 So, just to mention whatever we have said so far, a localized wave packet can be formed as a result of superposition of infinite ideal waves varying in their wavelengths. Wave packet, of course this particular part I have not talked about which I will try to mention now, that if I want to talk of the speed of the wave packet, because this wave packet is essentially the speed of the envelope and that we have said that the envelope travels does not travel at the speed of omega by k, but travels with the speed of mu omega by dk. So, what you will find out that this wave packet actually will travel with the speed of d omega dk. So, in fact what is most important in the physics is not the group velocity, not the phase velocity, but the group velocity which is defined as d omega dk. This is the only realistic velocity for any wave, because as we have said all real waves are always finite in dimension, they are always wave packet and they will always travel with the speed of d omega dk. Only if it so happens that omega is equal to constant times k, then group velocity and phase velocity will turn out to be same, but in the case of particle wave we will see that this does not happen. If it will actually be different, two speeds actually turn out to be different. But in fact there are many other cases in which you will see that group velocity and phase velocities will not be same, but as I said the realistic speed is always a group speed, the phase speed is only of mathematical consequence which if at all there was a infinite wave that wave would have traveled with that phase speed which anyway does not exist. So, there is no special physical significance of the phase velocity. The position of the wave packet is not completely localized, the wave packet of the wave length is not also not unique this leads to what we call as uncertainty principle. Now, let us look at the implications of the uncertainty, when wave nature and particle nature coexist, we have to allow certain tolerance in the way we look at the nature. So, this is what I mentioned just now that if both the things have to be together many type of thing which we are used to talk in terms of classical mechanics with certainty, uncertainty has to find its way, things we will not be able to talk in that certain manner in which we are used to talk in terms of classical mechanics. We have to allow certain tolerance in the way we look at the nature. It is not exactly the way we thought the nature is. Physical interpretation of the wave we shall discuss later, but at the moment let us look at its implications. So, this is a very interesting experiment which we always do and which we always describe when we are talking about uncertainty principle is a very, very beautiful experiment which tells how the way we look at the nature would become different if we are talking of uncertainty principle. So, I am pretty sure all of you know about the Young's Devil's Slate experiment is basically performed with light source, you have two different slits and you allow a monochromatic beam to travel and divide themselves. The wave front gets divided by these two slits and then these two things actually get an interference. So, once we, these two waves interfere, you start seeing dark and bright fringes that I am sure all of you know. And this is because of destructive and constructive interference because the path difference turns out to be different and depending on what is the value of the path difference, these two waves can interfere destructively, destructively or constructively, you may get a large intensity of light or you may get zero intensity of light. This experiment can be explained on the basis of wave nature of the light that we understand. Now, basic thing is that as I said this particular experiment can be very easily we understand about the wave, from the wave nature, so long we do not invoke the particle picture. Now, I want to invoke the particle picture and want to see what type of problems I face into it and that's what I will associate to the uncertainty principle. That's the reason I have made that statement last time, just now that uncertainty principle protects the particle to anything. I am trying to mix up these two ideas on an experiment which can be very well or much better described on the basis of wave theory, okay? So, I make a statement and let's accept that we agree with that. At the moment I am talking of only light. I have also mentioned that light consists of photons, light also is a dual character, like particles have a dual character, light also is a dual character, in some experiment it shows a particle behavior, some experiments it shows a wave behavior and in difference is one experiment where it tends to show a wave behavior. So, I make a statement that whenever I get a bright fringes, a large number of photons go and hit that particular point. See remember the intensity is always associated with the number of photons, its energy is associated with the frequency, okay? If I am saying that a light is more intense, it means it has larger number of photons. If it is light is more energetic, then its frequency is larger, okay? So, a place where you have seen a bright fringe, essentially it meant much larger number of photons came and hit at that particular point. If you see a dark fringe, no photon came and hit at that particular point. So, I hope you agree with this particular statement that in a photon picture, a bright fringe would mean a larger number of photons hitting that particular point or reaching that particular point and a dark fringe would mean no photon reaching that particular point. So, this is source of light, let us suppose a monochromatic source of light, you have two slits S1 and S2, one I have set blue and another I have set red. Now, this wave front gets divided by these two slits and then there is a constructive interference and ignore this particular part of the figure at this particular moment, okay? Just look at this