 Let us start, so what we discussed today morning basically first we saw that how equipartition law was not successful in explaining certain experimental observations, we specifically talked about the specific heat of gases, then we talked about specific heat of solids. We mentioned that many of the discrepancies which we mentioned at that time probably came at a later time when we had better experimental facilities to measure specific heats at various temperatures. But the problem which was considered very severe was the problem of black body radiation and we discussed that how this particular problem could be solved by Planck by making two different assumptions. One is that the energy of the harmonic oscillators are quantized and second that this quantization or quantum of energy is proportional to the frequency of the oscillator. Let me remind you again, these we are purely phenomenologically, this is purely hypothesis, there is no special theory from which these things were derived, it just they made a postulate and with that Planck was able to explain the notorious problem of black body radiation. Then after that we started talking about the dual nature of light, we did mention that you know the light can depending upon the experiment could show a behavior something like waves which we actually always knew. Then some of the later experiments indicated that light can also behave like particle and this particular concept of light being a particle was supported by special theory of relativity about which we will talk later which also said that there is a possibility of a particle which has a zero rest mass but this particular particle could possess energy and momentum and that particular particle with zero rest mass must travel with the speed of light and then we said that if we consider light to be consisting of particles then of course these particles would travel with the speed of light and then we said that a quantum of the or a particular particle of the light which is called photon can be ascribed by energy ash nu and a momentum of ash nu upon C. Then we gave some sort of philosophical arguments and said that in principle it is possible that if a light could have a dual character could show a particle nature as well as a wave nature why not the particles could also show a wave nature then at that particular time we introduced the Broglie wavelength and said that the Broglie proposed that particles would also behave like waves. At that time we mentioned that if you take a macroscopic particle for example particle like a cricket ball the wavelength that will turn out to be will be so small that to see any effect of that particular wavelength will be almost impossible. But on the other hand if you are talking of fundamental particles like electron this particular electron could possess a wavelength which is of the order of inter atomic spacings and therefore it could be diffracted by a crystal. I even showed some pictures where we said that one can obtain a diffraction pattern similar to the one we are used to finding out from light or from x-rays even when we are having electrons and I mentioned that electron diffraction and neutron diffractions are the experimental techniques which people use very often these days. Then we started discussing about pure wave we just discussed what a pure wave means and then we said we discussed how we assign a direction of motion to the wave. Then after that I started taking question now I think let us just start with this particular wave concept what I wanted to mention was about a realistic wave group about which I had mentioned here. This is a sort of realistic wave group which is limited both in space and time. For example this physical elongation of this particular what we call as a wave group or wave packet can be very long it could be some going right from here to here or much longer or could be smaller but nevertheless the disturbance that is created by any normal or natural wave or existing waves is always finite both in position and in time. So you can see this is a particular in a sort of a cartoon of a wave which is something like this where the amplitude is very high in between and then eventually decays and finally the amplitude tends to become 0. So this is somewhat a picture of a realistic wave which we call a wave group and many times we call this as a wave packet. Now if we have this type of wave packet which is associated with a particular particle or let us not even talk about the particle for any particular wave which is like that it could be electromagnetic wave or any other wave that we know the two questions are associated with this particular type of realistic wave. First of all what is the speed of this particular wave group? We just now have talked about that what is the speed of a normal or infinite or an ideal wave which we say that the speed of the wave turns out to be the phase speed which is given by omega by k which essentially means lambda multiplied by the frequency but this is a expression which now we know right from our high schools. The second question that I want to answer is how is this particular wave group formed? How do we form this type of wave group because eventually as we have said this is actually the realistic wave group and an infinite wave is not really a type of wave that we are going to encounter in actual real life. So let us try to answer this particular question of what is the speed of a realistic wave group or a wave packet and how is this particular wave group or a wave packet is formed? For that let us start with superposition of just two waves for again convenience in fact there have been some questions about why I have taken this particular wave of a particular form and I had mentioned at that time that does not matter in what form I take whether I take kx minus omega t or omega t minus kx or cos kx minus omega t or cos omega t minus kx this is the immaterial all of them represent a wave okay of course their phase could be somewhat different but just for