 Let us start, we had received some questions which we could not answer in yesterday's lecture. So let us first begin with those set of questions. So first request which has come, it came from Matha Vaishno Devi University, Katara saying that please specify the meanings of symbols in writing while giving the expression so that we can relate easily. The thing is that you know most of the symbols at least which I have used are sort of standard. So therefore I have not sort defined for example you know if I am using H it is very clear its blanks constant. If I am using C it is very clear the speed of light but still I will try to take care of this particular thing, K for example for Boltzmann constant but you know still never the less whenever it is possible I will try to incorporate these things. Now the second question is from Netaji Subhash engineering college from Garia Kolkata, say can you explain the term e to the power t to the power 5 in black body radiation curve. I am not very sure what actually the question means. The basic thing is that you know you have when we try to write a black body radiation curve universal black body radiation curve. So let me put it like that, let me explain something like this that if you plot the radiation which is being emitted at a particular temperature this is both a function of lambda as well as t. So if I take the total emissive power plot as a function of let us say lambda you will find that this is both a function of temperature and wavelength so it means at this wavelength the emission is slightly lower as we increase lambda emission becomes higher at this particular value of lambda the maximum power is emitted then it again keeps on going down. Now this is also a function of temperature if I increase the temperature then you will find that this curve goes somewhere like this and there is something which is this displacement law which says that this particular lambda where you see the maximum intensity lambda m into t is constant these are all the part of the black body radiation curve. Now as we can see that this is basically a function of both lambda and t however if we plot what we have written in that particular figure e lambda upon t to the power 5 sorry t to the power 5 as a function of lambda t all these curves can be merged into a single curve which becomes something like this. This is what is called universal black body radiation curve this is basically an experimental fact that all these curves which you are getting a different temperature this is let us say t1 this is temperature t2 okay if you further increase temperature it will probably shift it will shift the maximum will shift to lower values okay and of course you know the Stephens law which says that total area under the curve it means total power radiated will be proportional to t to the power 4 this is what is called Stephens law. So there are various laws which are pertaining to the black body radiation curve but the curve which I have drawn here was something which is called a universal black body radiation curve because in this particular case all these curves which you are plotting as a function of lambdas can be merged into a single curve. So this is just an experimental fact nothing else and this particular curve as I was saying that most of the other features one could have one could have explained by using classical theory other than the shape of this particular curve and only for the for explanation of the shape of this particular curve eventually Planck had to made his hypothesis. Now the second question which was there from College of Engineering Bhuvaneshwar does the flexibility of a bond depend on the pressure of the gas. Now actually the bonds are little more complex than what we have mentioned I am not 100% sure whether it does depend on the pressure of the gas but you know my gut feeling is that probably does not depend on the pressure of the gas or the same question is for the vapors also because whenever we are trying to approximate bonds by a sort of a harmonic forces that is an approximation the bonds are much more complex because it depends when you are bringing 2 atoms together what are the forces which are actually leading to the bonding of the atoms and also what are the forces which are repulsive forces when you are bringing the atoms to close to each other. In fact if you plot the force or potential energy diagram it essentially shows something like this curve something like this and this is inter atomic spacings and let us say this is potential energy and this particular curve you know is really not harmonic so it is much more complex it depends really on inter atomic forces. So normally I would not expect that once you apply pressure I mean eventually what could happen that the mean value of X could change but the shape of this particular curve I mean essentially is determined by what is this particular force which is the repulsive force and what is this particular force which is the attractive force which is essentially responsible for forming the bond. Now let me go to the next question which is actually a question about I mean this is of course a comment saying that instead of giving birds eye view over the quantum ideas please take an idea this is from Vellor VIT Vellor please take an idea and give a real time example so that we can explain to our engineering students. See basically I have the quantum ideas I have really given a birds eye view basically because I have a feeling that most of the students have studied this in their 12th standard and why I want to repeat these things because essentially I want them to follow a particular way of thinking and therefore I put them in my perspective so that I can eventually lead to the proper quantum mechanics when I actually come to the quantum mechanics which is a topic which is probably new for the students there I will be much more rigorous but here for example if I want to explain let us say black boundary radiation curve on its own self or for that matter let us say specific heat of solids or for that matter photoelectric effect or Bohr's model this itself becomes a part of the course while most of the students in their 12th standard in some way they are aware of these things and they are not really pertaining I mean they are not really very important to the main feature main theme of these lectures which are essentially leading to the quantum mechanics. The only one thing which I have mentioned and which I will be doing later is the Compton effect because Compton effect required a little bit of understanding relativity and because of our logistic issues we are going to do relativity later after we finish the quantum mechanics portion so at that time I will revisit the Compton effect other than that that would be my say I mean if you are really interested in some of the sort of basic quantum ideas to be explored I do not think this is possible as for the scope of this workshop is concerned maybe one has to conduct a different workshop to give details of let us say Bohr's model or you know black body radiation or whatever it is. Now next question is from Ranga Swami College of Technology Dhamakkal Tamil Nadu if light is considered as a particle how colour can be classified actually this is an interesting question but you know remember colour see like it is classified for a wave it can also be classified for the case of photons for the particle nature because see the colour perception is more of a physiological thing okay how the signal comes to the light and how the brain actually sort of decipher this particular thing and gives the signal of colour I do not think in any way depends on whether you are talking of the wave nature or you are talking of the particle nature see eventually we can always talk of the frequency also in the photon because it talks of basically of the energy okay now energy and frequency both are related as far as the wave particle and the particle aspects of the light is concerned so like we can start talk of perception of colour as far as the wave nature is concerned we can exactly talk in the same way of the perception of the colour of a photon. The next question is from Noida JPE Institute of Information Technology regarding the experimental evidence of black body radiation curve this black body radiation curve is only an experimental curve it is not a theoretical curve so there is no question of a you know sort of a experimental evidence this is what has been experimentally observed people had performed the experiment by taking bodies which can be approximately treated as a black body and for that black body was generally treated as a cavity in fact one of the ideal way of dealing a cavity was something like this where you have a sort of a small cavity with a very very small hole in this so any radiation which falls within this particular cavity always gets reflected back and this particular it never has chances of coming out so this is a body which absorbs all the radiation essentially which is coming out of it and then you have this particular hole because you want to wash what is the radiation which is being emitted by this particular thing so that is why we always talk of cavity when we talk of black body radiation but on the other hand the curve which I have mentioned here this particular curve is actually an experimental curve okay and what was actually required was to theoretically explain this particular curve and that is how the Planck's model come into picture. Now next question is from Vila Institute of Technology Durga Chhattisgarh is there any phenomenon which describes both particle and wave nature simultaneously I mean the thing is that this was generally called complementarity principle I think first proposed by Bohr's which says that both these models both light particle and the wave model cannot be simultaneously observed however as far as I know that this particular thing has been questioned quite a bit recently and there are some groups who have been doing some experiments to see whether this particular complementary principle really works but as of now general belief is that you cannot observe both the phenomenon together unless you know we are very very sure and experimentally it is proved conclusively that it can be done. Then the next question is from knowledge Institute of Technology Kaka Palayam in Salem is massive particle can emit de Broglie wave even though if the velocity of the particle is less than C. Of course you know the de Broglie relationship is correct even in the case of relativistic particles but you know here we always talk of non relativistic particle in fact the quantum mechanics that we are going to describe is purely non relativistic and this lambda is equal to h by p okay is valid both when particle is I mean when the it is a relativistic particle or non relativistic particle only thing the relationship between momentum and the velocity one has to take a relativistic expression about which we will describe discuss later when we come to the theory of relativity but whether we are using classical or whether you are using the relativistic expression this expression lambda is equal to h upon p is always correct. Now it says massive particle can emit de Broglie wavelength I know there is nothing like emission of the de Broglie wavelength it is supposed to be a wave which is associated with the particle so there is no question of emission of absorption this is just a particle possess that wave type of property. Now this we are going to discuss later probably in the tomorrow's lectures that when we are