 Alright. Good morning everybody. It's good to be back here knowing we're here in the Department of Physics, and we're trying to remember all the clocks are always some random numbers. So I'm just going to keep thinking that physics is perfect. We'd have clocks that are more different. Anyway, if you don't know me, I'm Ivan Doich and our TA for the course is here, Gopi. We're on the leader line, that's what I can do. But we just call it Gopi. And Gopi is going to be video-taking the course. The course will be, these lectures will be available online for you to review. And let me begin by just telling you how the information about the course is going to be distributed and how to access it and a little bit about the requirements, etc. So this is going to be a paperless course. So all the materials will be available to you through the course website. And you can reach it, you can just Google us or go to the department's classes schedule and it's linked there or you can just write it down on the URL. But you can find it and if you can't find it, just email me. So let's see, there's a few things I'd like to discuss in terms of the organization of the course to get started. Firstly, it's crucial that you've had, in order to take this course, the equivalent of the course here at UNM 491 and 492. That's our undergraduate level, senior level upon mechanics. And one way to get a feel for if you think, if you're not exactly sure if you're prepared for this course or not, I prepared here at the very bottom where we have our problem set for the course, a little diagnostic homework that you can see. I mean, yeah, it's okay, this is not going to be graded or anything, it's going to be you to get a sense of, you look at this and say, oh boy, this, I don't know whether this stuff is or the bad majority of this doesn't, you might think this course isn't for you. But on the other hand, I do know the nature of the way this education goes is we do subject matter and then we kind of do it again. So there's going to be a lot of overlapping some ways between what's happening, especially in this first semester and what you saw in your senior level upon mechanics classes. So we'll be sort of reintroducing all those concepts again, but at a fairly rapid rate and with more sophistication added on. The second semester of this course is really much more advanced and probably covered things that you haven't seen at all in your undergraduate courses. So let's see a few administrative details. We have, associated with the course, we have a problem session which is currently scheduled and an ungodly hour. You know, we're here at Fash and who gets back for this course at 9.30 and then we're going to have our problem session at 7 p.m. and we're not going to do that. So I will send around a little doodle poll to try to find another time that we can all have this problem session. It's going to be a challenge with about 15 people. We'll see if we can find a place and then we have to find a room. Because there's always Starbucks, but anyway, we'll figure it out. We won't, we'll try to find the right time for doing that. I think that actually the best date to do that will probably be Friday. So the way the course is going to be organized is that homework will be distributed on the web, on the course website on Tuesdays and then it will be due the following Tuesday in class. So my guess is it would be best to have the problem session on Friday so that we have the weekend to work on it and then I'll have an office hour on Monday where, you know, those last minute panic questions can be answered and then homework will be due Tuesday. That sounds reasonable. So we'll shoot for that. What do your Friday mornings look like right after the chair seminar? Anybody have anything at 10 o'clock on Friday? Oh, that would be terrific. Okay, so that's our tentative time. 10 o'clock Friday morning. And I'll see what we can do in finding a room. Maybe there's nothing in room 5 after the chair seminar, which would be just perfect. And we'll also try to find a time available for office hours that I hope we will hold an office hour as I will as well. All right? So in terms of the grading, there will be homework pretty much every week and it'll be about three or four questions. There will be two exams during the semester, sort of a little bit before fall break and a little bit before Thanksgiving. Not yet. It's not yet determined. Those will not be in class exams. We'll try to schedule a time in the early evening to take the class so that we're at 100 less time pressure. And then there is a final exam. I haven't looked at the date and time of that exam. Those are already scheduled by the university. So final exams are tied to the hour at which the course is taught. So the course is taught Tuesday at Thursday at 9.30. Then there is a slot for the final exam. So we'll look whenever we'll get to that. Let's see. There is no required textbook for the course because I'm a pretty idiosyncratic guy, as you'll get to know through the year and I like to do things my own way. So what that means is that I will be distributing lecture notes for the course. I don't promise that they will be available before the semester, I mean before the lecture, because one of my idiosyncracies is that I think taking notes is a good idea. I mean everybody should do what they like, but I find the process of trying to write something down helps me to learn the material. So just kind of zoning out, looking at the notes, it's not always registered so well. I won't always have to, but sometimes it's up to you. So you can download them, they're all in PDF format, and they look like this. And after, once we process the video, they'll be posted here, the lecture. We'll be linked here as a podcast, so to speak. But obviously it's good to have books. Books are your reference, and it's good to see