convenience I have taken here two sine waves okay. Now let us suppose there is one sine wave which is given by a sine kx minus omega t and let us assume that there is another wave which is having a slight different wavelength therefore it has a slight different wave vector and a slight different omega from this original wave so slight different frequency. So I am writing this as a sine k plus delta kx minus omega plus delta omega t. Now let us try to superimpose these two waves this is a phenomenon probably all of you know this is a standard what we call as a Beatz phenomenon which we have been talking in high schools in the case of sound waves it is the same thing but I want to give it a slightly different interpretation that is why I am talking about this particular thing or I am repeating this particular thing. Now if I have to superimpose these two waves one is this sine wave another another sine wave in which the two waves are differing slightly in their wavelengths and slightly in their frequencies I will use this identity sine a plus sine b is equal to 2 sine a plus b by 2 to cos a minus by b by 2 to superimpose these two waves and get a resultant wave this is what I have done in this particular next transparency you can see very easily that this particular wave will be represented or this particular disturbance would be represented by this type of waves this type of function which is just a summation of two sine terms here k prime is the average k which is 2k plus delta k divided by 2 and omega prime is the average of the two omegas 2 omega plus delta omega by 2. Now if you look at this particular type of expression this square bracket you can see is also like a wave like disturbance for this is a cosine term so there will be small amount of phase difference because no longer a sine term but you realize that this is delta k this particular thing for the wave vector has a delta k for frequency it has a delta omega which is the difference between the two wave vectors because delta k is likely to be small because I have said that I am assuming that the wavelengths differ only slightly from each other so this delta k being comparatively smaller it means they you know the wavelength of this particular this particular term will be much larger so this is square bracket will sort of modulate this particular sine term which the original sine term which has is the average value of k and average value of omega. So if I plot this particular thing this is the type of behavior I will get which is generally called a beats phenomenon because as you know the amplitude of the waves depend on the intensity of the sound depends on their amplitude and here you will find that the amplitude itself is varying as a cosine functions. So basically you can say that this whole term which is this in the square bracket acts as the amplitude to this particular term so therefore amplitude is also varying as a cosine function while this is your original sine wave which you can call for example as a carrier wave and this is an amplitude so this is an amplitude modulated a sine wave. If you look that this particular the envelope here which is actually not a part of the wave wave is only these the disturbances are only this thing this particular line that I have drawn is only for a better visualization with better easiness to understand so this is your cosine term and these are the original sine terms which are oscillating and you can see that the amplitude is actually varying as a cosine function. Now my first question which I wanted to answer was what is the speed with which a wave packet will travel so before I go into that particular question let us look at this particular figure and try to sort of understand this particular figure. Now I have two questions without talking about what is the wave the speed of the wave packet let me just try to look at this particular figure and try to answer some questions. Now this is what I have done this disturbance I have plotted at three different time. Now this has actually been done using a computer so this is actually a correct way of representing this particular oscillation so this is let us say at time t is equal to 0 this is at a slightly later time and this is at twice this particular time. Now if you look at this particular point which is a zero displacement now you find that at a later time this displacement has shifted here then again if I go by the same time this zero disturbance point has shifted to this particular point. So if I look at how in time delta t for example how much this particular point has moved I have to take this distance and this distance and take the separation and divide by delta t. On the other hand if I want to ask a question with what speed this envelope moves so look at here this is the envelope. Here at this particular point in fact I have drawn a vertical line which is blue with color I am not sure whether you can watch the color very clearly but I have drawn this particular line even if it is not colored and there is a circle here this circle actually presents a point when this amplitude of this particular envelope becomes zero here. So you look at this particular point the amplitude of this envelope has become zero. Now if I have drawn the same thing at a later time you can see that this particular point where the amplitude of this particular envelope has become zero has slightly shifted to the right. So this point is now shifted slightly to the right and if I go to this particular time this has shifted again by this much amount. Now what you realize here that this envelope moves at a different speed if I have to find out with what speed this particular envelope travels you say that this speed has to be found out by taking this difference from this difference divided by time delta t. But as you can see very clearly that these original waves are traveling with much larger speed in the same time till delta t the original carrier waves moved much larger distance but on the other hand this envelope traveled with