talking of the wave property what does this wave mean because see when we say for example the mechanical wave we know that mechanical wave it means the vibration of the atoms or vibration of molecules when we are talking of electromagnetic wave we know that the variation that we are talking is actually the variation of the electric field or the magnetic field okay but when we say that this is a wave particle wave or de Broglie wave associated with the particle what is the nature of that wave what is oscillating there this particular thing we are going to discuss in the our later classes. The next question is Shri Ramakrishna Institute of Technology from Coimbatore is it possible to combine two-way packets of course yes in fact this is one of the most important things which I will be discussing today is that you must be able to merge these two-way packets see if we cannot merge we cannot have interference we cannot have diffraction because essentially it is overlap of the two-way packets so the two-way packets can definitely be overlapped and this is one aspect which eventually will lead to the type of Schrodinger equation that we are going to discuss. So this is an important aspect and super position principle that two waves can superimpose on each other is a very very important aspect aspect and which has to be which has to be possible and this is possible. Then another question as again from VIT University Valor says we are aware that Stefan Boltzmann law holds good for all wavelengths in what way it matches with the religion's law as I had mentioned even in the earlier thing the religion's law essentially tries to tell what is the spectral distribution of the emissive power which means of course this is not really religion's curve but you know this is a sort of a experimental curve now this is actually spectral distribution it tells what power is emitted how much power is emitted for a given wavelength now Stefan's law talks irrespective of the wavelength it means it talks of the total power emitted irrespective what is the wavelength so if this particular curve is integrated then we will get the total power which is emitted at a given temperature. Similarly, if I integrate this particular curve this will tell me the total power which is emitted at this particular temperature and as I mentioned that this particular power which is emitted will be proportional to t to the power 4 sigma t to the power 4 and that is the Stefan's law. So Stefan's law talk of the total power emitted so essentially what we have been writing as e lambda d lambda I have to integrate it from 0 to infinity then what I will get is the total power emitted and that is what is turns out to be proportional to sigma t to the power 4. So Stefan's law again let me repeat is integral of all the wavelengths we take the total power emitted while when we are talking of religion's law or when we are talking of universal black boundary radiation curve at that time we are looking at the power as a function of wavelength we are looking little more finer we are going little more finer and trying to find out how much power is emitted at a given for a given wavelength. The next question is O. P. Jindal Institute of Technology a ragged from the experimental retains of hydrogen gas C v is equal to 3.5 R while for solids is 3 R why the gas is higher than solid of course this depends because it depends on the total degrees of freedom because in the case of diatomic molecule as we have seen okay we have basically 5 degrees of freedom if we have if this is rigid if this is not rigid then we have 6 degrees of freedom and 6 degrees of freedom one of them is vibrational. So you have 5 corresponding to rotational or translational and 1 corresponding to vibrational. So we have said 5 by 2 kT multiplied by Na of course plus this is kT multiplied by Na. So this gives you 7 by 2 kT which when you differentiate with respect to temperature you get 7 by 2 R on the other hand in the case of gas molecule there is only 3 degrees of freedom because each atom can only have vibration and it has 3 degrees of freedom. So obviously this specific heat turns out to be lesser. So there is nothing no contradiction here. The next question is from University BDT College of Engineering from Devan Gere in Karnataka. What is the substation for zero rest mass of the photon? This particular aspect we will discuss when we discuss about relativity because obviously in the classical physics there is no place for having a particle with zero rest mass but you know as we will be seeing when we are talking about relativity we can define a sort of a mass which depends on the speed when we are talking about relativity. So it is possible relativity gives an option where it is possible that where rest mass can be zero and this particular particle can still possess energy and momentum. Of course it could still not be present but there is a possibility that with m naught is equal to zero there is a I mean the particle could possess energy and momentum and then of course the only condition is that it must travel with the speed of light. So let me sort of postpone answering this question until the time when we are doing special theory of relativity and of course at that time you know you can ask the question again if you have doubts. Next question is from Don Bosco College of Engineering and Technology Azara in Guwahati. In the energy curve what is the physical significance of the wavelength temperature product? Again as I have mentioned that in this particular curve this is an experimental curve it so happens that using this particular way of expressing this particular way we are able to unify the all the curves which you which you would have obtained at different values of temperature. So let me just so all the curves that we would have obtained at various temperatures they can all be unified into one single curve if instead of this particular e lambda and lambda we plot e lambda upon 32 power 5 as a function of lambda t this is purely an experimental way of looking into it and I mean it is essentially makes it life simple because there is only one single curve which you have to bother about. Last question is from MGM College of Engineering Dandere CV is equal to 5 by 2 r for rigid molecules but you say that for flexible molecules is CV is equal to 7 by 2 r but according to the formula degrees of freedom is equal to 3 minus constraints that is 2 molecules degrees of freedom which is equal to 5 which I just now explained let me just try to once more see we have 5 degrees of freedom if we assume that this length does not change this length does not change because 3 degrees of freedom for this 3 degrees of freedom for this and there is one constraint which says that the distance between the center of masses of the molecules must remain constant in that case I will have 5 degrees of freedom in that case I will always get specific heat to be equal to 5 by 2 r on the other hand if I say that this particular bond is not very rigid and there is a possibility that the length of this particular bond can change it means it can vibrate the atoms can vibrate so it can the distance can go like this or can become larger while in the vibration then this particular constraint no longer holds good because the length of the bond is no longer constant so the distance between the two atoms will not be constant so that constraint is removed so you still have 6 degrees of freedom but one of the degrees of freedom becomes vibrational because in order that this particular length changes it can change only if the length of this particular bond changes and the length of this particular bond will change only when these atoms will vibrate okay and we have said for vibration there is also average of half KT energy for potential energy and half KT for kinetic energy therefore for that particular degrees of freedom that particular degree of freedom we will have on the average KT degree KT rather than 3 by 2 KT so for 5 degrees of freedom we will have 5 into half KT and for this last degree of freedom we will have 1 into KT this you must multiply by NA because they are total number of NA number of molecules so this will be the total energy and you take DE by DT this will be 7 by 2R on the other hand if this was not present and this term will not be present if the molecules are rigid because then this this constraint is applicable and then we have only 5 degrees of freedom in that case the specific heat will be equal to 3 by 2 5 by 2R okay so I think these are all the questions that we had received through the chat mechanism and there are some questions which had come by males which also I think I have tried to answer now let us come back to our lecture let us start with recapitulating what we did last time we first discussed the concept of localized wave packet and we said that it is a localized wave packet which actually represent a realistic wave okay an ideal wave is infinite both in position and time and that does not exist in nature what exists in nature is only a localized wave packet and that is why I call this as a realistic wave then we discussed that in order to generate a realistic wave I have to consider a large number of ideal waves which are obviously as just now mentioned are infinite but they must also vary in their wavelengths so if we mix such type of waves together it is possible to get a localized wave packet we also talk a little bit about the Fourier transform saying that how if I know a particular wave packet I can find out what type of waves it is consisting of what type of ideal wave it is consisting of then we sort of briefly mentioned that wave packet must travel with the group speed and this in the case of D Broglie wave turns out to be equal to the speed of the particle this particular aspect I have not discussed probably will discuss in one of the problem sessions that this particular wave packet would travel with the speed of the particle then we discussed about the double slit experiment and try to understand this double slit experiment which we understand very well on the basis of the wave nature try to mix up the particle model we try to think that these particles lab now we sort of discuss in the case of electrons these electrons can be treated as particle which we know that they can be treated okay then we try to and interpret the same experiment using the particle nature and then when we do this particular thing we landed up into difficulty we found out that there is certain point on the screen where when we had only one slit open certain number of particles were coming then we open both this this slits a larger number of particles a smaller number of particles start is starting reaching there which cannot be understood because if I would have opened the other slit if I have provided one more way for the particles to come and reach the screen the number of the particles reaching there must increase in other words that interference which is basically classically considered as a way property where it is possible to get a reduced intensity or zero intensity when the two waves come and interact with each other and have a destructive interference I am not I will not be able to understand purely from the particle nature because the particle nature tells that if certain number of particles are coming from one slit and if I have opened the second slit at least the number of the particles would remain constant but there is no way that it can go down so we have difficulty in understanding and then we realize I mean this is what we have said we cannot understand how