different perspectives. I mean books are expensive though, so it's not sort of up to you to something to read what assessments you want to and are willing to make. I've written down three books that sort of capture different pieces of what I'm going to be covering in this first semester, which is why I didn't pick any one of them. I'd say, you know, if you want to have one book, a really great book is the book by Sakurai, but it's really kind of more for the latter part of the course, and then for 522. It doesn't cover as much of the intro material as I will. The book by Kon Tanuji and all of you want to call it Kon Tanuji, because that one is never as poor as you. I speak French, I don't even know how to pronounce that name. But anyway, it's, you know, I was a postdoc in France, and Kon Tanuji taught at something about the College de France, which is college for other professors, basically. It's something that's been set up by Napoleon for the bourgeoisie to be continually educated and it can come in here about whatever the intellectuals were talking about. So people, I guess, will go in and talk to the social scientists, but the only people who go to the physics lectures or other physicists who go to this and you would be saying you go to church and you'd hear them listen to the Pope. That was Kon Tanuji, and in the audience were all the other big professors and you know, the bishops. Anyway, Kon Tanuji is known as the Bible, and that was the point of that story. And it's organized in a very particular way. Some people hate it, some people love it. It has these, sort of, there's the main part and then there's the compliments and it's hard to organize, but it's got everything in it, and it's a good thing to have. There's a lot of work-out problems. It's a two-volume set. I think the first volume is kind of better. So if you could afford it, get these two books. The book by Merzbacher is an old book that was, I don't know, we reached on about 15 years ago. It's kind of got everything in it, both the elementary and advanced topics. So it kind of combines these two. But I feel like the way the book was written is that it took each chapter, each chapter was stapled together, and then he threw it up in the air and then at the end of the site, the pounded it some random order because... So, as you can see, I like to do things my own way. But in any event, there is many references and I certainly recommend having some other textbooks on hand as references to get different perspectives, not just mine. All right? So the tentative syllabus for the course is listed here. We'll see what we do. I expect we'll get through all of this material. And my plan for the set of lectures is already sketched out to be modified as things go. All right? So that's what's on the agenda. Anybody have any questions, concerns? So we don't have a lecture the Tuesday before Thanksgiving? We won't. So there are some days that I will be traveling. I know that's one of them. Occasionally, there may be something else that comes up, so I do have some responsibilities from time to time and I try to avoid accepting invitations that overlap with our lectures, but sometimes I will be traveling. And if we will, we'll only have lecturers try to make it up at some other hour. Okay? All right, great. So, let's get started. This discusses, you know, quantum mechanics is a pretty big subject. What is quantum mechanics about? You know, the subject has really expanded in the last century. You know, we just passed the 100-year anniversary of the Bohr atom. Bohr atom was. Bohr model of the atom was produced in 1913. So here we are in 2014. So, originally, and I'd say, what the majority of quantum mechanics were quantum mechanics guys' birth and what many physicists of my generation and before always thought about what quantum mechanics was about was about talking about the structure of the matter. So that might include the birth of it was atomic structure, atomic physics and quantum mechanics were one and the same. And quickly, of course, you get a couple of atoms and then you talk about molecules and from there to a much broader subject of condensed matter. So talking about all of that structure of all of matter, common matter if you like, is a major structure, a major topic of quantum mechanics and first in some courses this is all you study. Of course, we then get into the subatomic regime whether we talk about nuclear physics or even particles, subatomic particles, and of course the fields themselves. So one should expand this subject beyond matter at the study of quantum fields is, of course, an important topic within the study of quantum mechanics. Now, of course, the way I describe all of these things you can think about this really as the kinematics says that there is this structure but it doesn't describe anything about the dynamics. And quantum dynamics is a subject that often and traditionally has gotten short shrift and that's because traditionally it's very hard to control quantum dynamics. Natural systems in the world are typically occurring in such a way that what we'd say is these dynamics are incoherent in a way that will come to understand a little bit later on in the course. Typically, traditionally what one learns about quantum dynamics it's all kind of couched in terms of Fermi's golden rule and a ray equation for jumps, quantum jumps for absorption or emission. But there's a whole world beyond that, the world of coherent quantum dynamics and what that's about. And that's a subject we'll spend quite a bit of time discussing a little bit later. But quantum mechanics and why many of us have been fascinated by it is not just about what is the structure of matter and fields and how they evolve in time but there are some of course deep foundational issues. So foundations of physical theory are such a shorty. So there are some very deep