much smaller distance. So what I want to say that in this particular situation the way I have drawn these particular figures you can very easily say and you can very easily see that the envelope this particular envelope would move with a different speed in comparison to the speed of these particular carrier waves. Now you can always ask me the question what has this speed of the envelope got to do with the speed of the waves but as we will soon see this is a very very important aspect. Now let us look and try to find out what would be the speed of this particular envelope if I have to look at what is the speed of this particular envelope I should go back to my original equation which is here remember this envelope is given by this particular equation which is written in the square bracket and this particular envelope is represented by cosine wave for which the wave vector is delta k and for which angular frequency is delta omega. As we know that for a normal carrier wave omega by k is the speed so if I want to find out what is the speed with which this envelope travels the speed with which this envelope travels will be delta omega by delta k. So what I realize that delta omega by delta k will be the speed of this envelope while omega prime by k prime would be the speed of the carrier waves. If you have understood this particular aspect then we can try to define certain other things we can define something which we call as a group speed which is actually given by delta omega by delta k in the limit tending delta k to 0 which we call as a group speed. Group speed of a particular wave disturbance is defined as d omega dk while remember the phase speed was defined as omega by k. Now question is that is this speed same as the phase speed of course it will be same if omega turns out to be is equal to vk which as I if you remember in the morning I have said dispersion relation relationship between the frequency and the wavelength or for that matter omega and k is called a dispersion relation if the dispersion relation is of the type omega is equal to vk where v happens to be a constant it means essentially in other words in a simpler words that the speed of the wave does not depend on the wavelength. If it so happens that in that case if you take d omega dk that will also be v and omega by k also happens to be v in that case the group speed and what we earlier called as the phase speed will turn out to be identical unfortunately in many of the realistic waves it does not happen that omega is equal to vk type of simple dispersion relation is obeyed many times you have many more complex relationship between omega and k which we call as the dispersion relation and in all such cases you may not find the group speed to be same as the phase speed. Now before I go and try to physically interpret how I am going to use this concept of group speed let us try to see how a localized wave packet could be formed. See remember we had started with an ideal wave let us just put time t is equal to 0 and instead of sine wave let us go to cosine wave just for convenience sake because as I said I can always switch over from sine wave to cosine wave the only thing you remember that for sine wave at x is equal to 0 and time t is equal to 0 the displacement will be 0 for cosine wave it will be maximum. Now let us put t is equal to 0 so that we do not worry about the time dependent part we have to only look at the x dependent part. So let us assume that we are visualizing the entire wave at time t is equal to 0 so time dependent part goes away and now let us assume that we have two different type of mechanical wave remember I am not talking of anything very fancy just simple wave because I want to reduce a very very simple concept. So I have a wave let us assume which has a wavelength of 10 millimeter and another wave which has a wavelength of 11 millimeter so 2 pi by lambda which is the k will be 2 pi by 10 of course x has to be determined in the unit of wave millimeter because remember this quantity has to be dimensionless so if this 10 has to be expressed in millimeter this x also has to be expressed in millimeters. Psi 2 now will be 2 pi by 11 because these are the two waves which are the second wave which has a wavelength which is equal to 11 millimeter. Now let us suppose we try to superimpose two waves together so we will get something like this a 1 cosine of 2 pi x by 10 and a 2 is equal to cosine 2 pi by 11 x without asking and looking at what is the type of disturbance it will look like unlike the case of beats which we have done earlier. Let us just ask a simple question that we realize that at x is equal to 0 this particular displacement will be maximum which will be a 1 plus a 2 because at x is equal to 0 this cosine term will be 0 this cosine term will also be 0 then this particular thing cosine of 0 will be 1 therefore psi will be equal to a 1 plus a 2 so we realize that at x is equal to 0 psi will be maximum and will be equal to a 1 plus a 2. My second question is that at what value of x I will get psi to be equal to a 1 plus a 2 so at x is equal to 0 we had psi is equal to a 1 plus a 2 what is the next value of x for which I will get psi is equal to a 1 plus a 2 now you realize that as far as this term is concerned after every 10 millimeters this particular displacement will become equal to a 1 but as far as this particular term is concerned after every 11 millimeter this displacement will become equal to a 2 because only when after it becomes not 11 but you know after 11 so then it becomes 2 pi cosine 2 pi now in that particular case let me put it at x is equal to 10 millimeter at x is equal to 10 millimeter this term will show you a 1 but this term will not show you a 2. At 11 millimeter this term will show you a 2 but this term will not show you a 1 and will show something less than a 1 so neither at 10 millimeter nor at 11 millimeter this term will become a 1 plus a 2 okay you can very easily see that is only when we go to x is equal to 10 into 11 which is 110 millimeter