the number of particles reaching a particular point on the screen can reduce when both the slits are open obviously means that we have to look at slightly different way into the nature and if we are talking to if you are trying to have a wave particle of duality then we will not be able to understand in that well as we can be able to understand that finally we of course course we talked little bit about the uncertainty principle we talked that how this wave particle of duality would lead into a uncertainty principle so this is the thing which I was trying to tell that if you are taking this particular point here initially a large number of points or let us say this particular point when only one slit was open you are getting a very large number of points here large number of particles here but when both the slits are open you suddenly start getting a very small number of particles which in the case of wave mechanism we can understand as a destructive interference but it is very very difficult to understand it on the basis of a particle model because here we are having very large number of particles by opening the second slit as if it appears that the electrons have decided not to reach there okay and even though I mean if the slit was close a larger number of larger number of electrons would have reached this particular point okay and as I mentioned that this experiment even if it is done so slowly that electrons are coming only one by one still you will see that these type of interference patterns will be seen. Now let us go ahead let us try to see whether we can wash these electrons in this particular experiment I mean watching is I mean if you want to watch electrons difficult as I say but this is sort of a thought experiment so let us assume that we can wash these electrons so what we can do that let us try to really see whether the electrons are coming this hypothesis the original hypothesis that we have put that electrons coming from slit S1 or slit S2 okay then I will let me put a light source here somewhere a light source and wherever there is a electron which is passing from slit S1 and I see a electron which is reaching here I can always find out whether this electron has reached from this particular slit if a light flash is seen and close to this particular slit then and an electron which is here then we know that this electron is the one which has passed through slit S2 so all the electrons which are reaching here I will be able to classify whether they are coming from slit S1 or S2 so let us try to do that particular experiment of sort of watching let you put what watch under under the quotes let us try to watch these electrons so can we verify experimentally if the particles indeed go through one of the two slits when both the slits are open so put a light source to watch the electron so just put a light source and try to see whether it is really going through slit S1 or through slit S2 when you do this experiment you will really find that any electron which is reaching on this particular screen has definitely come either through slit S1 or slit S2 it has something I mean some wonderful thing has not happened so that you can see that electrons are coming from somewhere else they definitely have passed either through slit S1 or through slit S2 in order that they reach this particular screen okay but on the other hand in this particular condition if you perform the experiment by taking by opening both the slits you will find that the interference pattern will be lost you will not be able to see the interference pattern we shall indeed find that the electrons will pass through one of the two slits but now interference will be lost interference pattern will be lost when both these slits are open you will actually notice that they are no longer interfering okay you will just see N1 2 will be equal to just N1 plus N2 okay then you will think what has happened you know earlier when I was not watching these electrons everything was fine I was seeing interference pattern but now was I have started looking at those electrons now I am finding that the interference pattern is lost okay then we suddenly realize okay of course photons also light sources also consisting of photons which are also particles and therefore these photons which have also energy probably can interact with these electrons and therefore they have changed the path of electrons or whatever has happened and essentially we have what we say that experiment has been perturbed the condition of the experiments earlier has become different from this because I have put a light source and this particular light source has perturbed the experiment and this perturbation has caused the interference pattern to be destroyed then we may think that okay let me try to do the other thing let me try to reduce the perturbation so what I do instead of using photons of higher energy let me go and try to reduce the energy of the photons it means I start using higher and higher wavelength because higher wavelength would be having a lower energy higher wavelength will means lower frequency will mean a lower energy so let us try to use lower and lower frequency or higher and higher wavelength photons in order to observe these particular electrons when you try to do that you will find okay so long you are seeing the flashes you will definitely find that the electrons are coming through slit S1 or slit S2 okay and you will not be able to see interference better but if you keep on reducing the energy of the photon the wavelength will keep on increasing and a situation will come that you remember the resolving power depends on the wavelength okay if the wavelength is very very large we cannot resolve two particles which are separated a distance which are typically order much lesser than the wavelength of the the light that is being used to wash them and in that particular case you will find that you see a particular flash of light coming from here okay because you are using very very large wavelength of light and you will not be able to detect whether the electron will is coming from S1 or S2 in that case only my perturbation will be lesser enough least enough so that I will be able to regain re obtain my interference pattern basically what I am trying to say that is not possible to perform an experiment by which you are sure that the electrons are coming from N1 and slit S1 or S2 and also observe the interference pattern together. So if I use low energy photons we can reduce the perturbation by using low energy photons if wavelength increases beyond a limit the resolving power would be reduced and this two slits may appear as one broad slit only in such circumstances we shall be able to get back the interference pattern the perturbation will be small enough so as not to disturb the original experiment. Now what sometimes it appears that uncertainty principle talks only about the perturbation of the experiment this is not really a correct concept because perturbation was known even classically for example if there is circuit electrical circuit this is some circuit if I want to measure let us say voltage between these particular point these two particular points then what I do I connect a voltmeter here okay but once I add a voltmeter here then you find that the current in the circuit changes when the current in the circuit changes then this is no longer the original experiment okay so you have perturbed the experiment similarly if I want to measure the current here in this particular branch okay then this emitter that I am putting will have a finite resistance and if it has a finite resistance this will alter the current so you are no longer having the original value of the current so you have perturbed the experiment so people think that this particular uncertainty principle is related to perturbation but remember in classical physics we always think that there is always a possibility of reducing the perturbation as much as we want and eventually get as accurate values as we want that is the way we always work it out in on the other hand in the case of uncertainty principle this is a sort of a fundamental principle this is not related just to the perturbation what is related linked between two different quantities one is position and momentum see as far as position is concerned we can also argue the same thing that we can always imagine that having better and better experiments I can I am able to measure I could be able to measure the position as accurately as I want but when I try to do that I will have to make momentum uncertain similarly if I want to measure momentum very very accurately then given the best type of instruments maybe I will be able to measure momentum as precisely as I want but when I make a measurement I will find that the position uncertainty becomes much larger so here we have two linked things in that way it is different from the simple perturbation that we are used to talking about in the case of classical physics so this is what I have said perturbing an experiment while performing in measurement is known also in classical physics but in quantum physics this comes as a basic requirement and it is coupled to two simultaneous measurements so I want to measure momentum simultaneously I want to measure the position okay if I want to measure position as accurately I could do it but at that time when I want to measure momentum momentum will become much more uncertain if I want to measure momentum very accurately I could still be able to do it but then at the same time I will not be able to measure position that accurately so these are two linked things and this is much more fundamental concept this is basically a fundamental concept of physics now let us try to I mean rather hurriedly try to get some idea of the uncertainty product what is the order of magnitude of the uncertainty product I will use two different methods get an idea of what is the uncertainty product so one method is using wave particle a wave packet approach remember when I am calculating this particular uncertainty here I am only evaluating the order of magnitude just to find out how large is the order of magnitude okay this is not really an exact way of calculation okay when we go I mean in fact immediately after that I will define uncertainty much more accurately and then I will try to talk about these things in a much more precise sense but at this particular moment we are trying to do just an order of magnitude calculation so let us go back to the old Beats problem and make certain assumptions some simplified assumptions to just get an idea when I am talking of measurement of position and momentum this product delta x and delta px uncertainty in position and uncertainty in momentum their product what is that order of okay what we will realize that the order is very very small which is typically of the order of Planck's constant which is what we call as h so let us go to the Beats problem now we take only one modulation of the wave packet so remember see this is what we had done about the Beats this is the old picture which I had shown yesterday when we superimpose two waves one corresponding to k equal to k another corresponding to k is equal to k plus delta k then we get this particular pattern but this is not really a completely localized wave packet because we have used only two wavelengths we have used just two values of k so it is I mean u it is not at all localized because this particular oscillations will continue right from minus infinity to plus infinity all right but let us assume we let us concentrate our attention