And this is done in some sense irreversibly. Now this notion of irreversibility corresponds, let me draw the six, records the outcome. The question of irreversibility is something that this, that we face in classical physics. Classically the microscopic laws of motion are reversible. They're time reversible and varying. We just said, we saw a set of differential equations when we do Newton's laws, where we're generally Hamilton's equations of motion. And if I have a flow along the trajectory, if I know what the pointed face face, I am at some time I can always reverse it and get back to the initial conditions. Yet we know in the macroscopic world there's all kinds of irreversible things that happen, right Amy? I don't see the chalk reassembling itself. So there's something that happens where for all intents, even though we might say that microscopically the dynamics of this pieces of chalk are following for all intents and purposes Newton's equations of motion. Effectively it doesn't. And the reason is because what's happening is that we have so many degrees of freedom. I mean what happened? Why doesn't that chalk reassemble itself? What happened? Because that's an extremely unlikely thing of happening whereas it falling apart and you throw it like that is relatively likely. Right, and why is it unlikely with the entropy? Well the entropy, but what's actually happened? I mean what happened is vibrations that when the chalk hit it fractured and it caused the floor to vibrate and those vibrations quickly dissipated into the floor. They're all there, but they don't reassemble because as Keith suggested it's just so unlikely for it to bounce in just the right way that it comes back to my hand that it just doesn't happen. The probability of that happening is just vanishing small. It's not that it couldn't happen. It's just that it's so rare and it's for all into the universe. It might be the age of the universe and by that time, or maybe it's not even the age of the universe in this case. Maybe it's some few millennia but that time we hope this physics building has adapted but I don't guarantee that. So irreversibility is something that is not microscopically true but can be effective. And that's true in the quantum world as well, as we will see. So in the quantum dynamics, although microscopically are unitary, macroscopically need not be. Now, so this question, there's really a third kind of evolution which I will call irreversible, predictable in the sense that there is a definite state that the system goes to after the evolution and this is what we would call open quantum system dynamics. So maybe there's my spin here but this is my system but it's open to the world which we might call the environment. So in that case, I can have the situation for example where a spin relaxes, dissipates it might decay, for example and the decay is happening because some dissipation happens. This kind of evolution is not unitary because we're losing information here. What Bohr would say is that the effect that the measurement happens when the device is sufficiently macroscopic that there is some kind of irreversible interaction that happens and the measurement is recorded and just like the chalk that's shattered for all intents and purposes it is irreversibly measured. And that's where the classical world is where you have that level of many, many, many degrees of freedom that for all intents and purposes we call it classical because it has that irreversibility. What if we make a microscopic meter? Well, that's a good question. Well, and people do try to do that. So it's the right question to ask where is that boundary? At what point does this become sufficiently macroscopic that we can call it a meter and when is it? And experiments have been done in this sense where we try to say measure a single atom with a single photon and there are properties that we can see in that kind of thing which are more akin to this where I put in the photon and that called that the closed system and situations where we're more akin to this. And this is a big however. This doesn't really solve the problem in the sense as we say that when I say there's a third kind of evolution it's not this in the sense that we can talk about dissipation we can talk about irreversibility and that would have to come out of taking a unitary evolution of the whole thing and then somehow losing some information to the environment sort of marginalizing our probability distribution over the stuff we don't know about but that's never going to give me a jump that's never going to give me another pure state. This went from one pure state to another in a completely random way. This goes from a pure state to something where I'm impure and that's what it makes. So dissipative dynamics isn't the same thing as measurement. So whereas the borer picture of this might tell us these are the kinds of interactions we want to call measurement. The interactions with the meter where the meter is sufficiently macroscopic that I can call it classical because it irreversibly reports the measurement outcome. It's insufficient to explain how you can't derive this from this. What did we say? Well, there's sort of in one sense one answer left and that is me. Me, me, me. The observer that in some sense and this came back to this question of what is collapsing? So what we discussed a little bit last lecture is this notion that we should think about the state as a state of knowledge for the state side and that the measurement rule is like Bayesian update. So you remember we discussed last lecture that if I have a probability distribution classically and I learned something new then I should update my probability based on what I learned. And in some sense this rule, the measurement rule is that so this perspective is what's called cubism today. It's a way of thinking about quantum mechanics, quantum Bayesianism. It's a take up on Picasso. And it's a perspective on quantum theory that kind of had its origins here at UN. It was something that came out of the group of Carl Paves and is really being developed and pushed forward by Chris Fuchs and Gruber Schach. That was Carl's student and Rudecker was his postdoc some years ago. Which basically says that quantum