then you will find psi 1 turns out to be equal to a1 plus a2. So, if at x is equal to 0, psi is equal to a1 plus a2 which it happens then next time when psi will become a1 plus a2 will be at x is equal to 110 millimeters. Now, let us try to do one more thing. Let us try to add one more wave to this particular term and let us assume that this particular wave has a wavelength of 13 millimeter. I have purposely taken things which are sort of odd numbers here so that you know we do not have any issues about those things. So, let us just go to this particular thing and let us try to add another term which is psi is equal to 2 pi by 13 x. Now, my question is that again at x is equal to 0 I will get amplitude of a1 plus a2 plus a3 or other displacement to be a1 plus a2 plus a3. Now, what is the next value of x at which I will get the displacement to be equal to a1 plus a2 plus a3 then you can see very easily that now it will be 10 multiplied by 11 multiplied by 13 millimeter. So, you are seeing that if my amplitude is very large then you will have some sort of disturbance here then next time my amplitude becomes larger. This distance is going away, becoming more and more when I am adding more and more term by adding my third term it has become 10 into 11 into 13 millimeter. If I add one more term let us say 10 into 11 I mean 13 into let us say 17 then and 10 into 11 into 13 into 17 millimeter I will get again an amplitude a1 plus a2 plus a3 plus a4. So, more number of waves I try to mix up you will find that this large value amplitude will become again high only at very very large values of x. So, you can sort of extrapolate it and sort of imagine that if we really mix infinite number of waves then once the amplitude has become very large the second time when the amplitude will become very large will be essentially pushed to infinite term and you will be able to get more or less a localized wave packet. So, this is what I have said a localized wave packet can be constructed by superposing infinite waves which are varying in wavelength. Now, in principle there is a technique which is called Fourier transform which can be used it is possible to decompose any wave packet into their component ideal waves. So, in principle any localized wave packet can be written as integral of 0 to infinity ak sin kx minus omega t again I have put sin but as I said sin and cosine can always be reverted back that is not a very important thing or in other words what I can say that if I take various sin terms which have different values of k and I mix them up with an amplitude ak dk and integrate from 0 to infinity I will be getting a localized wave packet which will be represented by yxt or I can do the other way whether we are taking Fourier transform or inverse Fourier transform if I know a particular value of yxt in principle I can always find out what will be the value of ak dk what are the waves which have been mixed up in this particular thing in order to generate a wave packet. So, basically the moral of the story is that a wave packet essentially consists of infinite number of ideal waves which are differing in their values of wavelengths that is the way we are going to construct a localized wave packet in this space. Now, let us try to come back to our classical particle and a classical wave if you look at the classical picture of a particle we always say that this classical picture is localized a particular particle is localized if a particular particle exist in a particular part of the space or let me write it here if a particle is existing here it exists nowhere else it is a completely localized entity if I have to detect the property of the particle I have to send some particular signal at this particular point for example, if I want to find out whether the particle is located there I have to send a light signal the light has to get reflected from that particular particle has to come and reach our eye then only I will be able to find out whether the particle exist here. I cannot detect the property of the particle I can if I have to find out whether this is a particle on this particular table okay for that matter there is a person in this particular room okay only I have to watch that particular person I have to look at that particular person only when the light reflected from that particular person comes and reaches our eye then only I will be able to detect it. Similarly, if I have to detect the particular property here, unless a signal goes here and gets reflected back or gives the information to our brain or to our eyes or whatever it is, then only I will be able to say that the particle exists here. I cannot perform an experiment here and detect the particular property of the particle here. The particle is here, it exists here, it does not exist here, it does not exist here. So, this is what I mean by localization, classically a particular particle, a classical particle is supposed to be localized which is existing only in a certain region of a space. While as you know an ideal wave as we have already said is infinite, it is represented by let us say sin term sin kx minus omega t irrespective of the value of x. So, a ideal wave exists everywhere for all values of x. So, if you have to detect a wave for example, if you are you can imagine yourself sitting by a beach in a near an ocean and let us assume that there is a long wave which is coming and hitting the, let us say beach, okay. Now you can, one particular person could be watching at one end of the wave and could say that okay there is a wave coming, another person could be sitting at another end of the wave and could find out that the wave is coming at that particular side. So, wave is an elongated thing, it is not really localized, it extends in a region. So, different people performing experiments at different ends can locate the property of the wave. Let us say that my wave exists, wave is not localized but a classical particle is localized. So, I have said in order to detect the property of a particle some