questions that reveal themselves in thinking about quantum mechanics. We have fundamental questions. Is physical theory deterministic versus random? We have the question is whether things happen in the world independent of us, are they objective? Or are we in any way at the center of it? This is a discussion we continue to have. Continues to be at the heart of our understanding of what we do as quantum mechanics. Are things objectively real or in some way subjectively real? And then that is the whole term real. This notion of art, is there a notion of local realism where an object has a property unto itself that is objectively real? And if not, what the heck are we talking about? What is even the alternative to that? So of course this is what motivates many of us to continue to probe and examine the nature and these issues are not independent of these. The way in which we understand the structure of matter and fields is intimately related by the nature of the physical theory itself. And yet there is another stream that has become very prominent in the last, I don't know, 80 years. And that's technology. So, you know, ranging from things like material science, optical science, communications, computation and the general understanding of information itself. All of these are part of the description require a deep understanding of the quantum world. In fact, this latter subject, which is the subject of my own research interest tied into opticals and comic physics again draws very deeply on all of these issues in the foundations of quantum mechanics. Our many of us who worked in this field of motivation was to understand the foundations of quantum mechanics and the process where that maybe has had something to tell us about information theory itself and so the interaction between those has been fascinating subjects that we can use to be. So, we're going to touch on all of these things throughout the course of the academic year. We won't, this stuff may seem a little sexier but, you know, to really understand all of it it's important as physicists to have all of these foundations as well we best understand how the heck is a molecule? What's the Born Oppenheimer approximation? These are things that are forgotten because they're not sexy anymore. We all want to learn about Belze qualities but we better understand something if we're going to do astrophysics about the spectra of atoms and molecules. I want to understand the reverse. We should understand the structure of matters and fields so we'll be spending quite a bit of time on that as well. So, that's sort of the big picture. Let's get into it. Why is this a subject that is challenging? Well, obviously we have these crazy questions that are related and we're going to make sense of that so what's the key idea? Well, I would say the most important idea that we won't have to understand at the most basic level is the notion of the probability half of the two. How are we going to motivate that? Well, let's just start thinking about the ideas of some glamour human. We're going to talk about events and processes. What do I mean by that? Well, what I mean by an event is and this is pretty hand-wavy I'll have to try to make this more precise in a little bit is something that registers a lot of semantics there. Think about what the heck that means to find that or it's a tough thing to do it's part of the whole problem why I'm not doing it yet but we could just you know I forgot where there's this famous law case where someone said well you can't define pornography but you know it when you see it it's the same thing with an event it's hard to define it but when you see it you know that's what you need. So let's just say I have some kind of photo detector that goes click and I have some light-fuel photons impinging on this detector and this detector is in some kind of diver mode it goes click when a photon is detected this is a random event so now this event generally we can't predict if and or when the event will happen generally a random event now because we can't predict this event with certainty what that means is we have to appeal to the ideas of probability probability is the calculus we use in order to deal with incomplete certainty so probability is what we use when we have incomplete certainty of events I won't get into this too deeply but again the notion of probability is a tough one is probability a subjective notion or is probability an objective notion there's kind of two schools of thoughts about that there's what are called the basions and the frequentists don't have a hero so they don't have a name person but the subjective probability school does have a hero that's base and that question of objective versus objective probabilities is something we'll touch upon when we get more deeply into trying to understand what quantum mechanics means but for the moment if we have incomplete uncertainty we assign our probability to an event so for example suppose I have a 50-50 beam splitter so here is a what back in the old days it was called a Hatsilver mirror and it is such that if I send this photon to this beam splitter that if I have two detectors on these two sides of the beam splitter all this that there is a 50-50 chance of this going to click or this going to click that's what we mean by a 50-50 beam splitter so what is true is that if I send this single photon onto this beam splitter then either this detector goes click or that one goes click but they never both go click at the same time if I were to get these outputs onto a coincidence counter and look for coincidence counts for two clicks simultaneously within the window of the photon I would never see both go click either this or that and each one is a 50-50 chance of that happening if this is a 50-50 beam splitter so this is a random a set of events so this kind of looks like flipping a coin a fair