mechanics is really just a theory about the observer being able to make predictions. And that what is happening is what the state is not a physical object it's just about what we know and when you learn something new you update it. There's no mystery there. Physical interactions happen. The meter and the system interact they could do so irreversibly. But I don't update the state until I look at it. Now here's why where I think this still doesn't really answer the question. This is like I look at this a little bit like Guido's theorem and mathematics there are formal questions you can pose within the formal system that are just not answerable within the system itself. Here is the issue. Quantum mechanics is a meta theory. What do I mean by that? That when we have a set of rules that are self-consistent within the theory itself we have states and we have unitary evolution. That's all perfectly clear self-consistence and we don't have to ask questions is this this or not. It just is the rule within the formal structure of the mathematics itself. What is different about measurement is that the measurement is stochastic. The basic rules of quantum are deterministic. Measurement outcomes are random. Where does this randomness come from? That's the fundamental question that continues to baffle us. The perhaps failed experiment is going to do it a little bit different now. You found that mic. I'll borrow that one. So, here we go. Schoch is in one of my hands. Which hand is it? Your pocket. It's in this one. Can you see it everyone? Steven check it out. That was random. That's not surprising. It's random. The way we think about randomness is that always is that randomness arises traditionally or classically because we have incomplete information. If the information was there we just didn't have it. There was information that was hidden from you. Once you've learned it then you have a particular result. But in quantum theory we can have maximum possible knowledge. And yet. Possibly. And yet. Random. Of course it's hard to know whether that's true or not. It's been one of the great triumphs of the last few decades last maybe half century coming up on that path. It's very natural to think the reason that random results are seen is because we just don't know everything. And if we just had that rest of that information we wouldn't be able to predict what would happen. So you know when we put the coin it's only random because of exactly how he inflicted and what the air pressure was. But if you knew all that you could predict the tunnels and whatnot. So intuitively we think about randomness as coming from incomplete knowledge. But what we know now is that if Einstein of course famously believed that this had to be the case that the randomness that we're seeing in quantum mechanics is of this sort. There were hidden variants that were there that were telling the photon which split to go through. But they were just hidden from us. And if we had access to them we could predict which way the photon would go through the beam slit. That's natural. That's what you think. And what we have come to learn is that if there are hidden variables in quantum mechanics they can't be local to the particle themselves. They must in some way be distributed non-locally and things at one side of the universe affect the other. And that's a completely inconsistent perspective on physics. So we just give up hidden variables. We say this is why quantum... So let me come back to this one when I'm talking about meta because I haven't quite defined that yet. If meta and outcomes are random and the basic... physics is deterministic and the randomness is not coming from just missing information then we have to import something outside quantum mechanics to extract meta and outcomes. Because we can't get quantum mechanics to evolve into random outcomes. So we have this theory and then we have this layer on top of quantum theory that's there outside the quantum theory itself that doesn't obey the laws of quantum mechanics that somehow extracts these random things. In the Copenhagen interpretation the meta thing was the classical world and the classical device. And you wield in the classical device you don't tell me what devices are classical or not and where that border is if it's three photons, four photons, five tucky balls but you just say somewhere it's the classical world and it's outside the quantum world and it extracts the results. In the cubism perspective it's you the observer the person the human being your consciousness what you know about that perspective I'm not saying I adhere to this but the cubists say really frankly is that your experience is outside the description of science itself it just makes that statement it's an oboe impossible it's a particular logical reasoning and it basically says that I mean so it's me, me, me but maybe me, me, me over here you know should be inside the quantum system too so it's not just about me looking at the meter so here's an experiment that I want you to think about suppose I have a single photon and that single photon is put through a beam splitter and that beam splitter now sends one path to this eye and one path to this eye well well how should I think about that I could think about that as you know my eyes and my brain are physical stuff there's red and A and optic nerves and native stuff and there's a quantum correlation between the photon hitting there and some activity maybe in this bald head and the photon going here and some other activity and from the point of view of quantum theory well maybe there's some dissipation in there but there's no way that that's going to be one or the other quantum theory of all the nerves in my head aren't going to make a jump so just by making it a human being rather than a meter does that solve the problem it just moves the problem there that's why I'm saying the problem unless you cut the cord and say this is outside I mean in Cubis we answer the problem by just stating and in some sense it's not consistent because my experience is no way for you to actually test that because you don't know my experience you have your experience but you're inferring about mine so it is a logical perspective but it's somehow maybe not completely