 Let us begin. So, what we discussed in our morning lecture that the position of the wave packet is not completely localized. The wavelength of wave packet is also not unique and this is what leads to what we call as uncertainty principle. A localized wave packet can be formed as a result of superposition of infinite ideal waves varying in their wavelengths. Wave packet travels with the group speed which in the case of D Broglie wave is equal to the speed of the particle. This I have not shown it specifically, but it can be very easily shown that the group velocity of the particle waves actually turn out to be actually equal to the speed of the particle, which is important because if the wave has to travel, it has to travel with the velocity, it has to represent the particle, it has to travel with the particle. So, it must have same speed. Okay, now let us go a little bit more into the implication of the uncertainty because this we feel probably is one of the most exciting and mysterious part of the entire quantum mechanics which brings the idea of uncertainty probably little more clearly and probably one of the best book which deals with this particular thing is the Feynman's Lectures of Physics, Volume 3. So, this is what I said when wave nature and particle nature coexist, we have to allow certain tolerance in the way we look at nature and how that is what we are going to discuss today. Physical interpretation of wave, we shall discuss little later. What we when we say that, see when we talk of a normal wave electromagnetic wave, we know that what is varying is electric field or magnetic field. If you are talking of sound wave, now we know that these are particles which are traveling in the medium. Okay, when we are talking of wave associated with a particular with any particle, what is moving, that is something which we have not talked about it and then we will talk about it a little later because this is essentially fairly complex thing, but at the moment just look at the implication of if wave particle duality exists, if a particular thing shows both wave character and particle character, how certain things which are commonly asked questions in classical physics, okay, become unanswerable when we talk in terms of the wave particle duality. So, let us discuss about this particular aspect. So, we will discuss what is called famous Young's Double Slate Experiment which almost everyone knows in the high school that there is a when you have two slates and you have a monochromatic source of light. In fact, it can also be done with the water waves, light is not really necessary and the beam gets split, okay, you have two slates and then eventually they interfere and what you see and the your screen is an interference pattern. I make certain statements, let us be clear that we agree with these statements. The experiments can be explained on the basis of wave nature of light. I think we will all agree that you know this particular thing interference can be very easily explained in terms of the two waves interfere with each other. It is rather difficult to explain in terms of the particle nature, wave nature is very very clearly explainable. Now, I make this particular statement, see at a point in the interference pattern where we see a large intensity, okay, in the wave nature I would say that there has been a constructive interference, alright. But if I want to invoke particle wave picture because we know that light also behaves like particles, I would say that a point where there is a maximum intensity, a larger number of photons arrive there and in a given time, do you agree? Place where I see a dark fringe, okay, at that time hardly any photon arrives there. So this is statement which I am making, let us suppose, let us see whether we agree with this statement. In photon picture a bright fringe would mean larger number of photons reaching that point and a dark fringe would mean no photon reaching that particular point. See, remember I am trying to mix the two ideas. So long I was talking about the wave nature, things were reasonably clean. When I start want to mix trying to talk of the two things together, let us see how this particular statement that I am making will land me into a problem, alright. But let us generally I mean everyone agree, if we say there is a more intense light, then more number of photons, even photoelectricity effect, that is what a more intense light means more number of photons. So there are there is a source of light here, there are two slits S1 and S2, there is one blue slit, another red slit, okay. The wave comes and there is what you see is basically on the interference pattern. This particular part at the moment, please ignore it, okay. We will talk about this particular part little later, okay. So this is what we are seeing in interference pattern. For this is not always uniform, there is also decays, all those things please ignore it. As I said it is a very very rough way of drawing, just I want to bring to the point that there is interference pattern. Now, we just now said in the morning that such type of experiment can also be done with electrons, okay. In fact, I showed even a electron diffraction pattern. Of course, the length scales would be somewhat different because when I am talking of electrons, the general wavelengths that I will be talking has to be much more smaller. In comparison to for example, photons I could have right from you know, 10,000 angstrom to let us say 100 angstrom or 1 angstrom, we have varieties of photons, okay. While electrons you know one has to have certain limitations, but let us at least at the moment talk in terms of the thought experiment that let us assume such length scales are possible and I am able to do a similar experiment with electrons. So instead of light beam, okay, I have now electron beam, okay. And let us assume that it is a monochromatic beam like a light beam, which is actually creating a monochromatic beam of electron, it is not all that problem. In electron diffraction experiments, you know, you have transmission electron microscope, scanning electron microscope, you always have a reasonably monochromatic beam of electrons, that is not a big issue, okay. So let us agree that this experiment can be performed even using electrons. So let me just read, let us assume that we performed the experiment with electrons by choosing appropriate length scales so that we can observe such an interference pattern. We know for sure that electrons can be diffracted like light waves. Now we have no doubt about this particular thing that electrons cannot be diffracted. I mean every day people do experiments to do electron diffraction, okay. Now I make a similar statement, when I see a bright fringe, okay, in electron diffraction experiment like photons, I mean exactly the same thing. Similar to photons, a bright fringe would mean a larger number of electrons would reach and when there is a dark fringe, it means hardly any electrons reach there. That is what it would mean, just taking parallel with the photons, okay. Exactly when I see bright interference pattern, bright fringe, a large number of electrons have reached at that particular point, okay, where I see a dark fringe, hardly any electron has reached there. Now let me ask a question. These are questions appear to be somewhat funny, but these are the questions which probably will come in every student's mind. How do the electrons reach the screen? See, remember this particular thing, okay. Here we are talking of photons, but you now assume that this source is not really a photon, but of electron source, okay. You are seeing lot of electrons here, okay. Here you are seeing very large number of electrons coming. Here you are seeing very few electrons coming. Here you are seeing very large number of electrons coming, like that, okay. My first question which looks rather funny question, that how the electrons reach the screen? Obviously, electrons were emitted from the source. Probably there was a wall there in which these two slits were created, okay. Very large number of electrons got absorbed in that particular wall and they could not reach transfer to the other side. Let me go back to my, see electron comes from here, hits this particular wall here, it is not able to go through. Electron comes, hits here, it is not able to go through. So, if the electron has to go to the right hand side, it must have crossed either this particular slit or it must have