coin you flip a coin and you have a 50-50 chance of it coming up as it counts there's almost nothing about this that's really the same as a coin flip why do we say that the coin flip is a fair coin I mean this is classical physics if I flip a coin I have Newton's laws Ethicals MA works pretty darn well for a coin why do I not know whether it's going to come up heads or tails initial conditions you don't know those but there's missing information about the coin the reason for a coin or a coin flip the outcome is random because and this is an important point information what are some of the things we need in order to specify that trajectory of that coin we need to know initial conditions we need to know where all the forces and torques are we need to know the air pressure wind we need to know the forces exerted the surface in which it lands in fact if we're good enough I'm not anywhere near as good enough but I've seen someone who can always flip a coin to land heads you have to practice a heck of a lot but if you torque it just the right way with just the right force it's going to make exactly the right amount of flips and you can kind of see you can cheat and catch it it's not a place so it's going to land heads but it's only random because we don't know anything if we knew all of these things plus Newton's laws we could, there's nothing random about the coin at all that's not true about this there's something fundamentally different here in some sense there's no information archive that if we just pure in and like this archive they would tell us which thing is going to happen and of course this is the kind of thing that gets at the heart of the questions of the nature of the quantum world is there some local hidden variables some hidden information that would tell us which of these things have, we just don't know it and that's why things are random to our understanding of things the answer is no it's impossible it's just actually inconsistent with experiment techniques that we can test that there is a local archive information local to the photon itself which tells it what to do in the same way as the coin and that's a very odd thing because there's no for the quantum and that fundamental difference means that the way in which events happen in the quantum world is just fundamentally different from the way they happen in the classical world so for example so this fact that there is some local information means that probability obeys different rules so the classic example of that is to think now of consider the following let's say, so here's my beam splitter again here's my photon incident on that beam splitter as we said it could be reflected from that beam splitter it could be transmitted from that beam splitter but suppose now we set things up that we take these two possible trajectories that the photon goes and we put in mirrors here and so now this mirror reflects these guys so now they're come back to this point and at this point I put another 50-50 beam splitter and now I put a detector here and another detector here and let's imagine that this was a balance, so this is a mock what's known as a mock center interferometer and let's imagine that it's a balance that's to say these two are going to start what's going to happen in that case it turns out in this situation what we find if we balance these two arms of the interferometer is that detector A always goes quick every single time the never receives a photon how in the world is that possible what this does is that the laws of quantum mechanics defy the laws of logic because the laws of logic the language of logic is probability so the classical probability I should say is there are two processes detector A so there's two possible things that you might imagine a lot of one possibility is that the photon reflected hit the mirror got reflected and then got to the second beam splitter and was transmitted thank you so this we'll call process one that's one possible process that could lead to a click on detector A alternatively it could be the case that in fact the photon was transmitted each one got reflected at this mirror and then was reflected we'll call this process two okay now logically what you would say is either process one or process two they're mutually orthogonal we're going to argue conjugation right and so if we want to say that either this or that happened then the probability A clicks if you have four how do you combine probabilities if you have two different alternatives you add them right if this or that happens and the probability the total probability is the sum of the probabilities right so that's the probability process one happened plus the probability process two because we have four now the the probability of process one happening well there are two things that have to happen this is equal to the probability of reflection of reflection at first beam splitter probability of transmission at the second beam splitter and how do you combine those probabilities multiply right so in this case there's a half and a half and the probability for process two happening is the same kind of thing is the probability of transmission at the first and then reflection at the second and so that's a half and a half quarter so classically logically we would say the probability that A goes quick is this or that which is a half wrong answer that's true that's logic and if that but in fact that's not the case because the mechanics doesn't operate under these two laws of logic it has its own calculus that brings us to our idea of the probability amplitude what the rules of quantum mechanics tell us quantum mechanics we have said is that if we have maximum mistake of a system like a photon we assign not a probability but a probability amplitude which I might generically call psi and psi is a complex number the magnitude of this which is less than equal