signal must be received from the place of localization. For example, if I have to find out what is the color of this particular particle, I must send the light and the light must come back to my eye then I can say whether this particle is red or blue in the color. An ideal wave on the other hand is infinite, people sitting at different places far away from each other can simultaneously detect the presence of a wave. So, as I just now mentioned that two different persons could be sitting at two different ends of the beaches and a long wave comes without knowing that people could be very far off they can simultaneously detect the property of the wave and say that oh a wave is coming and hitting us. Now, remember that we are talking of wave particle duality, these are two contradicting properties a localized entity and an entity which is an extended entity, an extended wave and a localized particle. These two things have to coexist in a quantum particle because when we are saying there is a particle's quantum, in a quantum particle we have both wave character and a particle character it means I must built in both these localized and being localized and being extended state together. What sort of problem does it create in our understanding, let us just look into that. A particle behaving like a wave we have just now said a realistic wave is like a wave packet and has somewhat localized. So, we have just now said that actually realistic wave I mean an ideal wave never exists what exists is only a realistic wave and a realistic wave is always localized. So, actually what should be associated to a particle should not be an ideal infinite wave but it must be a localized wave. So, we are sort of trying to sort of consolidate our views particle has to be localized wave has to be extended but we know that we can localize the wave also in a certain way and therefore and we have also said that is more realistic type of wave. So, probably what is a wave which is sort of associated with a particular particle is only a localized or a wave packet type of thing. We even said that the wave associated with a particle in the aspect of wave packet form is what we have mentioned that a wave associated with a particle is also has to be of the wave packet form. But nevertheless whatever is the extent of the wave packet it cannot be zero while a quantum particle for example electron could be localized to a very very small region which is extremely small but I mean a wave whatever it is we generally expect it to be somewhat extended. Now question is that whatever is the extension of the wave packet in principle it is possible to detect the property of the particle at any of the two ends. So, if you have a localized wave packet as we have mentioned earlier something like this if this type of wave packet is associated with a particular particle a person sitting here can also detect the property of the particle here a person it is just like a wave as I said a person sitting here also can detect the property of the particle. One can perform interference experiment or diffraction experiment here or some other type of experiment here and these two ends and here person sitting here also will be able to detect the property of the particle a particular person sitting here also will be able to detect the property of the particle just like an ocean wave. This could be one beach where one person could be standing here, another person could be standing here and if this wave is coming out here like this, one can really detect the property of the particle. This is just a pictorial way of looking into this thing. So now let us try to look at, so this is what I said. So the position of the particle becomes uncertain to what is the order of width of the wave packet. So this is what I am trying to picturize here. So let us suppose this is an ideal wave and I have shown that there are three rather five persons which are sitting at different positions and this is a wave which is sort of they are trying to watch this particular wave. Now this person sitting here can perform an experiment here. This person sitting here can also perform an experiment here. This particular person can also detect the property of the wave here as I mentioned just like ocean wave. This particular person sitting here can also look at the property of the wave here. So if you ask this person, do you see a wave or do you find a wave, he would say yes. If you ask a person here, he would also say yes. I can detect, I can find out what property of the wave. So one does not know where does the wave exist, wave exists everywhere. So if this wave represents a particle, in principle particle can be detected here, particle can be detected here. All the five persons can actually perform an experiment and detect the property of the particle there. About the position of the particle has become uncertain. It is not sure whether the particle exists here or here or here or here or here. But now let us suppose we try to create a wave packet which is more realistic and let us suppose we have created a wave packet something like this, which is this much in elongation. Remember it is just a cartoon, let us not try to go into the deeper in which direction the wave packet is moving and things like that, this is not the idea. The idea is only just to do a cartoon, just to give you the idea of what is the concept for localization. If this particular wave packet is localized by to this much amount, now this particular person sitting here, here this particular person says that cannot because the amplitude has become zero, the displacement has become zero here. So this particular person will never be able to find out the property of the particle. This person will also not be able to find the property of the particle. So these two persons at these two ends would not detect any particle. But on the other hand these three persons here, they would say that okay, they can see the particle here. If I shorten it further, then even these two people and these two