crossed this particular slit. So, electron must have gone either through slit 1 or slit 2 to reach to the other side. Do we all agree on this particular statement? So, this is what I said, they are emitted from the source. Some of them move towards wall or for some of them could be moving backwards, okay. If they hit at a point where there is no slit, they get absorbed. Some electrons happen to arrive at a point where there is a slit and they cross over to the other side and then reach the screen. Are we sure? Let us do the experiment. Let us close both the slits. If you close both the slits, okay, no electron would reach on the screen. So, we are sure that this statement is correct. Do we all agree? Close both the slits and we would not see any electrons on the screen. Hence, the presence of slit is necessary for the electrons to reach on the screen. Do we all agree on this statement? Okay. Very simple thing. Probably very obvious thing. But now I am asking some other questions which is going to land us into trouble. So, let us look at this particular questions. Do electrons reaching screen pass through one of the two slits? Obviously, you are going to say that, you know, you are, when in Hindi there is a statement that, you know, you are taking skin out of the hair. I mean, what is this? This is obvious. This is not this obvious because there are two holes. Okay, electron obviously has to pass from one of the two slits. So, this is probably looking a silicoid question. That is what I have written. Since electron is a particle which cannot be split, so isn't it obvious that it should pass through one of the slits? How silly is the question? Okay, it is almost asking a question that all those who are in this room, they have come through this door or this door? Okay, let us assume there are only two doors. Okay, obviously you have come either through this door or that door. You know, you could not suddenly appear here or you could not have passed simultaneously through both the doors and come here. Okay, there is always, you know, either through door one or door two. It is obvious. Had it been equally silly if we are talking about interference of two classical waves? If we are talking about two classical waves, let us talk about water waves. Okay, if they are coming through the state, then probably this question was not all that silly. Okay, the wave, you can always say that there is a wave which is simultaneously passing through that and there is a new wave front created and all those things and we know how to explain through the wave. Okay, if I would have asked this question, if I would have seen this interference pattern using a classical wave, probably this question would have appeared nonsense. You know, this was probably, this question does not mean it because then we expect the waves are coming simultaneously, they are getting passed through the fuselage and they are interfering. Problem is our concept of electron as a localized particle. Let us see how. Well, it is not so silly as we have seen. As we have seen, if this statement is true, we can classify the electrons reaching on the screen into two types of electrons. Number one, those which have passed through slit one, number two, which have passed through the slit two. Okay, so like we could classify people in this particular room, okay, you have entered from this particular door, that particular person has entered from this door. So, each particular person which is sitting in this room, we can be classified that whether that person has entered from door A or door B or door one or door two, it is as simple as that. Okay, so it should be possible for us to do this particular type of thing, a type of naming the electrons, okay, that once electron has reached on the other side, it must have come through slit one or through slit two. Okay, then what, how to ensure that? Okay, let us suppose somebody objects, no, no, no, no, the person who is sitting in this particular room, okay, could not have come through this particular door. Okay, there is some other way that particular person could have appeared in this room. Okay, what we will do? Okay, we will ask them to repeat the question, you go all out. All right, then close one of the door and say you come by exactly by the same way. Okay, close one of the doors, so those persons who are coming by one door, they will get stopped because that door is no longer open, so only half the person, probably half or whatever it is, person will remain in the room which have come from the other door. That is what would happen. All right, let us try to do the same experiment with the electrons. Let me read through it because I don't want to miss any point, that is why I am reading through this. Well, it is not so silly as we shall soon see. If this statement is true, we can classify the electrons reaching the screen into two parts, one, those who came through slit one, number two, those who came through slit two. Can we experimentally verify this statement? This is what I want to verify. Now, perform the experiment. For that we perform three different experiments. The first experiment is that we keep only slit one open. Like the one I said, keep only one door open, close the other door, okay. Choose a point P on the screen which I have chosen earlier. Count the number of the electrons reaching there in a given time, all right. Keep on moving the position of your P, then you can plot how many number of electrons are reaching at a particular point, all right. So, this is what I said. Keep on changing the position of point P and find N1. This is what I am calling as N1 as a function of the position for the same time. Let us suppose we choose, let us say, 10 seconds, we will count the total number of particles that are coming in 10 seconds, okay. Find out a different position of pi particles, see where my, let us go back. So, this is my particle P, okay. Initially I counted here, then I moved this particular point at different location and then I start counting total number of electrons which are reaching that particular point in, let us say, 10 seconds. Whatever time it is, all right. Then do the same experiment, now close the slit one and do the experiment in slit two. Repeat the same experiment, only slit S2 open, find N2 as a function of different position. And of course, that experiment is the same experiment when both the slits are open. Now let us look at this particular part of the screen. You know when there is only one slit open, there is no interference pattern. Electrons cannot interfere, okay. What at the best you will see is sort of small amount of diffraction, okay. If a wave comes only through one single slit, there will be small amount of diffraction. So, your N1 as a function of X is likely to look something like this, okay. Of course, the width of this particular thing will be larger, smaller this width. It is standard diffraction experiment, you know, there is nothing very important. If I perform N2, this N2 will also be exactly similar, slightly displaced with respect to S1 in the position, because this will be, this maximum will be closer to where slit S2 is there, okay. So, slightly displaced, otherwise there will be similar position. But when both S1 and S2 are open, then what you start seeing something different here. Now, it is very funny, because look at a particular point somewhere here, for example, or let us say somewhere here. Here, when only slit 1 was open, you are getting very large number of particles here, electrons. Now, when slit 2 has started, we have also opened, suddenly the number of electrons reaching at that particular point has gone to 0. We just now agreed that electrons either have to go through slit S1 or through go to slit S2. So, did the electrons which are coming from S1 suddenly realize now slit S2 is open? So, let us not go to that particular point, but go somewhere else, alright. Doesn't it look strange for us to believe that electron at a point, when the electron were reaching largely in number. Initially, when there was only one slit open, I have now suddenly opened slit 2, and suddenly at that particular point you find very few electrons reaching there. So, if this statement is correct, our earlier statement, the electrons are going through this slit or that particular slit, okay. Opening this slit should not affect those electrons which are coming from slit S1. At least those electrons should have reached that particular point, which