satisfying it puts your experience met whereas the Copenhagen interpretation puts the classical measuring device met I think that's completely unsatisfying because the classical measuring device is stuff maybe your experience is somehow different I don't know moreover if you want to take this Bayesian perspective completely for me here's a problem I mean when we learned that the shock was there so when we say corner state is a state of knowledge about what in this case it was knowledge of whether the shock was in my right hand or in my left hand and you learned that when you saw the answer but when you measure the photon you're not learning that it was in this path or that path because we just said that that's not really consistent with experiments you can do so it's still we're in this goinal circle we just keep pushing the problem out and all I'm saying is at the end of the day what we have to do is accept this we should continue to think hard about this problem you know every generation has made further progress in the spilling down the essence of this problem and doing the kinds of things that it was suggestive can we really push these boundaries make the meters more and more quantum and see what happens move them macroscopically understand is there a quantum classical boundary but at the moment I don't think we're there I hope one day before I die I will understand this but we're not really quite there yet so given all of that we're going to return I should say today's about local hidden variables and so forth at the end of the semester we're going to circle back when we have a little bit more of the formalism to address understanding why I can't assign local hidden variables to the objects why is that for now so let's just talk today and continue about an example of well what is an example of a measurement so let's talk about the most familiar and clear example of a quantum measurement so we are familiar with this but I would just remind you so the question is I want to measure the spin of for example silver atoms silver atoms have a spin like that because it comes so little bit like that so how is it done I want to measure that but what's done is so I have it up and there's some very familiar sodium vapor here and the sodium atoms that comes out of this and then I pass it through a region with a inhomogeneous magnetic field so the magnetic field we have this north and south pole and that has some gradient in the strength of the magnetic field it's closer together at this point here so the strength of the field is stronger up here than there and what one finds is that the beam separates into two groups and the way that this is interpreted is by the force that is given to the sodium atoms the kick either up or down that is given to the sodium atoms that depended upon whether they were spin up or spin down so what we know from classical physics is that the force on a neutral particle with a magnetic tonic well the energy is minus 2.0 and then the force is minus the gradient so the force is the gradient of u.v now let's call the direction of the gradient let's just call that the z direction the magnetic field strength is varying the function of z so this is equal to in this case mu z d by d z of the magnitude of z in this direction so fine this is the magnetic backbone what we do is we say in fact that the magnetic moment the magnetic backbone moment is proportional to the spin and in fact that proportionality constant we can find by measuring how much this deflection is and what we find is that it's equal to twice the board magneton times the spin or special half so this is the board magneton is e e e bar over 2 the mass of the electron c in my favorite Gaussian units 2 something special for electron it is for that's the so-called g factor and we'll review all that stuff again we should have seen what we're doing now of course what we say quantumly is in fact the spin observable is an operator so this is really a force operating it's not but quantumly it is we're treating the magnetic field here as a classical field that's the question exactly right or not in fact it's not effective okay so that means that and that spin is equal to h bar over 2 times the Pali matrix so this is equal to to you need I'm sorry this should be over h bar now it's got the right units and that's equal to h bar over 2 times the Pali matrices over h bar so what we have here is that the quantum and it's an electron so is that even minus equal to let's see what we got here we have minus uv the gradient in the magnitude b times sigma c and this is all in the direction this gradient is actually negative because it's pointing down so alright so that's the force and what we see is that if the spins here are spin one half then there are two possible trajectories the trajectory associated with the spin up atoms which go here, spin up along c and the trajectory along spin down the atoms which go along that and if I were to put a screen here that absorbs these silver atoms and makes a little black spot on the screen what I would see is I would see two spots I'd see a spot here and we would say that we've done a measurement and if it lands in here we would say it's spin up and if it lands in here we would say it's spin down so here this is an example of a measurement now is this a projective measurement now in some sense it is because it's saying that it's projective because the probability is to find spin up or spin down to find up or down probability is uniquely given by the ion detector it's not quite of the sort that we describe over there in terms of the post-measurement state we said after the measurement the state of the system is in that ion detector corresponding to the ion value is measured the reason is the sodium is gone it's been absorbed it's done a chemical reaction I can't do another measurement on the thing but it's not although the probabilities of being here or there can be calculated by the Born rule with a projector associated with that as to say the probability of being this so that probability of being that is the expectation value of the projector and this probability of that projector that's the Born rule the post-measurement state is not well it's no longer so it's something else