only on one of these modulations so let us just take which I have great this particular area that let us assume that this is the only wave packet we ignore other wave packet and let us assume that I am not using only two values of k but I am taking two limits one is k another is k plus delta k and I am mixing all the wavelengths between k and k plus delta k in order to generate a localized wave packet which is of this particular shape which is given by the grade area okay so I am assuming that whatever I am getting here is actually a result of combining a very large number infinite number of wavelengths ranging between k and k plus delta k so this is my assumption which I am not justifying because my idea is only to get an order of magnitude of the uncertainty product not doing any exact calculation as of now so this is what I say assume that one modulation of the wave packet can be treated by mixing waves of not just two wavelengths but all the wavelengths between k and k plus delta k just to give you as I said in approximation now I of course assume that this particular wave packet which I have generated is one of the modulation which has been obtained as a result of this particular equation okay so we estimate the order of magnitude by using this particular method. Now if I look at this particular envelope of this particular modulation we have seen earlier that this particular modulation is given by this quantity in the square bracket while this sine k term which is larger k and therefore smaller wavelength is actually are these oscillations or these oscillations so these are my sine terms and this is the term which is in my square bracket which has lead to a variation of the amplitude all right. Now using this particular amplitude let us try to estimate what is the value of uncertainty product so what will be our way of looking into it see because this physical wave packet I am assuming its extends from here to here so whatever is the envelopes lambda by 2 half of the wavelength that will be the physical extent of this particular wave packet let me repeat I assume my wave packet consists of only this particular modulation so my uncertainty in the position of the particle will be somewhere between this and this so the order of uncertainty in position is this because as we have discussed last time if I perform an experiment here I will be able to detect the property of the particle if I am able to perform the experiment here I will be able to detect the property of the particle but not here if somebody performs an experiment here does not see this particular modulation so they will not be able to detect the property of the particle there so it means the uncertainty in the position of the particle is given by this much amount all right. So let us calculate what will be the value of delta x which I can calculate very easily by knowing the wavelength of this particular modulation which is the wavelength of that particular expression in the square bracket this particular square bracket so I know what is the value of wavelength because I know the k is delta k by 2 so I can calculate what will be the lambda of this particular modulation once I know the lambda of this particular modulation I know that this is half lambda because this curve will go like this and then go like this so that will be lambda so here will be the lambda by 2 okay so I calculate this lambda by 2 then I know what is the value of k is that I have used I have used values of k starting from k to k plus delta k so I know what is the uncertainty in the value of k which is delta k okay which I will relate to the uncertainty in momentum using de Broglie relationship so let us just try to do that so first let me try to express this delta p you know p is by de Broglie relationship is given by h by lambda I multiply this by 2 and divide by this by 2 pi so I multiply by 2 pi divide by 2 pi so this becomes 2 pi h this becomes 2 pi lambda now h by 2 pi is called h cross or h bar which is here all right then 2 pi by lambda is called k so this express of course h cross or h bar is constant so this will just be h bar delta k so uncertainty in the momentum can be written as just h cross or h bar multiplied by delta k now as I said position uncertainty can be evaluated from the wavelength of the modulation now if I take the wavelength of the modulation this will be 2 pi divided by the k of the modulation k of the modulation is delta k by 2 so this is delta k by 2 remember this particular thing the k of the see expression is k x minus omega t so k is delta k by 2 so the wave vector of this particular expression is delta k by 2 so this is what I am using here so this is delta k by 2 and because I want to convert this into wavelength so this will be 2 pi by k and k is delta k by 2 so this becomes 2 multiplied by 2 pi divided by delta k this will be the wavelength and I have to use half of its wavelength so I just putting half okay if you just take this particular expression this 2 will cancel from this this will just be 2 pi delta k so I see that delta p will be h cross delta k delta x will be 2 pi by delta k if I multiply delta p into delta x what you will find out that this delta k will cancel this will be h cross multiplied by 2 pi which means just it will be equal to h the Planck's constant so we see this purely in estimate of uncertainty product that as I mentioned that uncertainties are always are uncertainty principle as always is always written as a product delta x and delta p of what is the order of delta x and delta p the product of delta x and delta p so we see that delta x and delta p is of the order of h so you can see that this particular product is exceedingly small and that is the reason that this uncertainty we are never able to detect in our real life the Planck's constant being so small not able to see this particular effect of uncertainty principle in let us say macroscopic particle bigger particle let us say in cricket ball or whatever it is it is only when we are going to that particular limit of small particles that I will be able to get a feeling of this uncertainty principle the effect of this uncertainty principle will be seen or the effect of wave particle duality will be seen only when we are talking in that particular limit now let me just estimate by using another particular way of doing this particular thing this is also another interesting way of doing it so let us just discuss this particular aspect which is the wave particle duality approach now we just now considered what we call as a double slit experiment now let us consider a single slit diffraction this particular idea I had used earlier if you remember when we are trying to talk about n 1 and n 2 see when we are talking about this particular curve of n 1 and n 2 at that time we said that this will show a sort of diffraction effect okay now let us consider that there is only one slit there are no two slits and therefore we are only seeing this particular diffraction effect let us try to understand this particular behavior on the basis of wave particle duality and this will also bring some interesting application or interesting consequence of uncertainty principle about which I have found generally most of the students being confused so now we consider a single slit diffraction of a beam of mono energetic particles okay these particles again we can treat electrons for our sake I mean these particles have to be quantum particles tiny particles like fundamental particles okay let us assume electrons as these those particles and interpret this on the basis of uncertainty principle and define the order of product that is what I am going to do next so this is our single slit experiment now I do not have any two slits I have just a wall in which I have a single slit and let us assume that the this particular width of this particular slit is d again like before I have a source of particles let us see source of electrons and then I have a screen on the top of it then as you know the single diffraction the if I mean in a normal situation the this particular this particular beam should directly hit but there is always a bending and you will find that this particular thing shows you what we call as a diffraction pattern and you will find the smaller is the value of d the larger will be the value of theta this the angular separation between the maximum and the first minimum and this particular separation is given by the standard expression b sin theta is equal to l lambda okay where d you know n will correspond to first separation second separation etc etc this is a very standard single slit diffraction experiment which normally people are expected to know at the time of high school now what I would like to interpret again like the the Young's double slit experiment trying to explain this on the basis of particle invoke my uncertainty principle trying to understand this on the basis of sort of both wave particle duality try to somewhere invoke particle nature somewhere invoke wave nature and eventually from that derive the not really derive but get an order of estimate of the uncertainty product that is what I will do now so let us first understand this particular experiment of course this is a consequence as you know that smaller is the slit width the larger will be the width of the central maximum the experiment can be understood well by the wave theory now let us try to bring the particle nature assuming that the experiment is being performed with a particle of course even light is that in that sense is a sort of particles but as I say for light we always have a very hazy view in our mind so we are not so much concerned but when we try to talk about electrons which we know classically as a particle then only we start getting this shocking effects okay so let us assume that you have some source of particle which is let us say infinite distance away so let us suppose that this particular source of the particle is very very far off it is infinite distance away so if the source of the particles is infinite distance away essentially the particles which are reaching here they will all be parallel okay let us call this direction as a y this direction as x direction so essentially it will mean that the velocity of this particular electron because I am assuming that this source is pushed to infinite so all the electrons will be having velocity only in the x direction and therefore the y component of the velocity v y will be zero and will be quite certain as or let me put it like that in the language of the calculus as this particular source moves to the infinity the y component of the velocity will tend to zero okay and then of course uncertainty when we are very sure that the y component is zero the uncertainty in the y component of the velocity is also zero so this is what I have written there assume that the source is infinite distance away therefore electrons are moving purely in x direction therefore delta p y which essentially means why because p y is related to mass into velocity okay delta p y is equal to zero and obviously delta y by uncertainty principle would mean infinite essentially means that in this particular case you are never sure whether the particle if the uncertainty principle has to be holding good because your delta p y is zero it means delta y has to be infinite it means the particle could land anywhere here it could be here it could be here it could come somewhere anywhere here the uncertainty in the y component will be infinite which also you can understand because this source has been pushed infinite distance away so a