suddenly they have decided not to reach, because I do not know what reason. The answer is no. When one slit opens, one sees a diffraction pattern. There is no interference pattern due to the other slit. There are places on the screen which receive much less number of electrons when both the slits are open, then when only one slit is open. Can electrostatic interaction change the path? You know, probably one can think, okay, maybe there is electrostatic interaction. After all, when there was only one particular pattern, there is another beam which was coming and now probably there is some influence, of charge influence, and that is what is likely to affect. But the thing is that this experiment can be performed when electrons are coming so slowly that there is absolutely no chance for them to have an electrostatic interaction with any other electron. You will still see the interference pattern. And that is the video which I want to show you, which is actually the experiment which has been done. Okay, I will show it a little later once we have discussed this particular aspect. Isn't it mysterious? Mystery. Even if electrons are emitted very slowly so that electrons reach one by one, still they would show interference pattern when both the slits are open. Both electrons know that there is another slit open so that they decide to change their path. Yeah, this is one very, very interesting question which I always ask to the students. Are we equally surprised if we had used photon in place of electrons? See, with respect to light, we have been explaining interference pattern, diffraction pattern, number of times. Okay, exactly the same statement would have been too for photon also. But if you ask student, photon has a, oh, after all, it's a wave. So no, okay, it can behave like a particle. Electron? No, electron is a particle. Okay, you have problems when I say that electron is moving like that. Okay, then you get disturbed. But if we say photon, see photon, same thing is true for photon also. See, photon has to either pass through slit one or through slit two, and suddenly in of the, suddenly find a very small number of photons. Okay, somehow have the photons decided that you know, don't go through slit two, go on very small number to that particular point. Exactly the same statement which I have said is also true for photons. But if I describe that way for photons, no student will be interested. Oh, photon, wave, okay. But when you describe with electron, people get shocked. Okay, how can electron, electron is a particle. It's a localized entity. Okay, photon is a wave which is an extended extended entity. So that's the picture we have, that's the classical picture we have in mind. And that's what is the picture which we want to sort of avoid or tell about that thing. Are we equally surprised if we had used photons instead of electrons? It's easier to imagine a wave front hitting the slit, making source of wave fronts. All those things with photons, people will be very happy. But electrons, they're uneasy. Some statements which I'm making, which personally I always like. So let me just read these statements. Our concept of particle is based on our observation of bigger particles. We can see them and we have never seen them diffracted. That's why we are surprised. Do you agree? See, whenever we think of electron, we always imagine this to be like a particle like this. Okay, this particle I have never seen diffracted. Because the wavelengths are so small, you'll never get diffracted. Okay, so therefore, I mean, I don't expect, I mean electrons also to get diffracted. We do not know how the particles appear when it is as small as a fundamental particle. Clearly, we do not see these particles the way we see bigger particles. We never see electron, okay? So we have, we don't know how electrons really behave. The way we can see a particle like this. It's a mysterious particle. Our impression of a smaller particle is based on the extrapolation of our ideas of bigger particle. That's what we thought. If we make it very, very tiny, after all, it should look exactly identical, except it doesn't become tiny enough. All right, that's the way we think. So we extrapolate that, you know, if we keep on reducing the size to very small, that's the way particles should behave, okay? These extrapolations clearly are not correct. The particles just behave mysteriously when it's small. It does behave in a very, very funny fashion. Now, one can, I mean, this is again a thought experiment, as Feynman says, that can we observe the particles, okay? It's not very easy to observe electrons in that sense, but you know, just imagine that's the reason it's called thought experiment. Put a source of light there so that whenever there's a particular particle which comes, okay, you can really see whether a particle is going through slit one or slit two. Put a source of light here, okay? Just before the normal screen. So, I mean, like if I have to see whether a person, you are coming from this door or from that door, what I will do, I put a source of light, okay? When you are entering here and the light gets reflected and then I see that you are coming, okay? So, let's imagine that I could do the same thing for electrons, okay? I put an electron beam of light there and then you find that electrons, where when electrons are coming, a flash comes and then I know that the electron really is passing through that particular particle. And if at the same time I see a counter there on the particular screen, then we know that this particular particle has actually come from this particular, I mean, the one which he just now observed is the one which has reached the screen. You can imagine such an experiment. Yes? In case of electron water, what about the coherence? See, like you can talk about any wave nature, you can also talk about the coherence of this particular wave. So, the fact is that electron waves, in that sense, are no way different from any other classical, I mean, for that matter, electromagnetic waves. You can talk about coherence exactly in the same fashion. But my idea at this particular moment is not to talk about coherence and what condition you can observe interference, okay? You have to create coherence beam in order to get interference pattern. If you also divide the beams exactly in the same fashion as you do in the case of optics to see interference pattern. My idea is that, suppose we have observed interference pattern, my idea is to bring out the mystery or the uncertainty which we are actually landing ourselves into, okay, in order to protect wave particle relative. See, either so long we talk only about wave nature, we have no problem. So long I won't talk only about wave pattern nature, I don't have any problem, okay? When you have an experiment which basically shows a wave property and there you want to introduce particle property, that's where you have problem. That's where you are supposed to be answering these questions, okay, which we do not know how to answer. And fortunately for us, there is uncertainty principle which actually involves all those things. You can say, it's uncertainty. If I want to do an experiment, I will never be able to find it out. That's what basically the uncertainty principle is. Okay, so we say put a light source. If you put a light source, you'll always find that electrons indeed pass through either slit one or slit two. Nothing else is happening. But in that case, when you open both the slits, you will never see the interference pattern. You mean the, you have to put a light source to watch the, that's precisely what I'm coming to that. See, the thing is that why interference pattern is lost because you realize that when you are putting light source, you are performing the experiment. Because this particular light source consists actually also photons, okay? And we know that photons, electrons can interact. We just now discussed about the Compton effect. So when a photon comes and hits electron, this is likely to deviate the path. When it's likely to deviate the path, I'm not getting interference pattern. So can't I reduce number of photons? If I reduce number of photons, there'll be a time when I will not be able to see an electrons and only those electrons will be able to show the interference pattern. You can also say that, okay, what I will do? I will do a, in order that perturbation that is created by these photons to watch these electrons is very large. I will