I'm sorry so alright let's find that it's really hard to do a projector but what about this suppose I come over here and instead of putting a screen what I'm going to do is this blocker here is that a projected measurement kind of again it's not exactly of this I mean what is true these atoms are gone the silver atoms are being absorbed by this blocker and they've gone on but these guys are free and since I know which this being moves is consistent with this force then I can say that with certainty the atoms that went through that are in this path are spin and so the post-measurement state regardless of what he imagines of and I haven't even explained yet what that state is what it is I can say I projected the silver atoms into this state okay it's not quite a projected measurement where I kind of were able to say well I got this or that and after the measurement it's projected into that guy because I blocked this guy but still okay and how can I check that in fact this was a projection onto this I'll run it through the same thing again so I can run it through another Stern-Gerlach apparatus and what I'll find is in contrast to this what I'll find is I'll just get in that case just one beam I'll do a repeated and I'll get only one result because it was definitely in that state and atoms that are spin up along z go get deflected along that direction of course it very hard for me to draw but I hate trying to perspective suppose that I put the second Stern-Gerlach apparatus along the x-direction so that's the x-direction okay barely and what you're going to see is those two beams correspond to let's call this the x-direction spin up along x spin down along x this is what I mean to say I have a pure state that pure state is I had the maximum possible information I can have it's that state spin up along z but nonetheless, even though I have complete information about the system there are measurements I can do which don't have definite outcomes that's the essence of the quantum world that's what distinguishes it from the quantum world in the quantum world if you have complete information you can predict everything about it whereas in the quantum world you can't now we know also the probability where the the thing is we're going to end up here or there and that's given by the born rule right now let me ask you another question is this a measurement well sort of depends suppose I have some way of having it accounting in the mirror which exists and I can take these two beams and bounce them back with some steering magnets these two beams come back together ok, so the two beams separate, they balance and they get recombined and now I put them through another Stern-Gerlach apparatus along with this what's going to happen I'll have one beam again so they don't lose that so in it kind of depends how good I think this experiment these two possibilities remain in superposition they have coherence they can interfere if there is if they are in principle indistinguishable just like this you can think about the Stern-Gerlach apparatus in fact the Stern-Gerlach apparatus is formally equivalent to a polarizing beam splitter in optics so this is an analogy a polarizing beam splitter in a polarizing beam splitter in optics you go through one you go one direction or another depending on what your polarization is so this is like a beam splitter with the beam and if you recombine the beams and the two beams are coherent they remain coherent they will interfere so just because I separated it doesn't mean that I've done a measurement of course if I scatter a photon I can look at it and see a photon scatters and I see it then I've done a measurement because the photon distinguishes the two paths and if the two paths become distinguished they may no longer interfere with one another they lose their coherence coherence means the potential to see interference it goes back to this issue of the measurement having to have some aspect of irreversibility that you put it here it will split back up into two it will I mean then you'll see spin up and spin down along z absolutely or along x or along z as he said if I put it along z this is spin up along x spin up along x is a superposition of spin up along z and spin down along z and this beam is put in two but the fact that I have these two beams and I've combined them in such a way that I couldn't tell if there was no way to tell if I had to align the interferometer perfectly such that there's no way the interferometer can tell which path it came from then as we said in the very first lecture if I came up in no way distinguish the two alternatives they interfere I was just going to say that putting sg at the top is the same as breaking the blocker from the bottom one it is of course you could again if everything remains coherent then I could try to recombine all those three beams in some way and see the interference between the different alternatives is there some sort of phase shift with the that too and I could you know try to phase shift this such that I can think about this analogous to our remember we drew in the first lecture we drew the optical interferometer the mobs ender where we send something through a beam splitter found stored off and what we found was that if we aligned it perfectly it went this way right but I could phase shift it and then sometimes it would come out this way similarly there were two outcomes here and which outcome is depends on how these two paths interfere with one another so the general state that I see here is some superposition c plus and c minus between those two paths and what that is it started out in this case as 1 over root 2 and 1 over root 2 actually this is x but I could try to phase shift one of the other and get the beam to go the other way just like here let me remind myself where I was headed with this let me go back to this picture, I'm sorry I'm using the same word over again but I'm too lazy so here's my apparatus suppose that I wasn't a very good experimentalist and I didn't calculate this force well I didn't do my modeling properly and I didn't know where to put the screen so I decided I'm going to put the screen right here well before those two beams separated from what it is so what I'm going to see if I