particular particle which is being emitted from here will land up at this particular point may land up in this particular point will end up anywhere here so therefore as per uncertainty principle the y component of the velocity is zero and uncertainty in its position along the y direction is infinite so this is what I said hence we are not sure about the y position of the particle at all now what happens when suddenly these particles encounter a slit up till wall it was perfectly alright but suddenly these particles encounter the screen this particular wall and they encounter a slit now when electron is passing through this particular slit suddenly it finds that its uncertainty has been limited to something of the order of D now we are very sure that a electron which has reached from here could have gone must have gone from this particular slit because now there is no other way unlike double slit experiment where there was another possibility of electron being going from somewhere else the electron is only going from here so when at the instant when the electron was passing through the slit suddenly it finds that the uncertainty in the y position has become finite because we are very sure that the particle's uncertainty in position is of the order of D of course the particle could have gone slightly up could have gone slightly down but it has to be within the slit and therefore at that instead of time the uncertainty in position has been reduced from infinite value to d value what would that mean in in the language of uncertainty principle that delta p y which was earlier infinite has also has to go down so delta y is finite so delta p y cannot be 0 it means now when the particle is coming here if you suddenly because its y component cannot be 0 so therefore some particles may start developing totally in an unpredictable manner a y component of the velocity some electrons may acquire a y component of the velocity in this direction some may acquire in this particular direction so earlier when we are very sure that v y was 0 its no longer sure as we have as the particles have just started passing through the slit see delta y has suddenly become as particles are passing through the slit delta y has suddenly been made finite hence delta p y cannot also be 0 anymore hence the particles I mean this is very very important sentence to realize hence the particles would pick up a y component of the momentum in a totally unpredictable and uncontrolled manner these two words are extremely important okay it means I see in classical mechanics we are always talking of predictability okay if we know the initial conditions then we know at and we know where the particle is at t is equal to 0 we know precisely if I know the forces what will be the position of the particle at t is equal to 1 second it is 100 percent predictable in classical mechanics in quantum mechanics that is what does not happen okay things are not predictable okay that is what is uncertainty about okay so once the particular particle is coming here then I do not know what value of y component of the momentum it will pick up some may pick up larger some may pick up smaller some may go positive some may go negative okay but on the other hand it is totally unpredictable and totally uncontrolled I cannot control okay in what value of y component of the momentum this particular electron will pick up so let me read again hence the particles would pick up a y component of the momentum in a totally unpredictable and uncontrolled manner smaller the slit width smaller will be the uncertainty in y and therefore larger will be the uncertainty in the momentum all right so essentially as you can see that this falls in line with whatever we have seen from the wave theory it means the p y component will become larger and larger so the velocities in the y direction will tend to become larger and larger of course you can always ask me a question which is probably a correct question that if I know electron has reached here okay cannot I find out the y component of its velocity very very exactly I can of course find out because I can find out how much distance is this and how much time this particular particle would have taken to reach this particular distance okay if I know v y v x then of course I can find out what will be the value of v y and I can find out pretty accurately from my calculation but that is what that is not what uncertainty principle talks about uncertainty principle talks about when this particular particle was passing through this thing could I have predicted what will be the value of v y the answer is that I could not have predicted okay one particular particle would have picked up this particular value of velocity so that it landed the particle here another particle would pick up a velocity of y component so that lands up here third particle may land up here fourth particle may land up here and I cannot predict I cannot control I cannot prejudge and say that okay this particular particle which is now passing through this slit will now pick up this specific value of v y and therefore it will land here or here or here once it has landed there of course we can find it out but I cannot predict so that is the reason I had emphasized this particular aspect of predictability and controlling I cannot predict and I cannot control while in the case of Newton's law of motion I could have predicted the thing so that is why I have put the question what about Newton's law in Newton's law I would have predicted if there is no force on the particle the particle must keep on going straight okay if it has deviated from its path okay obviously there has to be force on it and then once I know the force I can predict where this particular particle will land up that is what was the classical mechanics that was that is what was Newton's Newton's law but now because of this mixture of wave particle duality I hope you can appreciate because there is a mixture of wave particle duality therefore you see that this particular particle will totally in a random fashion totally in uncontrollable fashion will pick up a y component of the velocity which I will not be able to control now let us use the wave formula to predict the order of uncertainty product let us assume that electrons land themselves only in the central maximum of the diffraction pattern again I am doing order of magnitude calculation so I am ignoring these other wings here which I am getting here so I am ignoring these particular wings here okay so I am assuming that the electron is landing only here anyway this particular part is the most intense part so what I will try to do how much is the component v y required to put this particular particle let us say here okay I will calculate from this this will be my delta y from this I will try to calculate what will be the value of delta v y from that I will try to calculate delta p y and eventually take the uncertainty product delta y into delta p y so first let us try to calculate the time taken by the electron to reach the screen after passing through the slit okay l is supposed to be the distance between the slit and the screen let me just redefine things this was l this is the distance between the wall and the screen remember this distance this is the slit width this is d theta is the angle corresponding to the angular separation between the maximum and the minimum y naught is the distance in the screen on the screen between the maximum central maximum and the first minimum okay so we have to remember these things this theta y naught d l there are four quantities I have defined l distance d is the separation between the width I mean the slit width theta is the angular separation y naught is the distance seen on the screen so if I assume that the particle we are coming with a velocity v x in the x direction then the time it will take for the particles to reach the screen after they have just crossed the slit will be l by v x this is the time these particles will take during this time the electrons also gets displaced along the y direction due to the random velocity acquired after passing through the slit due to uncertainty principle this is what we have just now discussed that uncertainty principle causes these particular particles to pick up randomly a y component of the velocity so because of that there will be displacement of the particles along the y direction if vy is the y component needed to take an electron to the end point of the central maximum this is what I have earlier said so y naught is the distance of the first minimum from the central maximum and let us assume that vy is the velocity component in the y direction which is required for the electron to reach at that particular point then I can find out this y naught must be equal to see this l upon v x was the time and vy was the y component of the velocity developed okay so velocity multiplied by time will be the distance that it will travel along the y direction okay and this must be equal to the y naught so this I am writing y naught is equal to l vy upon v x now I am multiplying by mass to convert into momentum because uncertainty principle is expressed in terms of momentum so I have multiplied by m so this m v y becomes p y and in denominator we have m times v x so y naught the distance between the central maximum and the first minimum on this screen will be given by l p y divided by m v x now I have to also use this particular I will also calculate this y naught using the wave theory because that is what is the concept of wave particle reality this y naught is being calculated using uncertainty principle I will calculate also y naught from the pure wave theory I will equate the two and then I will find to try to correlate and find out what will be the uncertainty expression so now we know that for the first maximum the angular separation will be given by d sin theta is equal to lambda that is the standard diffraction formula that again we know at the high school now let us assume that theta is very small then sin theta can be approximately written as theta okay of course theta has to be expressed in radians so this expression becomes theta d is equal to lambda now theta d I as you can see that theta will be given by y naught by l let me go back to this theta will be given by y naught by l remember theta has to be expressed in radians if I write sin theta is equal to theta that is valid only when theta is in radians and this will be given by y naught by l so this is y naught upon l into d this must be equal to lambda this lambda by de Broglie relationship has to be given by h by momentum so I am assuming that v y component is very very small so momentum is totally given by only the x component so this I can approximately write as h upon m v x using this relationship I can calculate what is y naught I will multiply this by l and divide by this d so y naught will be given by h times h times l divided by m v x and this d will also go into denominator so this will become h l upon m v x d so I have obtained two values of y naught two expressions for y naught this expression for y naught was using purely a diffraction purely a wave property earlier I have found a y naught assuming uncertainty principle of course I have not used any particular form of uncertainty principle I have only used that how this uncertainty principle will cause a y component of the velocity to develop now I equate these two y two then try to look at the uncertainties and obtain what will be the uncertainty product so we equate the two values of y naught so one I had obtained earlier using