use a very, very low energy photon. If I use a very, very low energy photon, it means I'm using a very large lambda photon, okay? Then still you'll be perturbing the experiment until the situation comes when your lambda is so large that your resolving power has become so bad that you'll never be able to know whether the flashlight has come, it has come from this particular slit or that particular slit. Because after all, resolving power depends on the wavelength. Basically the question is that whatever experiments you do, you'll never be able to do this particular experiment. That's what Feynman writes very clearly that that's not uncertain. You'll never be able to perform an experiment where you are sure that particles pass through slit one and slit two and also retain your interference pattern. It's not possible. If you're, I mean, of course this particular experiment can be explained on the basis of uncertain difference which probably I'll not be able to do it, but. We shall indeed find that the electrons pass through one of the two slits, but our interference pattern would be lost. The photon interaction with the electrons could have changed their path. We can reduce the perturbation by using low energy photons. If wavelength increases beyond a limit, the resolving power would be reduced and the two slits may appear as a one broad slit. Only in such circumstances, you will be able to get back the interference pattern. This statement I would like to mention because this is very, very important because many times when we teach uncertain difference, the way it's basically introduced, sometimes students have confusion. Uncertainty principle is fundamental. Perturbing the experiment while performing measurement is known also in classical physics. I mean, many times people will always give this example which personally I do not like. For example, if you want to make a current, measure a current in the circuit, okay, you have to add an emitter. When you are adding an emitter, that itself is going to perturb the experiment because this particular emitter will have always a small resistance and therefore the current in the circuit would change. Suppose you want to measure the voltage between the two points, you have to add a voltmeter and this voltmeter will also draw some current. When it will draw some current, the voltage will change, okay? So whenever you are measuring something, you're always perturbing the experiment, okay? People will take this particular example and say that this is uncertainty. I personally do not like this particular thing because the thing is that in normal standard thing, we always believe that if we have better and better precision, you can keep on measuring things to more and more accuracies, okay? Here when we are talking, we are not talking of two couple things. Uncertainty principle is a fundamental quantity. We can make still a statement that if I have to measure the position of a particular particle, I can keep on measuring if I have better and better equipments, okay? I may have a normal ruler which can only measure plus, minus one millimeter, but I can go to better and better resolution, I can have better and better apparatus and measure as closely the position as we want. In that sense, we are agreeing. But as and when we are trying to measure positions very, very clearly, I will make momentum measurement more and more uncertain. These are two couple things. It's very, very different from just perturbation of the experiment, okay? I can measure position as precisely as I want, okay? But in that case, I will make momentum more and more uncertain. Similarly, momentum, if I want to measure momentum, if that's my focus, I can measure momentum as precisely as I want. But in that case, I will make position more and more uncertain. So here we have two couple things. So in that way, it's very different from just the perturbing experiment, all right? You have two couple quantities if you measure one of them more accurately, second one goes off. If you want to measure this more accurately, other one goes off, okay? In that sense, it's different. So let's go ahead, where I essentially try to use this particular thing to calculate the order of uncertainty principle. Just to convince the student that this uncertainty product is very small and comparison to H. It's of the order of H. And I always refer to a very well-known book. I think, you know, this is a very nice book. I'm not sure whether you are aware of that thing. It's a book by George Gamow. Mr. Tompkins in the Wonderland. I'm not sure whether you've heard this particular book. It's a very, very beautiful book, okay? This book, I think, I have been recently republished and it's just called Mr. Tompkins, I think, if I'm not sure. The name has been slightly changed. And it's by George Gamow. So the book is actually a funny description of modern physics. And it's like a student who is going through a course on modern physics and then in the night he dreams and he goes to a different world. And there are various type of words. In one word, where he finds the velocity of light to be of 20 miles per hour. So that person starts seeing relativistic effect in daily life. So he finds that, you know, as he's driving very, very fast, the market is contracting and things like that, all those, very, very beautifully described the entire effect. There is also an effect on uncertainty principle. It's called quantum jungle. See, a person reaches to a particular point where Planck's constant happens to be very large. It's of the order of one joule second. So, you know, then he starts seeing uncertainty principle everywhere. He goes to the jungle and there is a land which is coming and he does not, because the uncertainty is so large so he does not know whether the land is coming from this side or it's coming from that particular side. It's a very beautiful description of the book. You know, some of the students or for that matter, some of you are interested. You know, it's worth reading. There is a different approach which I normally like to do also. It's a single state diffraction experiment. So let me just try to tell a little bit about this particular approach of evaluating the order of uncertainty product using this particular experiment. So let us assume that we perform now a single state experiment. Consider a single state diffraction of a beam of mono energetic particles and we try to interpret this on the basis of both wave and particles and try to get an idea of uncertainty principles because uncertainty principle is the one which as I said protects the two. Therefore, whenever you try to mix up these two ideas you have to invoke about the uncertainty. So let us assume that there is a source of particle. Let us assume that this source of particles is very, very far off almost at infinity. If it is at infinity, the beam of particles which is coming will be essentially parallel along the x direction. They will all be moving along the x direction and they will be essentially parallel. So it means v y or that way for that matter y component of the momentum would be zero. We are very, very sure because this particular source has been kept very, very far off. Do you agree? If that happens, my uncertainty principle tells that position along the y direction would become very, very uncertain. It means I would not, I am not sure where my particle is going to land up along the y direction. So this particular particle may come here. This particular particle may come here. This particle may go there. It may come here, it may come here which also is understandable because you have put your source so far off. Okay? So a particle may land up on this particular thing anywhere. So it's perfectly all right that this particular uncertainty principle is, is valid. Now suddenly this particular beam of particle meets with a slit which is here. The width of which is d. All right? Now once it enters through this thing, suddenly the uncertainty in the position y component of this particular beam becomes finite because only those particles which pass through this particular slit will be able to pass to the other side of the particular screen. All right? Now suddenly the uncertainty in y component position by position becomes finite. Now my uncertainty principle says that it means my uncertainty in the y component of the momentum will also no longer remain zero. It will also become finite. All right? It means these particles you are