to one and the phase is arbitrary if you're rusty on your complex number you better redeem it and the rules of quantum mechanics say that if there is an event probability amplitude to that event the probability of the event is then given by the square of the amplitude this is the so called the famous born rule and this is the whole story so the counter-intuitive behavior that we see in the interferometer follows what I'm going to call Feynman's rule of course Richard Feynman was one of the greats of the last century developing quantum field theory understanding of quantum physics as a whole and Feynman's rule let's see how I want to state to processes to the same event and the processes are indistinguishable I'll explain what that means in a moment then the total probability amplitude for the event of the amplitudes so what that means is that whereas classically if we have logical or we said we added probabilities quantum mechanically if we have logical or we don't add the probabilities we add the probability amplitudes so that if there is an amplitude a complex number associated with detector A clicking that total amplitude is the sum of the amplitudes for the two the logical or now involves the amplitudes rather than the probabilities which means that the probability for A to happen according to the Born rule well that's the square of the amplitude associated with that which is equal to which is equal to the number times its complex conjugate that's how you get the magnitude square of the complex number which is equal to psi 1 squared plus psi 2 squared plus psi 1 star times psi 2 plus psi 2 star times psi 1 this of course is the probability for process 1 and this is the probability for process 2 so if we contrast that to what's written on the backboard the probability for event A to happen is not just the sum of the probabilities of the two distinguished I mean the two alternatives but involves something else and this is quantum interference this of course this quantity itself probabilities are still probabilities probabilities are real numbers they are between 0 and 1 each one of these numbers is complex generally but this is the complex conjugate of this and a number plus complex conjugate equal to twice the real part of that number so this is a real number but that real number can be positive or it can be negative so that here then is for the balanced for the balanced Mach-Zenger interferometer for event click A what one finds is that this number is negative it's negative a half so that whereas P1 plus P2 here according to what's written on the backboard is a half this is equal to minus a half so that the total probability for A to go click oh it's the other way around sorry this is this is positive this is 1 Rp whereas for event click at B this thing is negative the probability for B is 0 this is what we would call constructive interference and this is destructive so that's in some sense all the strangeness of quantum mechanics really is here it's the fact and the idea that if you have different processes that lead to the same final outcome you can't think about one or the other as we do classically, logically we have to think there's a potentiality there's a possibility that this or that happened and to each one of those potentialities we assign an amplitude and then to get logically the total probability for the outcome, the logical or we add the amplitudes and then square it to get the total probability and the result of doing that is that we will get these cross terms we will get these interference terms the phylogic now there's a point that I one term that I threw out here which is indistinguishable indistinguishability is important here so processes are said to be indistinguishable if there is no information anywhere which tells us which process happened so if in principle there is something that tells which of these processes happened then the processes are no longer indistinguishable and in that case they no longer interfere so if the processes are distinguishable as opposed to indistinguishable well then those two alternatives no longer interfere with one another so for example let's draw our moxendron interferometer here it is mirror this path reflected if it was balanced what we just described was a situation that all the photons would come out here and they'd never come out here but suppose I modified this a little bit and I place in the path of one of these arms of the interferometer a pockel cell the pockel cell is a crystal which when life passes through it develops a voltage across it so I can put a little volt meter across the pockel cell it's a transparent crystal the light passes through it but when it does it there is an interaction between the light and the crystal such that a voltage develops across it well in that case if voltage develops across this crystal we know that the photon went along this path so what do you think happens in that case 50-50 in that case because this now is distinguishable paths this goes 50% of the time this goes click and 50% of the time this goes click because then if I don't have two paths, if I don't see voltage here then I know I have process 2 and when I have process 2 there's only one possibility it went along this path and then when it goes along that path it splits 50-50 if I do see voltage here it does one happen and if process 1 happened it went like this and then when it gets there there's no two indistinguishable alternatives which can either constructively or destructively interfere with one another now of course the notion of interference has its birth in classical physics we're familiar with the notion of interference of classical waves so if I look at this interferometer for example I can think about this as a interferometer that interferes classical electromagnetic waves so it doesn't have an electromagnetic wave suppose I have