people will also not be able to say where my particle is there and only this particular person will be able to detect the particular property of the particle. So basic idea is that, that if I shorten the wave packet, more I shorten, more I make the wave packet, uncertainty in the position go down. I make the position of the particle certain. Let me repeat, an ideal wave will have an infinite uncertainty in the position because everywhere a person sitting there can detect the property of the particle. So one can never find out where my particle is actually located. Once I have localized it to some extent, then I am in a better position to specify the position of the particle. If I localized it still better, then I am still in a better position to tell the position of the particle particle. So my position of the particle is making more and more certain as I am localizing the wave packet more and more. So this is what I said, can we shorten the wave packet? Yes, we can shorten the wave packet to any amount, but in case I have to shorten, what is the prescription that I have just now mentioned that I have to superimpose more and more ideal waves in a longer and longer range of the wavelength. For this particular statement, I am not qualifying it very well, but statement I am just trying to tell from the Fourier analysis, one can always see this particular thing because as I said for every disturbance you can actually do a Fourier transform and find out what are the waves which have been mixed up in order to generate that wave packet and what has been there the weightages. So as we said that if I have to shorten the wave packets, then I have to impose larger and larger range of wavelengths. Once I impose larger and larger wavelengths, then I will be able to make this particular particle, this particular packet localize more and more. So if I want my particle to be much better localized, then the wave packet that has to be associated with that particular particle has to be extremely short. But in order to create that extremely short wave packet, I have to use much larger range, much larger number of ideal waves with varying wavelengths in order to actually localize the particle more and more. Now at this moment let me try to ask a question, what is the wavelength of this particular wave packet? See because this is an ideal sine wave or cosine wave, the value of K that I am using in this particular equation is going to be a fixed value and K is equal to 2 pi by lambda. So this particular wave is being given by a fixed value of wave vector or fixed value of wavelength. So wavelength of this particular wave is precisely defined. It is just one single value of wavelength which is going to give me this particular wave type of disturbance. Now once I have said wave packet, then as we have just now said, this wave packet is created by superimposing a large number of ideal waves which have different wavelengths. If they have different wavelengths, then I do not know what is the wavelength. I can of course approximately calculate what is the wavelength but this is not an ideal wave. Therefore I cannot say it has a fixed wavelength. So its wavelength range has changed or its uncertainty in wavelength has become larger. More and more I try to localize the wave packet. I have to use more and more number of waves with larger range of wavelength. So its wavelength values are also becoming more and more uncertain. That is what I am asking value of wavelength. A unique value of wavelength can be assigned only to an ideal wave. Larger range of wavelength means that their wavelengths have become more and more uncertain. They are not very certain what is the wavelength. Only for an ideal wave you can define a precise wavelength. But we have just now seen rather in the morning class that this wavelength is actually related by de Broglie relationship to the momentum of the particle. So if the wavelength of the particle or wavelength of the wave associated with the particle is not very well defined, it means its momentum is also not very well defined. So there is a uncertainty in momentum. Now we have landed up into very very funny situation. If I want to define its position very very accurately then I have to localize it to a very very large extent. It means its momentum, its wavelength has become more and more uncertain and therefore its momentum has become more and more uncertain. On the other hand, if I do not mind I want to define very very clearly its momentum then I must use only a unique wavelength. And if I have to use a unique wavelength I will get an infinite wave and then its position becomes very very uncertain and it becomes infinitely uncertain. So more I want to localize my wave packet, my momentum becomes uncertain, more I want to localize my momentum, my position becomes uncertain. This is what is called uncertainty principle which Heisenberg has realized. This is the basis of uncertainty principle. An ideal wave has a precise wavelength and hence its momentum is very precise. But then this wave extends right from plus infinity to minus infinity hence it has infinite uncertainty in its position. More we want to localize the wave packet we make the position of the particle more precise but then lose control on the precision of the momentum. Momentum is no longer precise. Uncertainty principle is the one which is needed to protect wave particle duality. We will give more examples just now to say how this uncertainty principle protects the wave particle duality. If you have to have wave particle duality uncertainty principle is inherent there it has to be present there because if something has to have both particle character as well as wave character then uncertainty principle must exist because we are trying to mix up two contradictory things of one being localized and another being extended together in the same entity. And if they have to exist together then I have to tolerate uncertainty principle. It means the way we are looking at the nature has to be slightly changed.