actually coming straight may suddenly not come straight may decide to change their part and they may develop a y component of the momentum. That's what is diffraction. All right? Now you can always argue it out. This is one thing which again I try to illustrate that once I find a particle let's assume that this particle is electron. I mean it can be done for any other particle. If this particular particle reaches here I can find very precisely what should have been the y component of the momentum okay for this particular particle to reach here. But what uncertainty principle talks is about predictability. When it was passing through the slit was I sure that this particular particle is going to pick up a particular component of momentum. Okay? Uncertainty principle talks about that particular thing. So when it picks up the component of the y component this particular particle could go straight may turn upwards, may turn downwards it may turn anywhere. Okay? This predictability cannot be there. Classical mechanics always talk about predictability. Classical mechanics if I know force if I know the position if I know the initial conditions I know the final condition precisely. Okay? But this is what is something different when we are talking of quantum mechanics or for that matter wave particle duality that you are not very sure when the particle is passing through the slit you are not very sure whether it's going to go this way whether it's going to go this way whether it's going to go straight what angle it will make what y component of momentum it will pick up that you are not very sure. That's what this particular uncertainty principle talks about. Of course this is the angle of position where we see the minimum will be given by d sin theta is equal to n lambda this is well known diffraction experiment. The smaller is the slit width the larger will be the width of the central maximum that we know about it. The experiment can be understood by the wave theory. Now let us bring the particle nature assuming that the experiment is being performed with particles. Assume that the source is infinite distance away therefore electron move in y direction delta P y is equal to 0 therefore delta y is equal to infinity that's what we just now discussed. Hence we are not sure of the y position of the particle at all. What happens when the particle passes through the slit? Delta y is suddenly made finite hence delta P y can also not be 0 anymore. Hence the particles would pick up a y component of a momentum in a totally unpredictable and uncontrolled manner. Smaller is this slit width larger is the uncertainty in the momentum component is consistent with what we expect. Yeah. I say it's totally unpredictable, totally uncontrolled then it gives a miss kind of creates a confusion uncertainty principle gives a measurement of uncertainty. How much will as we know delta y so we have an estimate of delta P y. That's right. So how can it be become totally unpredictable? See that's what it means. See when we say delta P y in fact you can define using standard deviation you know all those things. See quantum mechanics is always about unpredictability you can always say that there is a finite how much is the probability of finding particle here? Yes. We are never talking about surety. So see basic differences in the classical mechanics is that if I know that this particular particle was here at x is equal to 0 and this is the force then I will know at time t is equal to 0 at time t is equal to 1 second where this particle will be exactly located. Now what I am trying to say that I know the same conditions but I am not very sure whether my particle will be here my particle will be here. It's represented by a wave function and that describes what is the probability that the particle will be found here what is the probability that the particle will be found here what is the probability will be found? It's a predictable. The probabilities are predictable but not the position of the particle so that's what I am trying to say what I am talking is the position of the particle the momentum that it picks up okay this whatever you are going to get get the width that is predictable the width that you are going to get is predictable okay but if I am talking of an individual electron or an individual particle what value of P y is going to get that is unpredictable like in a wave function it's unpredictable whether this particular electron which has gone through this particular force will be exactly found that okay that's not predictable it could be here it could be here it could be here probability is predictable okay similarly here what is the width that it's going to get that is predictable in fact that's given by d sin theta is equal to n lambda alright but what is not predictable for an individual particle when classical mechanics for all individual particles since we are predictable a particle goes this particular way it's going to land up here nothing else it cannot do any other thing I mean this particular thing is best described in this step potential a particle comes with a particular energy okay in a classical mechanics if this potential energy is larger all the particle has to be reflected back if it is smaller all the particles have to go okay well quantum mechanics there is a possibility that some particles also get reflected back but which particle will get reflected back you don't know okay there is a probability certain number of particles are going to go back okay but which particle is going to go back that I do not know I cannot predict once it has gone back then I can say oh I know this particle has gone back alright but I cannot say that this this particular particle before it has placed its path that this particular particle is going to go straight or it's going to come back that's something which we cannot say in that sense this is unpredictable and uncontrollable smaller the slit width larger is the uncertainty in the momentum concept can't we find out the momentum after an electron reaches the screen that we can do but that's what the uncertainty principle does not talk about of course we always saw Newton's law Newton's law of course you know we always have equivalence of Newton's law but this applies only on the averages as probably you are all clear this is actually the way we derive we define uncertainty which is standard deviation as you know the averages are generated by the angular brackets and there are various type of absolute uncertainty minimum you can show that it's going to be only for a Gaussian wave packet which happens to be h cross by 2 which you get in harmonic oscillators and there are various type of uncertainty principle there is also a time energy uncertainty principle these are the various uncertainty principles that happen Sir what is the difference between minimum uncertainty and uncertainty principle well see uncertainty principle uncertainty let's say as I say at the moment whatever we have talked about uncertainty principle is only very very qualitatively we have never talked about quantitatively we just say it should be happening it's to make again the lecture a little more glamorous a little more attractive for the students to pick it up okay but all these uncertainties can be actually defined and this is what I have said that this is the way to define uncertainty okay it means for example if I have to define the uncertainty in position so you see what is the mean position of the what the mean value of the position so that's what I am calling G under angular bracket okay so the thing is that this particular uncertainty is defined in this particular fashion and you can get a quantitative information about the uncertainty principle okay now this product if you take delta x delta px standard deviation in x and standard deviation in momentum and take the product of this two this particular product will depend on what type of wave packet that you are talking about or what is the wave function of the particle okay now this product may be much larger than h cross by 2 depending on what type of wave packet that you are talking about it what you can show mathematically is that absolute minimum is h cross by 2 and that happens only for one type of wave packet which is the Gaussian wave packet okay which happens for example in the harmonic oscillator in the ground state uh okay that's actually the Gaussian wave packet wave function actually is a Gaussian and that's where you