particular thing. Of course, intensities of these fringes are not always same, but we have just taken, as I said, pure cartoon. If you go to a point P, this particular point P here, there is a destructive interference, the intensity has become 0. At this particular point, the interference has been a constructive interference, the intensity has become maximum. Here again, there is a destructive interference, the intensity has become 0. You see this destructive, constructive, destructive, constructive and you get fringes. These are generally called fringes, okay? So, when I say here, there is a constructive interference, a large number of photons have come and hit this particular point. Here, there is a dark fringe, it means no photon has come and hit this particular point. So, this is standard experiment. Now, let us perform this experiment with electrons. We know that electrons can be deflected, okay? Now, of course, these experiments now can be done in ways in which you can perform these experiments even with electrons. But you know, the length scales that we are talking, but let us just take at this particular moment as a sort of a thought experiment. So, let us assume that we have exactly similar type of experiment. So, remember for electron diffraction, you have to have really slit distances which are of the order of angstrom, okay? While in the case of normal light, you know, these distances can be much larger. But as I said, let us suppose we are able to perform this experiment using this type of length scales appropriately. So, we will observe exactly similar type of pattern, okay? Because we know that electrons can be deflected, exactly like to light, okay? And therefore, we would also see instead of this particular light source which I have given here, if I had a monochromatic or rather mono energetic source of electron, okay? I should be able to form exactly similar type of fringes using electrons. And let us also accept, let us also agree to the fact that when we said that a dark fringe here means no photons, okay? And a large, bright fringe means a large number of photons. In the case of electron, it will also mean the same thing. If I am having a dark fringe of electron, it means there is no electron which is reaching here. If I am having a bright fringe of electron, it means a large number of electrons are reaching. So, all that I am using instead of the word photon, I have replaced with the word of electron. Otherwise everything has to be identical. Of course, length scales have to be different. But let us assume that we are able to perform an experiment with an appropriate length scale. So, this is what I said. Similar to photons, a bright fringe would mean larger number of electrons reaching there and a dark fringe would mean a very few electrons reaching there. Now, let us go into some funny questions. See, let us ask the question, how does the electron reach on the screen? Electron goes from here. How does it really reach the screen? These questions are looking very sort of naive questions and very sort of simplistic questions. Or sometimes you can say some sort of mad questions. But let us see that they have a lot of meaning. If we say how the electrons actually reach the screens, so this is probably the question that I am going to answer. They are emitted from source. Some of them move towards wall. Of course, there is a source which can emit electrons in all the directions. But only some of them actually will move towards the wall. And if they hit a point where there is no slit, they will get absorbed. Some electrons will arrive at a point when there is a slit and they will cross to the other side and then reach the screen. So, there is an electron which are coming. Some electrons will go and hit here. Some electrons may go in this particular direction. They are of no interest. Some electrons will come and hit this particular point. They will get absorbed. Then again of no interest. Then some electrons will come here and they get absorbed here. They are also of no interest. It will some happen that some electrons will come through this slit and then they will go and hit here. Then some electrons are coming from this. Then they will go and hit here, okay? So I am all I am trying to say is that the way that these electrons can reach this particular screen is only by going through slit S1 or slit S2. So this is what I let me reread this particular slide, they are emitted from source, some of them move towards wall, if they hit a point where there is no slit they get absorbed, some electrons happen to arrive at a point when where there is a slit and they cross over to the other side and then reach the screen. My question is that are we really sure, let us perform the experiment and try to see. So what I will do, I will close both the slits, if I close both the slits I will not find any electron reaching on the screen, so obviously this slit was necessary for the electrons to make them reach the screen, if there was no slit no electron reaches there, so it is very clear that electron has passed through the slit then only it has reached, okay. Question looks silly silly but let us see it has lot of meaning, close both the slits and we would not see any electrons on the screen, hence presence of slit is necessary for the electron to reach the screen, so let us be very clear that this slit was really necessary for the electrons to reach there. Now again this is a silly question, two electrons reaching the screen pass through one of the two slits, we just now said that there are two slits and if we close these two slits no electron reaches there, so presence of slit is necessary that is for sure but does it happen that the electron really goes through one slit or the other slit, so it looks very silly, since electron is really a particle which cannot be split, so is it not obvious that it should pass through one of the slit, how silly is the question, it is almost like asking a question that if we are