the electric field associated with the say a linearly polarized plane wave my electric field with some axis with a function of time say it's got some linear polarization e not cosine kx-minus omega t which we often write as the real part of either the i kx-minus omega t this we often call the complex amplitude of the electric field it's a complex number but the field itself which gives rise forces is the real part of that in the case of the classical electromagnetic theory we introduce these complex amplitudes for convenience the field is fundamentally real we do that because it's easier to deal with exponentials than science and cosines but we could do it always in terms of science and cosines nonetheless this is a convenience and it gives us the basis for where some of this stuff comes from of course we know the intensity of the field which there is the prime average the electric field is given here by most proportional to this it's got some epsilon-nons or something like that but basically that so this kind of looks has some flavor of the bond rule the intensity is the square of the amplitude and of course the flavor of a part of the Feynman rule which is that if I have I can have linear superposition that is to say if I have two waves impinging at the same point the total intensity at that point is not the sum of the intensities of the two waves it is the sum of the amplitudes the sum of the fields so that if I have that situation my total is the sum or the amplitude is the sum of the two fields and then the total intensity is the total electric field amplitude squared which of course is the sum of the intensities and then an interference term coming from the interference of those waves so this is familiar in some sense but yet quite completely different because in the case of the classical field if I send in a wave at this beam splitter half of the intensity is reflected and half of the intensity is transmitted but if I have a single photon at here it's not like half of the photon is reflected and half of the photon is transmitted it's that there is this corpuscule this quantum this particle like thing which is either completely transmitted or completely reflected and if we were to try to look for both detectors clicking simultaneously we would never see that so this classical idea what would happen here if I send in my laser beam here where half the intensity is transmitted and half of the intensity I'm sorry is reflected and half of the intensity is transmitted well that's in some sense an average phenomenon because if I were to re-look at this what I would say is that I might think about this as some random stream of particles impinging on here which click sometimes go here on any event I either see a click here or click here never both but it's a 50-50 chance right? so what I would see if I were to do this experiment is what? suppose I have n photons impinging on this beam splitter and I want to know what is the probability of seeing x of a photons at this detector what would you say? well no any one photon is a half but the problem is assuming n of those is what? yeah that's true if I had in the limited and it goes to infinity but let's think about this for a moment flip a coin that's what's going on I have n flips of a coin there's some fluctuations there it's a binomial question remember that? from your probability theory there's a number of different ways I can choose n sub a photons out of n photons this probability times a half to the power n a half to the power that's a binomial distribution so but as in the average number of photons that I would see at this detector is what was suggested this half of them will be transmitted and half of them will be reflected but there will be fluctuations as monitor suggested about this because this is counting statistics right that's the fluctuations associated with the binomial issue so the fluctuations about that are uncertainty if you like will be equal to the square root of n now in the limit as n is very very large fluctuations are tiny compared to me which means that if we were to do a very very very very very very large number of photon counts what we would find is that the number we would detect on here would be n times the probability so one way of thinking about this is there's a relationship between probabilities and fractions frequencies of outputs and that's in some sense where the born rule came from originally although it's a little confusing because as you say that gives a sense that probabilities are objective in fact that's not really the right way to think about it but nonetheless I could say that the probability becomes concentrated on the frequency so that the frequency of a here is equal to the number a over the total number or I can think about this as the electric field coming out at port a over the total intensity so if I think about the fraction this is the fraction or frequency of the intensity that comes out that converges as n goes to infinity to the probability and so the born rule has its birth in thinking about interference of electrical waves but it's something deeper than that and something that we're going to grapple with over the course of this year and that's really the most important idea but we're going to use all of this to talk about all the ingredients I discussed the structure of matter technologies etc so we'll get started with this next time what we're going to do starting from the next few lectures is to review and make more sophisticated this amplitude calculus through the theory of Hilbert space linear algebra operators all of that stuff and we'll be doing kind of math for the next few lectures I'll be emailing everybody to try to determine the times for our office hours etc and I'll confirm that we will have our problem session on Friday but we'll start next week alright have a good rest of your day