uncertainty principle this is using wave nature if you put this particular thing what you will find out this l into m v x will cancel so here you will have only p y and here you will have this l will also cancel so here you will have will be equal to 1 upon d and there will be h here which does not cancel out so l cancels out m v x cancels out so you have here p y here you have h upon d so this implies p y into d must be equal to h now if both these relations have to be same that this expression must be valid now this d I can identify it as the uncertainty in the y direction and this p y is the momentum component of the y in the y direction which has been picked up due to the uncertainty in the momentum so this I can identify as the uncertainty in momentum so we can see that delta p y into delta y must be equal to h so take approximately d as the uncertainty in y and p y as the uncertainty in the momentum then we can write delta y delta p y is of the order of h so this is what we we see that the value of Planck's constant turns out to be very very small maybe some of you would like to mention to your students a very wonderful book by written by George Gamow which is it is called Mr. Tomkins in the Wonderland and this is about a student who starts learning about the modern physics and there are various chapters in this particular book one of the chapter talks about a quantum jungle where a particular person goes to hunt in the jungle and there the value of Planck's constant is very very large and then the loam comes and because of uncertainty principle the person is not able to find out where the line is there is a very wonderful reading book a very nice way of representing what would have happened if Planck's constant would have been very very large in our real life and we would have started seeing uncertainty in our daily life similarly there is another chapter of relativity where it talks of a bird where the velocity of light is exceedingly small and in that case you will start seeing relativistic effects in your daily life then how the life would look like it is a very very interesting picturization of that particular thing now let us actually define uncertainty because in fact some of the people have even asked this question that you are talking in a very very hand-waving way okay now this is the time when we will start going from quantum ideas to more formal quantum mechanics of course once we actually use a wave function we will also talk about the uncertainties at least in one specific case we will definitely talk about it but now let us go to little more definiteness a uncertainty product is defined as the standard deviation in whatever well the particular expression that we are talking if you are talking of uncertainty in position essentially it means that you make measurement of position of that particular particle n number of times then you take the standard deviation of the position value when you take the standard deviation whatever is the standard deviation that you get will be called this the uncertainty in that particular component or uncertainty in x in this particular case the example which I have given you all right so this is the way we exactly define uncertainty principle uncertainties then it can also be shown which we will not be showing that of course uncertainty product always depends on the type of the wave packet that we have got and different wave packets will have different minimum uncertainty product but it can be shown that the absolute least value of uncertainty product that you can get for any wave packet is delta x delta px greater than or equal to h cross by 2 and this happens for a Gaussian wave packet which you get in the case of harmonic oscillators which probably is beyond the scope of this particular course but just to mention that h cross by 2 is purely absolute minimum so you will not be able to find out wave packet in which delta x and delta px product is smaller than h cross by 2 then there varies other uncertainty principle one of the other important uncertainty principle is time energy uncertainty principle. So, remember when we are talking about the wave packet we were talking of it being limited both in position as well as a time all the examples that we have given so far we have limited the wave packet only in position we have not bothered about limiting the wave packet in time. If I limit the wave packet in time domain in that case almost whatever we have said about position also becomes true for the time and therefore we get because remember in the case of when you are writing the expression for wave k and x comes together. So, there is a uncertain relationship between k and x this k is related to wavelength x is related to the position omega is related to frequency and t is related to time. Now, this omega is also related to energy because of h cross omega is supposed to give you energy therefore this if you if you limit the wave packet in time domain this creates uncertainty between energy and the time exactly in the same way as it creates uncertainty between position and the momentum. So, you have other uncertainties let us come back to the lectures. So, for example, if an electron spends a very short time in an excited state before an spontaneous emission then the time spread of the emitted wave cannot be larger than the time it has spent on the excited state. Therefore, it causes a spread its energy and then you find the value of energy of a photon which is emitted is also uncertain and it is also so to say broadened. So, this is called time energy uncertainty principle. So, we have various types of uncertainty principle which I have listed here delta x delta p x is of the order is greater than equal to h cross by 2 delta y delta p y greater than equal to h cross by 2 remember delta x is always related to delta p x delta x is not related to delta p y similarly delta y is related to delta p y it is not related to delta p y p x or delta p z. So, delta x delta p x is greater than equal to h cross by 2 these are the various uncertainty principles which are essentially useful as far as this particular part of the course is concerned. Now, let us go a little ahead before I start taking questions we have essentially completed what we call as the the the quantum ideas we are coming now more closer to the formal quantum mechanics. See remember when we talk of the classical particle we talk of Newton's law of motion that describes the behavior of the classical particle when I talk of the wave nature we generally talk of a wave equation. Now, we have something here when we are talking about quantum particles which behave something like particles and sometimes like waves what should be the equation which should describe the behavior of this particular type of a dual particle wave. So, let us like to look at this particular thing what I am saying look out for new laws let us try to look out for new laws how to go ahead I mean whatever we have talked in the quantum ideas most were phenomenological Bohr model angular momentum quantization which of course is not correct today was given purely phenomenologically there was no logic it was never derived from a theory. For example, if I have to discuss the energy levels of not hydrogen atom but some other complicated atom I do not know how to how to do because it so it was not derived from any basic equation. Now, I would like to form my laws I would like to know what is the equation which governs this particular particle so that in general I can say that like Newton's law of motion will describe the behavior of classical particle it is this equation which will describe the behavior of wave particle dual something in a particle which has a significant wave properties significant or rather observable wave properties what is the equation which governs that. So, we are essentially in dark we do not know how to proceed. So, we are trying to form our ideas let us be very clear the equation that we are going to land up as all of you know the Schrodinger equation we are not trying to derive Schrodinger equation. So, many of the arguments that I am going to give are not probably unquestionable there can always be questions which can be raised but on the other hand why we believe on Schrodinger equation because from Schrodinger equation whatever we obtain at least as of now are experimentally verified that is why we believe in Schrodinger equation I am not deriving Schrodinger equation I am just putting some arguments how I am reaching at that particular equation a real test of any theory is whether it can explain the experiments. If it explains the experiments then we take that particular theory if it cannot explain we reject that particular theory. So, now let us look for the two I have lost this is what is called Newton's law of motion force is equal to mass into acceleration and acceleration you know in the differential form can be written as the second derivative of time of position vector d2r dt square. This is the standard wave equation which I mean if you are not very familiar this is the wave equation which you normally will obtain in the case of let us say electromagnetic theory by using Maxson's equation I think Professor Ghosh will cover probably I am not very sure about the electromagnetic waves. The thing is that can I pick one of these two equations and say that this is the equation which will also govern the behavior of the particle nature of I mean a particle which has a wave nature. But before I do that let me just sort of convince first of all I will say that this particular equation I will reject because there is nothing like wave property here it is purely particle behavior but this particular equation does have a wave behavior. So, let us first try to understand let us first try to convince ourselves that this particular equation will satisfy is satisfied by the standard wave equation because remember in some of the earlier transparencies we have just written the wave equation psi is equal to a sin kx minus omega t. But let us first convince ourselves that that particular expression is actually a solution of this particular equation. So, my question is first which is the equation that governs a non-relativistic particle this I must emphasize again shooting an equation talks of only in all non-relativistic limit. We are always assuming that the speed of the particle is much smaller than the speed of light in non-relativistic particle with observable wave behavior. The first one does not imply wave nature at all. So, should I check it let us check at the second equation let us see from second equation first of all whether the second equation will work if it does not work can I get some hint from the second equation and arrive at an equation which probably will be valid for the particles with observable wave behavior. Now before I go ahead I mean I have written four different forms but there can be many more forms in which a wave type of solutions are possible. A sin kx minus omega t I have used some of you had asked that why I do not use minus psi kx minus omega t that is also another form except in that particular case the wave will have a phase difference with respect to this particular wave. This particular wave cosine kx minus omega t is also a proper wave again there is a phase difference between this particular and this particular wave if you want to plot let us say at a given time as a function of x you will find that these two waves will differ in phase by pi by 2. At this moment I would like to introduce exponential form of representing a wave this is actually a