Let us start. Let us begin with some questions. See one question is about can you tell Schrodinger equation of kinetic energy of a particle in a one dimensional box? See in a one dimensional box because potential energy is 0. So basically what you are doing is only taking the kinetic energy consideration. Even if you take kinetic energy operator, essentially you are trying to find out the eigenvalues only of the kinetic energy operator. So that makes no difference because the potential energy actually is 0 in this particular case. Then another question is that you know do all electrons accept hydrogen atom in one orbit has one wave packet associated with them. See question is that you know for every different quantum number you have a different wave function and a different wave function means different wave packet you know because what we had been calling as wave packet is something what is equivalent of wave function and that is what is called wave function in quantum mechanics. So you might have if you take any book on modern physics you might have seen these all these wave functions drawn in r theta phi space. So if you are going to any other level you know you know what is the wave function and that is what is the wave packet representing that particular type of state. Now how the density of particle is maximum at x is equal to L by 2 no that is what we get from quantum mechanics or we cannot probably argue because once we agree that Schrodinger equation gives me the correct results or expected to give me the correct result you know that is what you obtain from them. So it is correct that you know when you have n is equal to 2 and is equal to 3 state the maximum position where you have maximum probability of finding the particle will keep on becoming different. Then next question is that if you are studying the wave nature of particle then in particle in one dimensional potential box particle in three dimensional potential well probability to find the particle why the word particle is used. Well this is something which is as I said is the interpretation you know I agree that you know it can sort of create a confusion because when we are I mean let me put like that when we say that I mean when we are talking the quantum ideas we are talking of wave being associated with a particular particle and then eventually we said that this wave is sort of a probability wave this is not really the type of wave that we are normally used to it where something physically is vibrating and then you can associate oscillating at something that we can see physics you know here and then we said that this is a particle I mean it is a probability wave and when we are trying to take a particle see the fact is that if you are looking at the particle particle can also be visualized as a particle and this probability wave actually represents that if you want to I mean once you perform the experiment you can try to you can find out what is the probability of the finding the particle. Of course this particular thing can also be looked in a slightly different fashion I mean one can also look at the position operator and take its eigenvalues and one can also interpret in a slightly different fashion but you know this goes slightly beyond the thing but I understand that there could be some amount of confusion in this thing but I hope you know one can understand that in a particle can be visualized as a particle also so when you are talking of this wave associated with that and once you try to make a measurement on that particular particle okay it essentially tries to collapse it to that particular position and you can always look into this particular fashion. Now the next question is that in a potential well if e is equal to v0 what will be the wave function you can always solve that particular thing in fact one of the gentleman has I am not sure gentleman already has always sent me a particular thing which I think is a correct way of I do not have time to look into it that you can always write e is equal to v0 then you will get d2 side dx square is equal to 0 and then when you solve the equation you will get cx plus d and then you put the boundary condition that x is equal to plus infinity the wave function has to be finite so it will just be constant. So it can exactly be done like we have been doing normally there is no difference in that. Then next question is from some Hindustan called shoot of technology and management is about is it quantum mechanical correct that the particle having e less than v have r is equal to 1 I think that we had shown that this actually turns out to be equal to r is equal to 1 so this is correct that it will have a reflection function of 1 there is no net transmission. Please explain a real time problem where the development of shooting a wave equation can be used when I mean whenever you talk of transitions etc and all those things we are using a time dependent shooting a equation so I cannot other than that I cannot immediately visualize any of the issues where you can show it. Now let us come to yesterday's lecture I mean yesterday's questions there are some questions on crystallography where we are talking of x-ray diffraction is reflection of x-ray from this particular plane see as I have been telling that you know actually speaking this a diffraction phenomenon but because this reflection phenomenon also obeys this particular condition of specular reflection which means angle of incidence is equal to angle of reflection so many times we call it x-ray reflection one should not get confused about this particular terminology actually it should be x-ray diffraction but many times people call it not correctly to be very honest to call it x-ray reflection so one should not worry about it so that we understand so long we understand this actually a diffraction phenomenon. The second question was about the Scherer formula because I had explained about Scherer formula that about a particular size determination see thing is that you know the question is that calculated from Scherer formula is an average grain size or particle size that is what I said depends on what type of sample you have okay it is if you have a powder sample this will be a particle size if it is you know for example having a polycrystine let us say material like it for example thin film this will give you the grain size if it is an average grain size try to confirm the