get delta x delta px greater than equal to h cross that's the absolute uncertainty product minimum but if you go to any other wave packet in fact for a particle in a box we do derive and probably I will not be able to do particle box but at least I will give you those particular expressions you can actually use this particular expression and find out what will be the uncertainty product for example for a particle in a box wave function which will be much larger than h cross by 2 okay all right yeah what is physical interpretation of uncertainty in time uncertainty in time well the way it's interpreted of course generally time and energy uncertainties are measured in a slightly different okay let me give in a slightly different fashion see like the way at least I have introduced uncertainty principle is that saying that because everything is limited in position and therefore if a wave packet is limited in position okay therefore delta x delta px has to have certain values that's the way I have introduced it now you can also talk of exactly same thing in time domain see a particular wave packet also has to be limited in time it cannot be limited for infinite time a particular wave packet cannot exist for infinite time okay there is only a certain finite amount of time where this particular wave packet will exist and because like in the expression of wave you have kx you have omega t okay if it is finite and t this will create a finite amount of you know uncertainty in omega so if I want to limit this particular wave packet not in the real space but in the time domain okay exactly the similar type of arguments will take place the one which I have taken in the momentum and the position domain so either I can make an infinite in time that I can measure omega accidentally precisely okay or if I have to limit in time my wave packet is my pulse is for a very very short duration okay then delta t becomes very small then the spreader omega will be much much larger exactly the same arguments which I have used in terms of position replace position by time see wave I mean remember wave can always be represented for a given time as a function of x okay or for a given value of x as a function of time okay so all I am saying that whatever I have all the graphs that I have put as a function of x replace them by time you will get time energy as a principle but the physical meaning of the time uncertainty principle generally often comes in what we call as the emissions okay because if for example there is an excited state of an atom and which leads to an emission of a photon okay you know that there is only a finite lifetime for which this particular state can exist in excited state and then either it will be stimulated emission or there will be a spontaneous emission when there is an emission of the photon okay this particular photon in the time domain cannot have a larger time domain then the time at which the atom has remained in the excited state okay therefore there will be spread in its energy by that particular amount so delta e delta t will become of the order of h cross so it means the beam that you are going to come out can never be mono energetic because you are not given in finite time for the wave packet to develop because this wave packet which has come out as a result of transition had only a finite amount of time to get developed so this will be limited in time and therefore by the same thing because I am going to make a wave packet in time domain therefore energy this omegas will get spread out all right therefore you will never be able to make this particular beam perfectly monochromatic there will always be a small spread in energy okay so in fact that is what come there is always a natural line width for any source okay and that depends on the total amount of time at which this particular atom or nucleus depends on what type of transition you are talking okay remains in the excited state so you can never make a beam 100 percent monochromatic just because it will avoid uncertainty dependence pool that is only possible if it had an infinite time for this particular wave packet to develop okay it will never have infinite time it will always have finite time therefore there will always be small amount of width in any beam as far as the monochromatic is concerned any other questions yes any detail research paper is regarding that particular I am not very sure see I am not very sure that they have published it I am not but I mean it requires a little bit of more of study which I have not done it if you are interested please go through this thing maybe they have done it I think the experiment was done by Tonamura if you do a google search on Tonamura maybe you can see whether he has published this particular paper or some other paper you know from Hitachi so it's possible to see I mean if you want to really work on this particular thing it should be possible to do but I have not done it now let's come to the Schrodinger equation now we look for a new law and try to see what should be the new form of the law which can be used for something which is neither a particle nor a wave okay or in between or both the way you want to put it all right now we had two classical equations first equation I am sure all of you know I am pretty sure you also know the second equation first equation is Newton's law of motion which is force is equal to mass into acceleration second is the standard wave equation okay this wave equation you know even if you are doing electromagnetic wave you can derive this particular equation in fact that's the way you derive it and I mean for any ordinary general non standard wave this particular equation is used generally called wave equation okay nothing is that whether one of these equations will be adequate to describe something which is both particle and a wave which is the equation that governs a non relativistic of course let's be very clear that when we are talking Schrodinger equation it's non relativistic particle with observable wave behavior the first one does not even imply a wave nature so we'll ignore the first one let us examine the second one which actually has is a wave equation let's see what are the problems and what are the advantages these are various waves ways in which a wave can be represented one of you agree okay sine kx cos kx some of you had said that cos kx is better than whatever it is and many times you use imaginary numbers when we use imaginary numbers you can also use a e raise power i kx minus omega t or e raise power minus i kx minus omega t remember I have written all those ways which propagate in plus x direction see today morning we discussed how to define whether it's moving how to find out whether moves we move in the plus x direction or minus x direction because it depends on the sine relative sine of x term and t term because the two sines are different here there is a kx which is plus kx and there is a minus omega t so all these waves represent they have a motion in the plus x direction now first of all you must convince yourself that all these four waves satisfy the wave equation okay which is very very simple all you have to do is put this particular thing into that particular equation which I am doing in next two three transplants which I will avoid here because I mean this is for our first year students where I want to tell them no convince them that you know this particular thing is actually satisfied by this thing so let's take in fact I take one of these particular things and one of these exponential terms and say that whether you use any one any of the four this will satisfy the wave equation which I have just now written so you take psi then you take d psi x d psi dx then you take d 2 psi dx square then you take d 2 psi dt square then you you can see that the equation get satisfied and c square turns out to be equal to omega square by k square which is the phase velocity which we have we have already discussed you can take an exponential term it becomes all the more simple cosine and cosine term there is a problem that once you differentiate sin becomes cosine and when you differentiate second times then only it becomes sine okay while in this case there is no problem of exponential always remains exponential all right so you can see that this also satisfies the wave equation with the same c is equal to omega by k now the second thing which I want to insist which is probably very very important is the superposition of