complex form a e raise power i kx minus omega t and as you can you know that this thing can be written as a cos kx minus omega t plus i sin kx minus omega t alright. Now this is actually a mixture of a sin term and complex term but more surprisingly this sin term and cosine term but more surprisingly it has also i and this i is equal to under root minus 1 as all of you know which makes it imaginary okay. Now one can always thought of a wave something which is always a real so how can I represent it by complex number but those people who are students of electrical engineering they know that is very very commonly done in electrical circuits even we are talking about LCR circuits etc we definitely represent these oscillations in the form of an exponential form with a i with an imaginary number okay and this is I mean the thing is so long it is the observable things are real it does not matter whether I am using a complex number okay there of course we are always talking about the real part and the imaginary part sort of represents a phase so it is very very common at least in many of the circuits to represent a wave like solution by an equation of this particular sort. Now first thing which I would like to show that both these all these solutions will satisfy the wave equation which I have just now written this which is this equation you remember this links second derivative of x with the second derivative of time and this c is supposed to be constant which I will actually identify this as the phase velocity of the wave of course this c will be actual velocity of light c only when this is an electromagnetic wave otherwise this c will represent the velocity of an appropriate wave okay so let us just take two of these terms just to satisfy its yourself that this is actually a proper way of representing the wave. So let us first take sin kx minus omega t so I take first the position derivative so remember these are partial derivative so when I am taking derivative with respect to x okay t has to be treated as constant so when I take the first derivative you will get my cosine will become sin and because you are taking derivative with respect to k so there will be a there will be k which will become derivative with respect to x sorry so a k will come out here so you will have del psi del x is equal to minus ak sin kx minus omega t when you take derivative for the second time this sin will again become cos okay another k term will come here okay it will become minus ak square sin kx minus omega t remember derivative of cosine will have a negative sin but derivative of sin will not have a negative sin so this negative time will sin will always remain. Now this is the second order derivative with respect to position let us try to take the second derivative with respect to the time okay so I did take derivative with respect to time so again cosine becomes sin okay but there was a negative sign here and time term there is another negative sign here so this becomes plus so this becomes aw sin kx minus omega t I take the second derivative now again because of this negative sign a negative sign comes here is minus aw square cos kx minus omega t I substitute in this particular equation the wave equation which was del 2 psi del x square there is equal to 1 upon c square del 2 psi del t square okay you will find that all these terms cancel out and all you get is c square is equal to there will be omega square and there is a k square here will be equal to omega square by k square and as you know omega by k is the phase velocity of this wave so this c represents the phase velocity of this particular wave so as you can see that this particular solution actually satisfies the equation the wave equation so this is actually a solution of this particular equation now let us take one exponential form this particular form this is actually much easier once I take the derivative with respect to x exponential form remains in the exponential form only thing this ik will come out here when I take the second derivative will be minus ak square into this particular thing now I take derivative with respect to time when I take the derivative with respect to time first time when I take derivative I get minus ai this exponential form remains second time I take the derivative I get minus a omega square this term I substitute in this equation I get c square is equal to omega square by k square exactly the same thing so we are actually getting exactly the same speed of fly same speed of the particle same speed of the wave sorry okay irrespective of whether I am using this particular term or that particular term similarly I could have taken the other two terms and there all also I would have found that these solutions also satisfy the wave equation now I will just stop if you have some questions you know we will probably try to take some questions for another 10 minutes and I will will some choose randomly from some of the centers some of the questions those centers which do not have question probably if they want to leave they can leave and then other questions you know people should post us and like before we will try to answer in the next lecture okay MES Pillay sir sir my question is yes please go ahead can we get interference pattern with single electron in double slit experiment as electron is associated with the wave is associated electron there is no question of interference pattern because this electron is going to land somewhere see in order to see interference pattern you have to have multiple electrons okay in fact I had a I mean I can refer to a website which probably I will upload separately in model where you can see the video of this particular you know the experiment which has been done which has been recently done I mean not very recently but something like 10-15 years back by a group in Japan okay where they have performed this particular experiment where electrons really come one by one and when it comes by one one it lands somewhere okay when the second electron comes it again lands somewhere so you will not be able to see interference pattern in that sense but when you are starting seeing a very large number of electrons then only you will start seeing that this how the interference pattern starts building up so for first few electrons you will not be able to see because they are essentially coming and falling there randomly in fact using cold gases you know my friend there is a great work of professor Kohantanujee etc I mean using the cold gases it has been possible to perform this type of diffraction experiments which originally were thought only as they thought experiments but recently it has been possible to perform these experiments and you can really show that once the particles are coming one by and they are coming so slowly that there is no possibility of them interacting with each other okay so one particular particle comes and lands somewhere second particle comes again lands somewhere okay because there is some probability of landing it somewhere okay unless you have very large number of particles you will not start seeing the interference pattern. Don Bosco I have two questions yes yes the first question is can we interpret the electron diffraction pattern as some sort of probability distribution function that is right that is what I am initially going to interpret it like that because probably I am not very sure in this particular lecture or in the next lecture I mean tomorrow's lecture I will try to interpret what is the wave function and then I will interpret this as a probability wave so actually it is a representation of the probabilities why you are 100 right okay sir. So my second question is the electron diffraction pattern I am just assuming that it is a sinc square function yes which is we know that it's a very well behaved function yes that's right so in the slide number 18 you had mentioned two words that is random and unpredictable and uncontrolled yes that's right you are right yes sir so these two so now if if the sinc square function which is a well defined function this is so that means and if you interpret it as a probability distribution function then this unpredictable and uncontrolled behavior holds good only for a single electron so am I to understand that the unpredictability holds good for only a single electrons but when we have a statistically large number of electrons then the behavior becomes more and more predictable that's right you are right in that sense also see things that you know when we are talking about a single particle you are right see the question is that when I am trying to talk of unpredictability there okay it's essentially related to the class comparing with classical mechanics in classical mechanics if you have one million particle and they have same initial condition they have exactly the same force all of them are going to land up exactly at the same position after a given time okay there is a complete 100 percent predictability in classical mechanics now when a particular individual particle is coming here I do not know where this particular particle is going to land up so in that sense there is unpredictability what you are sure you are perfectly right that if I perform this is basically statistics if I perform this experiment all it's a one million particle okay then you will find that the probability that you are getting are well defined okay so I am getting a well defined it's almost like I mean any probability experiment okay if I perform you know if I just toss a coin I will either get head or tail okay so I cannot find out whether the probability of getting head is half or probability of getting tail is half okay it's also possible that you toss a coin three times okay and all the three times you may get head okay it does not mean the probability is not half okay you have to perform this experiment over a large number of coins to really see that the probability is half so well defined probability whenever I am talking is not because of an individual experiment I have to perform this experiment over a large number of particles then only I will be able to get the idea of probability but what you get you are 100 percent right again that when I what I am getting eventually also from Schrodinger equation is a probability and these are the well defined probabilities so like in classical mechanics I get a predictability about the position or momentum here I get predictability only about the probability okay Thakur college hello yeah go ahead yeah sir good afternoon good afternoon I have two questions yes first one yeah can you elaborate Heisenberg uncertainty in principle that is applicable to angle and angular momentum that's so how it is okay second is yeah explain probability density function see probability density function I will explain in my subsequent lectures so I will avoid answering this question at this particular moment okay because I am going to spend quite a bit of time explaining what I mean by probability what I mean by the wave function okay so therefore let us postpone this question about this thing about the theta and angular momentum okay you are you are perfectly right there is uncertainty principle lies okay there are a large number of pairs among which uncertainty principle can be seen okay what I am trying to see show here is only a very very elementary way of looking into quantum mechanics quantum mechanics once you go in advanced quantum mechanics not only quantum mechanics becomes you know fairly you know you know let us abstract okay it okay it also becomes mathematically fairly complex there