were to make a histogram of the counts, I wouldn't see two spots what I'd see is this spot that's two spots that's not they're not resolved right? I can think of it, I can try to fit and say well there's this Gaussian over here and this Gaussian over here but they're not separated from one another so I've measured the spin in some way but I can't distinguish perfectly the spin up spot from the spin down spot they overlap with one another I can't answer yes or no are you spin up or are you spin down when they were resolved I could say the ones in that spot were spin up the ones in that spot maybe the ones out here I can but in here there's some tails that overlap so I can ask questions but sometimes my answer is maybe so projected measurements are yes or no are you spin up or are you spin down and the way we see that mathematically is that the eigenvalues of a projection measurement or a projector are zero or one true or false but this kind of measurement does have maybe in and so in order to determine are you in this as an experimenter what would I do I would fit this data to two gaussians and if it lies in this gaussian I call it spin up and if it lies in this gaussian I call it spin down or I call it the plus result or the minus result but in fact it doesn't really can't say sometimes it will give me a maybe so there are these two alternatives corresponding to these two gaussians but they're not projectors they're something else and what they are are positive operators so this alternative this is alternative plus and this is alternative minus there are those two alternatives they result in the identity and I associate a positive operator corresponding to these two overlap gaussians to each of them and my probability of seeing up or this alternative is the expected value of this the probability of seeing minus is the expected value of this it's the generalization of the projective measurement this is an example of what's sometimes called a weak measurement because I haven't really projected the system onto one or the other I've let things a little bit uncertain and this is of course a general POPM it's kind of maybe you can suggest a weak measurement what do you mean by that? that this is a weak measurement but I've put in the screen a little farther absolutely it is exactly what it's called so a strong measurement is the same thing as a projective measurement so there's a complete continuum here at this point I've learned nothing the two peaks are not resolved at all and as I move the screen farther and farther away the measurement becomes stronger and stronger and stronger there's five peaks where you can't do that though does it always become a weak measurement? every measurement physical measurement happens over time in some cases here the measurement the physical process that allows to do the measurement was the force exerted on and that it takes time for that force to accumulate in order to separate the two alternatives so it's always the case always that if you don't give the leader enough time to resolve the different alternatives the measurement will be weak so this is an example that is not discussed have you seen this example before? no it's kind of obvious isn't it this is not a projective measurement and most measurements are very hard to do a projective measurement in quantum mechanics most measurements are not now one of the problems in work is to see an example of that have you seen the mathematics of that to discuss that now before I quit for today in this lecture what is the state of the spin coming out of the I don't know is it a superposition? good so now there are two good answers two good popular answers we don't know and it's a superposition so let's think about those two alternatives coming so one possibility is to say well we made a measurement along the z so we don't know which would say for example if we have complete no knowledge, it's completely random we would say it's 50% spin up along z and 50% spin down along z another possibility is there is a quantum superposition of up and down along z so that would say the state of the system was one over to spin up along z and with some phase spin down along z now good questions the same thing I guess certainly we couldn't distinguish those two things if we did this experiment right? if we did this experiment we're going to see 50% if we did a bigger measurement a stronger measurement we would see equal intensity of these two peaks and 50% of them would go up for example what if instead I measured this along the x direction? well if this phase were 0 let's say we did this this was the superposition this state and this state would show the same measurement results for this but suppose instead I measured this along the x direction well what would I find if the state were this plus that's right because this is equal to plus x I would see one beam whereas in this case I would say it's half of the time it's been up so the probability of seeing up along x is that and half of the time it's been down along z so the probability of seeing spin up along x is that so that's the 50-50 mixture and what's that probability? well this is a half this is a half so this overall is a half so this is the right answer this is the wrong answer because why would x be preserved after e per over z it wouldn't exactly so that's why this is the right answer it's always going to be no matter what direction if it's an oven completely random state then no matter what I'll see two beams 50-50 this state is not a pure state this state is what we would call a mixed state a mixed state as we will discuss next week is a situation where we're actually missing information we don't know how the spin was spin out of this subject and because of that we need to average over missing classical information quantum mechanics needs to allow us to include missing classical information as well so the most general states of the system are ones where I don't have complete information about the quantum system there's temperature here there's thermal randomness that's different from this intrinsic quantum randomness it comes out as completely incoherent mixture it's analogous to polarized light versus unpolarized light alright we will pick that up next time