wave this particular wave equation allows this particular fact that two waves can superpose means if there is a particular wave which is given by psi 1 another wave which is by psi 2 this particular thing can superimpose see it means they are also a solution of the equation if this equation is actually a valid equation which describes the wave motion and if superposition is not allowed by these waves then we can say that superposition will not be possible okay but we know that the traditional waves do superimpose they ought to be solution of this equation and that I want to convince the students that they actually become the solution of this equation we know the electromagnetic waves even of different wavelengths superimpose all we know I mean we have then in the morning we experiment we talked about we experiment their wavelengths are different they superimpose on each other if we say that wave equation is the wave governing the electromagnetic wave it implies that superposition should also be allowed by the wave equation it means if A sin kx is a solution sin kx minus omega t is a solution then A sin k prime x if I add to this A sin k prime x plus omega prime t that should also be solution of this equation if that does not happen to that solution and if this equation is a two equation which governs them then that particular superposition will not be possible at all okay this wave will go like this this wave will go like this they will never interfere so if psi 1 and psi 2 are solutions of the wave equation psi 1 plus psi 2 also has to be solution of the wave equation which next thing I will show in fact it can be in generalize A psi 1 plus B psi 2 will also be solution of this equation okay so let us verify as this is the key to interference formation of wave packet everywhere we are interfering even wave packet we are saying that you know you have infinite number of waves which are interfering with each other okay so this I give a typical example that I suppose they are two different size which once satisfied this equation another is with this equation okay then we take the solution we add these two okay this equation can be written in this particular fashion psi 1 plus psi 2 so psi 1 plus psi 2 is also a solution provided C is same for light C is same okay therefore this equation can superpose we see that psi 1 plus psi 2 is also solution of the equation but note that C has to be same for both the equations now if this particular wave has to describe particles that is where we are going to end up with the problem a particle for example can travel with any speed all right that restriction that wave has to travel with C is only for electromagnetic waves okay if you are talking of a particle okay particle can travel with any speed so if I have C is different for the two equations they will not superpose that is what is one of the problem so in fact the wavelength of particle depends on the speed that is what we have said the wavelengths and the speeds are you know are interdependent if you take two particle waves corresponding to the same type of particles for example electrons with different wavelengths they would not superimpose if the standard equation was governing it that is what it comes to it but we have seen interference of electrons we have experimentally seen it I know that electrons can interfere therefore this particular wave cannot describe the motion of those particles which have wave characteristic all right because we are now looking for a wave equation so can we form a wave back we probably need another equation that is what I am doing this is where we have to look into one interesting aspect if the constant appearing in the wave equation was not dependent on dynamical properties like speed momentum energy the superposition could have been valid remember this equation what was the problematic thing is the C which is coming here the C which is coming there you know this particular thing okay these things can add only when C is same if you go here and this one C1 and C2 then superposition will not be possible okay this is the situation which we are landing up when we are talking of particle because there is one V1 and there is another V2 okay therefore superposition does not become possible now this constant which was appearing here this particular constant whatever is this constant if somehow we can make it that this does not depend on dynamical properties like momentum or energy or speed or something like that okay if it is possible then superposition becomes possible so let us try to look for an equation in which this particular constant which is related to C what is this constant this constant is relating the position derivative with respect to position and derivative with respect to time this constant is relating derivative first derivative second derivative whatever it is but derivative of wave function or of the wave okay with respect to position and trying to relate to derivative with respect to time so this constant which relates the two if it is independent of the dynamical properties then it is possible that superposition will be valid so let us look how is it possible to make this constant independent of the dynamical properties if the constant appearing in the wave equation was not dependent on dynamical properties like speed momentum or energy the superposition could have been valid now we know of this equation in the non relativistic case kinetic energy is equal to p square by 2m for a free particle where there is no force so there is no potential energy involved okay so let us talk of only a free particle under the influence of without the influence of any force so it obeys in this particular relationship kinetic energy is equal to p square by 2m all right now we have a wave equation which is of the form e raise power or sin kx minus omega t there is x term which is related to k okay there is a p term which is related to omega all right now if I differentiate with respect to x okay I will get k out okay if I have sin kx minus omega t if I take partial derivative with respect to x it will be k times this thing do you agree if I take derivative with respect to time omega will come out omega times psi do you agree now you also realize that in the case of particles k which is dependent on wavelength depends on momentum all right omega h cross omega is supposed to depend on the energy all right now the equation that we have under our control is k is equal to p square by 2m now suppose we would have taken derivative which was in the following way I take derivative with respect to x twice partial derivatives derivatives are partial derivatives d2x d2 psi dx square so I will get k square times psi agreed k square is related to momentum h cross k is supposed to momentum but try to relate it to first derivative with respect to time delta psi del psi del t when I take derivative only omega will come out because I have taken only one derivative derivative once so on left hand side you have k square on right hand side we have omega this omega can be related to energy k square can be related to momentum and because we have this equation p square by 2m is equal to k this thing will cancel out so probably the equation which I should think about is something which relates to the second derivative in position and tries to relate it to the first time derivative then probably it will work and then the constant that I am getting may turn out to be independent of dynamical variables and therefore probably I am a land up I mean remember we are in dark we are not deriving Schrodinger equation we are not deriving nobody can derive Schrodinger equation we are just putting in arguments like Lorentz transformation we did not derive you know we put certain arguments okay so we are in dark we do not know where to go so we are trying to find our way all right so why not try something like this why not try an equation something like del 2 psi del x square is equal to gamma del psi del t then maybe this particular gamma would not depend on velocity may not depend on momentum may not depend on energy therefore particles with different type of velocities can superimpose therefore I may be able to generate my wave packet I may be able to have interference pattern is this point clear all right but there is one big problem if I am taking psi as a sin kx minus omega t or a cos kx minus omega t once I take derivative twice with respect to x first time sin become cos second time cos become