sitting in this classroom and there are two doors okay and I say the person who is sitting in this particular room has either come through this door or through that door, obviously a person cannot come through both the doors together okay, a person cannot be divided, his head cannot come from one door and leg can come from the other door, I mean asking this question is almost as silly as that and electron is a particle, we always thought that is a particle and if it is a particle it in principle cannot be divided, so half the electron or head of the electron cannot pass through one door and the leg of the electron can pass through some other door, it has to come either from this door or from that particular door, so how silly is the question, had it been equally silly if we are talking about the interference of two classical waves, remember classical waves had if we are talking probably we will not be wondering about this question because remember waves in extended property, wave is an extended thing okay, in principle remember you know this wave can simultaneously in the in the sense we have totally different concept, this particular slit is dividing its wave front and these wave fronts are sort of interfering, this is not localized, this is not a wave which can I mean only going through this slit or through that slit in this electron we know it is a localized entity, so it has to go either through this slit or through this particular slit, so I mean this particular thing that I am describing would not have been that dramatic if we are talking about photons because photons we have always a wave picture okay, photons also consist of photons, particles also light is also constituent of photons, but photon we have always a very funny meaning, you know sort of jittery feeling about the photon, oh after all photons are waves okay, but when we talk of electrons we get shocked because we always thought classically that electrons are particles and particles cannot be divided, in that sense electron is a fundamental particle how can electron be divided into two parts so as we see that it is not a very silly question, as we will soon see if this statement is true then if you perform two experiments one week we will have only slit, we will open only one of the slit then find out how many electrons are passing through that particular slit then we close that particular slit and open the other slit okay, then or let me put it the other way, see any electron which is reaching on the screen in principle now can be classified the one which is coming through slit S1 and coming through slit S2, now let us try to close them one by one and try to see what is the effect, so this is what I have written we will classify the electrons reaching the screen into two parts number one those which came through slit S1 and those which are coming through slit S2 can be experimentally verify this particular statement we perform three experiments, first keep only slit S1 open we open only one slit, choose a point P on the screen count the number of electrons N1 reaching that point in a given time then keep on changing the position of point P and find N1 as a function of position for the same time see we did exactly the same experiment when both the slits were open together but now I am performing the experiment by opening one slit at each time then we will look at when both the slits are open okay and then we try to see whether we can really qualify our statement that this particular electron has either gone through slit S1 or has gone through slit S2 okay, so first we keep slit S1 open then choose a point P on the screen and then vary its position find out the total number of particles which are coming and hitting that particular part of the screen repeat the experiment with only slit S2 open now close S1 have S2 open then find out what will be N2 okay find out the total number of particles which are coming as a function of different position of the point okay at different locations let us say in a given time and find out N2 then we keep both the slits S1 and S2 open and then calculate what will be the N12 or rather measure what will be N1 and N2 actually I must get N12 is equal to N1 plus N2 if whatever I am saying is correct if I have to find out the total number of persons sitting in this particular classroom okay I first close one of the door repeat the experiment those persons who are coming from door 2 will get blocked there okay only persons coming from one door will come okay I count their number then open the other door then let the people come from the other door if we count the people okay we will always find out that the number has to be matching so total number which people who are coming from door number 1 total number people who are coming from door number 2 they must add up when both the doors are open that is what is as simple as the question so if both the slits S1 and S2 are open then N12 we must get equal to N1 plus N2 now let us try to look at this particular picture which in the earlier case I have sort of not shown if only slit S1 is open which is this one which I have said mark blue I am not sure whether you are able to see color very clearly then what you will find you will see not really an interference pattern but what we call is a diffraction pattern where you see a broad maximum very close to the slit S1 there is some sort of small interference here I am sure you know a single slit diffraction experiment this is what you will be seeing here only slit S2 is open then you will see behavior something like this which is the red one here this particular maximum will be very close to the slit S2 and this will be some sort of a oscillation some sort of maximum because of the interference particles come from different parts of the slit this is smaller this slit this particular second maximum will be put larger distance away but when both S1 and S2 are open then these two interfere as we know and what I will get this particular