you can define the pairs okay okay on which there will be an uncertainty principle which will be existing or you define a two dynamical quantities a pair of two dynamical quantities in which uncertainty principle will exist and there is a definite way of looking a commutative operator how it can be done okay but that is again a part of the advanced quantum mechanics course hello yes please go ahead good afternoon sir good afternoon sir in order to give an example of application of uncertainty principle we could able to prove delta x delta p is equal to h straws yes hello yeah please go ahead why we can now can we prove this the another conjugate also that delta e delta t is also using the single state diffraction or any other kind of what explanation applications you're talking about delta e delta t uncertainty principle are you talking about delta e delta t uncertainty we have three delta x delta v delta e delta d delta theta and l so can we prove using only one single example all these three pairs uh i'm not very sure with it can be a see let's also understand that you know the way i'm sort of introducing these are sort of an approximate way because or let me put it like a hand waving wave so the examples which i'm giving are purely more or less hand waving as i said you know using advanced quantum mechanics all these things are much better formalized but they become a little more abstract so uh at least i i cannot think immediately of any particular example which can simultaneously give all the uncertainty principles at one go my means sir if you have the momentum like in classical mechanics we could able to find out some energy kind of parameters whereas in quantum mechanics if i'm having uncertainty of momentum why can't we find uncertainty in energy we can definitely find uncertainty in energy products see the thing is that's not made this yeah please go ahead the pairs the conjugate pairs which we made here yes as a product are just to satisfy the dimensional analysis it is formed or really they're giving three of us together any collective information or productive information about a quantum mechanical system all these six parameters together because i'm interested about our system regarding all these parameters simultaneously or individually however but all six i want how to find out that see i'm not very sure whether i have understood your question properly but let me try to explain whatever i have understood see thing is that let me first talk about energy and time uncertainty and energy and time and uncertainty most commonly it's used when we are saying that whenever you are taking a particular beam of monochromatic light or x-rays for that matter can never be 100 percent monochromatic there is always a width in the energy and therefore i mean and that's related to the basically the lifetime spent the lifetime of the state the de-excitation of which has resulted into that particular photon so but if i if i understand i mean see as i again say that these conjugate pairs that you have talked okay can be defined okay it's not only the dimensional analysis of course the dimensional analysis of is an important thing because once they are of the order of h cross dimensionally they have to match the product has to match h cross but they are defined from little more fundamental way which comes from basically commutator rules in the in the quantum mechanics okay which is a part of a little more abstract and little more you know you know mathematical so these the these are the uncertainty which you can define simply by simply by looking at the wave packet thing see as probably you are aware that the wave mechanics way of representing quantum mechanics also one of the ways okay we need not talk about the wave mechanics we can talk purely about from the operator point of view okay we can talk purely operator mechanics which is a different way of looking quantum mechanics okay which is much more general and much more you know sort of you know profound but see here what i am giving is a purely from the wave mechanics aspect a feeling of high uncertainty principle should arise but you are right in that sense that a more rigorous you know valuation of uncertainty principle okay has to be done by taking more advanced quantum mechanics and taking proper conjugate pairs you are right thank you sir welcome yes n i t raucala yes please go ahead sir my question is why the wave function is expressed both in real and imaginary form why imaginary part is required that i am actually in the afternoon lecture that is what i am going to describe because the equation that i am going to get can only be solved by a complex number it cannot or we cannot be solved by a real number so automatically the all the wave functions become complex so this particular thing aspect i am going to describe quite a bit in detail in the afternoon lecture megnath sir institute sir good afternoon good afternoon in single slit diffraction experiment there is an expression y 0 is equal to l v y by v x that's right how can you get this so can you explain once again yeah sure this was my slit and this was my screen and the distance between this is l okay so if a particle starts from this particular point with a speed v x let me write little bit bigger v x then time that it will take to reach here will be l divided by v x is it alright alright now when this particular particle is moving from here and going here at that time it is also getting displaced in this particular direction because of the y component of the velocity so if i have to find out if i have a particle which has velocity v y component in the y direction then the total time in which it travels is this much time okay how much it gets displaced so this must be multiplied by v y so this will be the displacement of the particle in this particular direction here which i am calling as y 0 okay this is the total time the particle will take to travel from this particular slit to this screen during this time it's not only moving this direction but also moving upwards or downwards in the y direction okay and how much it moves it depends on what is the velocity component in the y direction okay multiplied by the time so this is the time multiplied by v y good afternoon sir good afternoon coming to uncertainty relationship yes in case if we take del x del p on one side yes on the other side we are seeing many representations like h or h cross h cross by 2 that's sometimes approximately equation we are using or sometimes the greater than or equal to we are using yes which is the exact form of the uncertainty relation see as i said the exact form depends purely on the wave function and eventually i will derive for particle in a box the uncertainty expression okay so because in fact a large number of problems that we do on uncertainty principle especially at the first year level are actually using an approximate expression just to get a feeling of the uncertainty principle so we do not bother about what expression that i am using only thing which i want to say that it can be shown that for irrespective of the wave packet that you use you will never get a uncertainty product which is less than h cross by 2 okay but depends once i solve Schrodinger equation i will get a particular type of wave function or what we call as a wave packet okay and corresponding to that wave packet there will be a corresponding uncertainty principle good afternoon sir good afternoon yeah i had a question regarding the dual nature of the light yes is it always necessary to analyze particle and wave nature for the applications that we are using today and which is the nature which dominates for most of the applications involving light nowadays see that as i said you know it always depends on the type of application that you are talking okay now domination is not really a correct word because it depends on what type of device dominates in our life okay so it is not as i mentioned there is supposed to be a sort of complementarity principle which says that you know both the particle nature and wave nature cannot be simultaneously observed in an experiment and as i have also mentioned today morning that this particular complementarity principle is under observation there are some groups who have been working researching on this thing in fact i have also seen some papers where people are talking that this particular principle does not hold good but as far as i know this has not yet been established that this complementary principle does not necessarily work so as of now i am taking that it works and if it works then you cannot in a device see both the aspects together now it again depends on what type of device that you are talking okay first of all if i am talking of semiconductor device i find it very very convenient that there is a photon which gets absorbed and creates let us say electron whole pair okay and we do not talk in terms of really wave nature so again depends you know and depends on what type of device is much more useful to you to talk which effect is more dominant but let me put it like that depends on what type of experiment and what type of device that you are working whether you will have a wave nature or a particle nature yeah you have one more question yes yes sir i have one more question yes please as we know that electromagnetic wave is different from a mechanical wave associated with the particle yeah so can we say that wave particle duality which is applicable to electromagnetic wave is analogous to uncertainty principle used for particle well i am not very sure whether i have understood your question see if you are talking of mechanical wave i am not talking of mechanical wave okay when i am talking of wave particle duality i am talking specifically of electromagnetic waves okay i am not sure whether we can talk about wave particle duality in the case of let us say water wave or for that matter let us say sound wave okay i do not think that we can do it so when we are talking of these duality i mean when we are talking of this particular see for example sound requires a medium to travel okay this is very very different from the wave they the electromagnetic wave okay so when we are talking of this dual nature of photon and particles and in terms of you know zero-res mass particle okay we are talking specifically in terms of the electromagnetic waves sir one question yeah does the uncertainty principle imply the existence of particles that exit the speed of light no there is nothing no relation with the speed of light see the particles that we are talking never travels with the speed of light in principle a massive particle cannot travel with the speed of light the speed of light has no role to play in quantum mechanics in principle the Schrodinger quantum mechanics is always non relativistic so i am i am little surprised why you are talking about the speed of light at all i mean maybe i have not understood your question but you know so you are can you repeat your question i am a little yeah does the uncertainty principle imply the existence of particles that exit the speed of light that exit that exceed the speed of light no no no it is nothing to do with the speed of light no i am sorry absolutely nothing the uncertainty principle is always talking of only the probability of finding the particle and probability of finding its momentum okay there is nothing no role of speed of light here okay i think we will sort of stop here then any other questions you have you know please send it by in the chat mode we will try to answer in the next lectures thank you very much