sin okay well I am taking first derivative respect to time okay sin become cos it will never become sin again so on left hand side you will have sin term right hand side you will have cos term that psi will not cancel at all so a sin kx minus omega t or a cos kx minus omega t will not be solution of this equation at all all right but instead of that if I take an exponential term a e raise power i kx minus omega t that does not have this problem because if you take either you take derivative twice with respect to x or you take derivative once with respect to time okay it will only have exponential term that can cancel out therefore if we are looking for this equation if this is the correct equation then my psi has to be always complex okay it can only be represented in the form of a e raise power i kx minus omega t that is what makes wave function very very mysterious and basically complex quantity but are the standard displacement equation given by pure trigonometric equation a solution of it traditionally we take the solution into this particular form this is what I have said it is a simple differentiation you take differentiation twice with respect to x this term will cancel it out okay put into that particular equation the trial equation that we have done you will be able to get the value of gamma all right as we have seen that gamma just depends on fundamental constant h cross and also of course depends on mass of the particle but that is okay because I have never seen a electron beam diffracting with a or interfering with a neutron beam always the particle of the same type interfere okay either we have electron beam interfering or only neutron beam interfering okay I have never seen that one particular electron beam is coming and neutron beam coming and they interfere with each other all right so let mass be there I don't care all right so I have solved my problem I have landed up into an equation with this particular value of gamma which probably would work going back to our old equation I put this equation here this particular thing and eventually land up at this particular equation put the value of gamma in fact this is the way this equation is put the reason we will discuss later probably you are already know in terms of operator we define this particular quantity therefore this equation is written in this particular form which is the equation which we expect if whatever our argument our thinking process is correct that this is the equation which should represent a free particle there is only one thing which I would like to sort of point your attention to this while coming to this equation I have taken this wave equation always in this particular form when we will come out and take what we call as a time independent Schrodinger equation this generally called time dependent Schrodinger equation okay then the time term will go away but we must remember that in all these derivations of Schrodinger equation the time dependent term was always taken to be of the form a e raise to power minus i omega t then only we can describe in which direction the particle is moving otherwise I will not be able to describe that particular thing if this would not have changed if I would have taken any other way I could have taken in a different form I could have taken this term as positive the sign of gamma would change but this is one of the very common problems that we give to our students what would have been the Schrodinger equation if instead of e raise to power i k x minus omega t I would have taken e raise to power minus i k x minus omega t but time dependent term would have been positive nothing nothing of physics will change none of the interpretation nothing will change okay but we take a convention and the convention which has been taken is to take the time dependent term negative the only one thing which I would like to mention because we have also done a course on relativity just now that here this particular energy you know whenever we are using in this particular thing energy is meant to mean the classical energy because same e I have used for e is equal to m c square which is the relativistic energy should not get confused because the Schrodinger equation also we use e so let us be clear that you know this particular e that we are talking is going to be somewhat different if you have any questions then please you know in the experiment that you have shown was the source of this electron coherent or incoherent I mean does that matter it doesn't matter because electron is actually coming only one by one so I mean I mean it's hardly able to form a wave in that actually what I was thinking of because it if you look into this experiment actually it looks like that there is an inherent time scale for watching that pattern I mean it's not that you can because finally you need to observe it for a certain amount of time or more than that to really observe it so if you are not really observing it within that time limit or more than that time you cannot see that so your interpretation of the experimental result will depend on the time scale of what you are actually looking at so what I was thinking is that it's somehow if you look into in totality of this experiment somehow an electron which has already crossed that slit and hit the screen or and the electrons which are supposed to go through that all of them somehow know each other that where they have either ended up or somehow where they have to end up so that finally over a that time scale after that time scale they should actually create that pattern so that is not somewhat looks in possible if they are completely incoherent if they don't have that information then it looks like if a motion of an electron is completely random of the motion compared to the motion of the other electron then I don't expect mean that's a kind of natural expectation that I cannot expect that pattern to be seen see that's what it's that particular experiment says and that's the statement which is made very very clearly that this particular beam has been so slow that the probability of finding even one electron is so low that normally you will not even find one electron okay now chances of finding two electrons was anyway very very small so it's really coming one by one so you can imagine that this particular electron has gone and reach this slit and by that time the second electron has not even started what I was thinking is that instead of sending the second electron from the same source if we use a different source that I don't know whether they have tried no I mean does that look a valid question I mean it's a definitely valid question you know in the sense that you know when the quantum mechanics was being developed people always thought I mean there was a lot of possibilities which had been discussed yeah and the one of the possibility was that are the particles intelligent yeah okay can the particles think okay can the particles really know that you know as we say that you know I mean of course most of these interpretations are now not accepted yes I mean if you look at many of those things you know there's I mean the history of this thing development of quantum is fairly interesting yes type of arguments which you are giving are fairly interesting arguments but at least that's what's not believed no I mean I was trying to think of that about the time scale is there some relation between the time scale and the what is the interpretation of that time scale I don't think at least that's what they claim that you know these we are so slow that chances of having any information exchange is not possible that's not okay thank you thank you any other questions I have a question in double slit experiments yeah we know that electrons behave like they have wave nature yeah in the double slit experiment when we carry out the double slit experiment then we should expect the diffraction right yeah but here no no in double slit there is the interference yeah yeah interference so that's what you expect interference that's what we got until you perturb the experiment see so long the electrons are coming one by one and you are not really disturbing them they'll form interference pattern that's what this happens and if the electrons pass only through one of the slit then all they form diffraction matter so does it happen because of the wave nature of the particle the precise precise that's what I'm saying I think you know this has been too heavy so let's just stop