pattern is what we have discussed earlier is will be N1 and 2 also very very funny thing if only one slit was open I was getting very large number of electrons here but when both the slits are open suddenly at that particular point the number of electrons are coming here have become very very small here of course they were large here also they were large but at this particular point where a very large number of electrons were coming only through when only one of the slit was open when second slit got open a very few electrons are essentially no electron is reaching there as if the electrons reaching you know going through slit S2 has realized that now slit S1 is open so I must have the particles should not the particles should not go there I mean this makes ridiculous sense as if they can talk to each other and say okay now this particular particle this particular thing is open this particular slit is open so I must pass through that slit or I must not pass through the slit they can decide okay normally we do not expect this type of thing alright so when both the slits are open in fact you will not find N1 2 is equal to N1 plus N2 so what has happened what has how things have changed see the answer is no we will not get N1 2 is equal to N1 plus N2 because now I have interference pattern with one slit open one sees a diffraction pattern there is no interference pattern due to the other slit there are places in the screen where which receive much less number of electrons when both the slits are open then when one slit is open can electrostatic interaction change the path when both the slits are open the fact is that you can perform this experiment extremely slowly when only one electron is coming at one time but still you will find that there will be an interference pattern this interference pattern does not depend on the number of particles which are going through this particular thing at a rate so I mean there is no possibility that one particular electron another particular electron there is electrostatic repulsion and therefore they have changed their path okay this will not happen okay if you really see and such experiments have now been it has been possible to perform now when you have only one electron going through slit at each time okay still you will find when both the slits are open you will see an interference pattern okay this interference pattern will not be visible if you have only one of the slit open okay obviously electrostatic interaction is not the reason which could have changed the path even if electrons are emitted very slowly so that electrons reach one by one still they will show interference pattern when both the slits are open as I said do the electrons know that there is another slit open so they decide to change their path okay human being we consider as intelligent so they can probably talk to each other that okay if this slit is not open can we change our path okay but do we expect the electrons also to do the same thing that they can talk to each other saying that okay this particular slit is not open so let us not go to that side let us go to some other side now things that this looks somewhat shocking but as I said probably we would not have been equally surprised if I would have repeated the same experiment with the case of light light also consists of photons I can argue exactly the same thing instead of electron I replace the word photon same arguments can be made there also but when I say photons whether they go through slit 1 or slit 2 you are not very surprised again oh after all photons were waves you know that is the way we have always thought so you have very hazy picture in mind but we get a shock when we talk about electrons because we always thought electrons are localized particles and these localized particles have to behave in a different fashion okay they can go only one by one only one particular electron can come from one particular state it cannot come from the two electrons cannot simultaneously or one electron cannot simultaneously come from the two slits together what I want to say our concept of particle is based on our observation of bigger particles we can see them we can see these particles and we never see them diffracted because wavelength is too small in fact as we have said the wavelengths are often minus 74 meters for a cricket ball now obviously if you have interference pattern you have to measure fringe rates of the order of 10 power minus 74 meters which is never possible okay we do not know how the particles appear when they become as small as the fundamental particles clearly we have never seen an electron far with our eyes we do not see these electrons the way we see the bigger particles a bigger particle we can see very easily okay person sitting in front of me cricket ball is in front of me I do not see electron that way this is what I have said mysterious particle our impression of smaller particle is based on the extrapolation of our ideas of bigger particles the way we think that bigger particles are behaving we expect exactly the smaller particles to be behaving exactly in the same fashion obviously this extrapolation is not correct we cannot think of small particles exactly the way we see I will say under quotes we see bigger particles these particles do behave mysteriously when they are small okay I think I will put this thing in the next class we have 10 minutes left so let us go to the questions if you have some questions yes Bhagwan Basuram Institute 1164 yes sir I want to ask you about the photon and the phenomenon of superposition for the photons okay because on behalf of wave nature we can say when two crust or two top will match okay superimpose that is we will find construct interference yeah but one crust or one top will spin pose you will find destructive interference yes okay yeah but that will stand the probability of photon and the intensity is zero that is correct so they know anything at that point yes yes that is you are right that is what I am trying to say that this particular picture of interference having destructive interference is not consistent with the classical concept of particles where we say that these particular particles must go either from one slit or the other because if that happens then the total number of particles which are coming from one particular screen when I open the second screen that number cannot go down so whatever you are saying is consistent with the wave picture but is not consistent with the particle picture so that is all I was trying to emphasize that when I try to mix up the classical picture and the quantum picture together I will land up to this particular problem I will not be because classically I would have expected an electron to follow a particular path go through either one slit or go through the second slit okay and it goes through that when I keep the slit one open for a particular duration of time a certain number of electrons reach at a particular point if I open the slit to also that number cannot go down okay that can go down only if you understand interference and that interference is difficult to understand or it is not we cannot understand from the classical picture of the particle which is a localized concept that is all I am trying to say. Sir thank you sir slide number 6 please repeat it in photon picture a bright field would mean larger number of photon reaching at that point a dark field would mean no photon reaching but principle of interference with conservation of energy okay so on here they know any photon in the and that is the interference then surely they know any photon at that point no see conservation of energy is totally different way of looking into see total number of photons which are reaching must reach somewhere that is what involve conservation of energy so at a particular point there will be a larger number of photons coming or a particular point a lesser number of photons will be coming so conservation of energy will be taken care of that that is not an issue the issue is that what we are trying to say at this particular moment that whenever intensity of photons that in the picture of photon means I mean see intensity we define in terms of the wave nature okay whenever see we say that a light is more intense essentially it means a larger number of photons okay so if I am seeing a bright fringes it means a larger number of photons are hitting there okay conservation of energy will always take place the total number of photons have to be conserved okay they will either get absorbed or they will whatever will happen I mean I have to be all accounted for okay so conservation of energy will take care so it is not bothering about the conservation of energy it is only saying that we are trying to mix up two ideas one is the wave aspect where we know how to define intensity intensity is defined as amplitude square then when we are coming of particle nature then we say that the intensity depends on the total number if you look at the explanation of photoelectric effect experiment okay the intensity of the light is given by the number of photons so similarly when we are having a bright fringe a larger number of photons are reaching that point that is what I am trying to say thank you sir thank you sir welcome MES I have two questions yes please go ahead first one is if particularly stationary its momentum will be zero then lambda will go in finite what it means actually and what happens to boundary conditions I mean see things that you know this is as we once we go to the quantum mechanical actual quantum mechanics course if the particle is actually bound in that particular case the particle cannot be stationary it will always have a finite energy the ground state energy cannot be zero only when the particle has infinite extent it means particle is not localized only then it is possible to have its zero velocity so that particular problem is taken care of but let's at this particular moment just accept the fact that if really you have lambda is equal to I mean if you have p is equal to zero which as I said in quantum mechanical case you will not find to be the case if it happens to be zero then in principle the lambda will be infinite so we are never doubting at the de Broglie relationship okay we are only talking about the uncertainty in the value of lambda or uncertainty in the value of x but lambda is equal to h by p is still obeyed now when we come to the quantum mechanics we will discuss this particular aspect that for any bound system the ground state energy cannot be zero so for a particle in a box you know the ground state energy has to be a non-zero energy okay I have a second question yes please when we are talking about wave nature of particle means particle is represented by waves that's right okay then is the wave is going to give you information about the position only what is happening to mass of the particle means is mass is having same classical meaning in quantum mechanics okay well in the traditional quantum mechanics you know because this is supposed to be non-relativistic mass is exactly the same meaning it has no other way okay okay this will come into the picture like for example you know when you are talking about the energy kinetic energy etc all those things because you know as you will see in Schrodinger equation mass will always appear so mass of the particle will always be there in fact this about this particular mass will talk also quite a bit when we are talking about what we call when we are coming to the Schrodinger equation. So mass the concept of mass does not change until of course you talk about the relativistic quantum mechanics okay as you know in relativity mass I mean there is something called rest mass and this mass could change but here the concept of mass is exactly identical as earlier. Thank you. Welcome. Yes Mehta Jay Subash. Sir particularly I enjoyed the cartoon that you have shown. Thank you. Where you have one wave travelling and the five cartoon character actually identifying them. Yes. And where you have the interferences mixing up two waves. Yes. Where you have localization of a wave packet. Yes. Then they could not identify and some people identifying it with larger amplitude. That's right that's right. Interesting question is I mean I just want to know. Yes. Classically the amplitude square is the energy. Okay go ahead. Yes. I mean classical picture. Yes. We amplitude square is energy. Yes. So where the person standing if there is amplitude there is a node sort of node. Yes. Okay so they don't find it as there is any wave and some character it will be helpful if you can show that particular cartoon page. The person who is actually standing beneath the amplitude. Yes. We will observe that a wave is with larger energy. Yes. But the person who is standing at the particular beneath the node. Yes. We will find that there is no wave at all. That's right. If the wave is something like that I mean can you elaborate that I mean really interesting to know actually. So sir the thing that in the classical way what you are trying to say is perfectly right because in principle the energy would depend on that particular amplitude and you can talk in terms of what is the amplitude that you are talking. I mean there is a perfectly I mean there is a harmony in that particular thing. There is no sort of contradiction there in that sense. See only when we start talking about a quantum picture in that particular thing that's where we start getting contradiction. So long we are talking of pure wave nature so long we are talking of pure particle nature we have no issues. Okay but whenever we are trying to mix up these two issues or an experiment which can be described by let's say wave nature we want to describe using particle nature then we will have problem or an experiment which we want to describe basically on particle nature let's say photoelectric effect experiment which can be much better described by particle nature. If you want to bring in your wave nature there you will have contradiction. So I mean the cartoon which I showed was a purely classical cartoon and whatever you are saying is perfectly right there. Thank you sir. The question is the wavelength the more the number of wavelengths that you add the particle gets more localized. That's right. My question is what role does the amplitude of the wave function does it play in localizing the same wave function. Okay. See thing is that you know basically Fourier transform if I go back to that particular transparency. See let's look at this particular equation which I was trying to write. See if you see sin kx minus omega t is basically the sinusoidal form of the term akdk appears to be the amplitude of this particular thing. Okay. Now a in principle is a function of k it means depending upon what k you are using the amplitude can be different. So ak can be a function of k of whatever form you want to use it probably we will try to add one or two problems which will give you one or two specific forms of ak just to tell you that depending upon how ak varies this yxt can have different type of dependencies. So essentially this amplitude will determine how yxt is looking like yxt is actually the realistic wave packet which we are seeing. So this wave packet can be thought of infinite because I am integrating from 0 to infinite all ideal waves which have amplitude akdk. Now depending upon what is a as a function of k yxt will be different. So this amplitude plays a role in the actual type of wave packet that I am taking. If ak happened to be reasonably large and over very large extent okay it means it essentially doesn't peak but you know goes or the distribution of case for which ak is significantly large is very large then in that particular case yxt will be fairly localized. So this is basically what I am trying to say that this amplitude will have a role to play in this particular functional form. Raj are you college? My question is regarding the shortening the wave packet. Yes, yes. So we are in super position principle to shorten that wave packet but in the example we have seen as the number of waves we are increasing. Yes. In super position the difference between the maximas it is increasing. Yes. Obviously the difference between minimas is also increased. So how is it shortening the wave packet? See the basic thing is that you should understand that see so long if you are mixing a finite number of waves this particular maximum is going to be displaced by a finite amount of distances okay. If I am mixing infinite number of waves okay this particular maximum will be displaced to infinite distance. It means once it has decayed it never rises again okay. This is a let's say qualitative way of putting it to it but there is a mathematical way of putting it which I have just now shown that by an equation you can actually mix a infinite number of waves and find out what is the value of yxt okay. Once you find out that value of yxt you will know that this turns out to be actually localized. This is a very very standard problem in Fourier analysis as we call it or Fourier transform where any finite disturbances can be decomposed into infinite number of ideal waves. So if one knows about Fourier transform it is a very very simple question but this is only way of qualitative explaining to the students because when the first year students come they do not know about the Fourier transform and we have to sort of convince them that by mixing a large number of waves we can really localize it. So the way we can put it like that that so long we are mixing finite number of waves once there is a maximum that maximum gets displaced by finite amount of time finite amount of distance okay. If I mix infinite number of waves this get displaced infinite distance away it means once it has gone down it never rises again alright. Okay thank you Welcome I think we will just stop now. Thank you bye bye all the best.