size of the nano particle from Scherer formula I have always said one has to take all these things with the pinch of salt I mean large number of students do take use Scherer formula to calculate the average grain size okay let us also be very clear that when we are saying average grain size we are not very clear what we mean because the grain sizes are fairly odd in shape I mean I mean you can watch this particular thing using PM or for that matter also somewhat from SEM and you can see that these grains are sort of very very odd shaped so how do you define the size so it is a some sort of average which is taken which gives you the value of Scherer formula as I said if this is not exceedingly critical for you okay and if you are not looking at a very very exact value okay then you know Scherer formula is a sort of a good estimate of the grain size I have even said that if you want to calculate the relative grain size for example if you want to heat the material and you expect the grain size to increase and you want to find out how much grain size has increased okay these relative changes probably one can get to a reasonable accuracy with Scherer formula but if it is a really very very critical for you to find out the grain sizes exactly okay or rather average grain sizes exactly then you know you have to I mean the best technique for grain size determination as I had told you is the dark field imaging in the transmission electron microscope which is not very easy because to make the sample for a transmission electron microscope is not all that trivial job when people spend weeks to make one sample and if the sample breaks away you know the whole effort goes waste so it is not that you know see while x-ray can be done very very routinely if I prepare 10 samples I can do for all the 10 samples x-ray today and get the idea but if I have to take a tm of all these 10 samples you know I mean I will spend huge amount of time so one has to really make a judicious choice when you are doing research to find out you know how critically you need that particular information and if you need it critically then probably it is worth spending that much time if all that you want to have an idea as an sort of estimate okay then Scherer formula is probably alright as I said that a large number of research papers still use Scherer formula to calculate the average grain size if I have also told you that you know things like stresses etc could also change they can also broaden the lines okay while when I am using Scherer formula I am sort of assuming that this you know the width is purely because of the finite grain size and there is no other effect so there are lot of you know ambiguities there but as I said people still do it because you know they do not want to use I mean getting that information is not very critical for them all they want is an sort of an estimate of what is the typical grain size okay so the next question is that in crystallography for body centered cubic you have shown that the position vector A has a negative sign with I okay that is actually correct I mean I mean of course I have told you that there are multiple ways of defining primitive vectors of any lattice so for BCC also you could have taken a slightly different type of primitive vectors which would have also served the purpose equally but on the other hand you know I mean this is the convention that people use it this convention has been I mean because it gives you certain amount of symmetry in the way ABC are represented so this is what is generally taken but nothing will change if you decide to take a different set of primitive vectors okay now there are some questions on relativity so let us start with these questions on relativity first question is that what is the difference between special theory of relativity and general theory of relativity as I mentioned special theory of relativity we are dealing only with inertial frames general theory of relativity is much more complex where we are sort of also including non-inertial frame of reference that is also an attempt to explain gravitation and it talks about you know I mean basically it looks at that what the inertial mass as we call as and see there are two masses which we come which is one forgot the name of the other thing which comes in the Newton's law of motion f is equal to ma another which comes into the gravitational constant g mm upon r square the fact that these two masses turn out to be exactly identical it is essentially based on that particular concept so I mean it is much more complex so but at least at the moment we can understand the difference is that special theory of relativity talks only of those frames which are inertial frame of references and while general theory of relativity in a slightly different fashion take tries to take into account the non-inertial frame of reference is also then there is some question about the I think there are two or three questions that I thought you know people got little confused by what I said see when I was discussing in the first lecture of relativity I had only talked about what was being thought about 100 years back or more than 100 years back before special theory of relativity came as the correct thing but as I will be showing in today's lecture this is something which actually Einstein objected in fact there is nothing like ether there is nothing none of these things can be can be shown if it will discuss the Michael's and Michael's and more in experiment today and talk about the results of this particular thing so that particular transparency what I have used the concept of ether and say that ether can be treated as an absolute rest and all those things this was all the thoughts which we are with before special theory of relativity those thoughts are no longer believe to be correct so we should not get confused because there are lot of questions where people have been asking is it that ether is is absolute rest and things like that see but in today's I mean the way we understand what today there is nothing like you know ether we cannot define velocities absolute velocities okay we have to we have to define velocities only with respect to something real okay you have to erect your frame of reference on something real and only then you can talk about the velocities in fact this is the point about which I am going to discuss about it speed of light is c in ether and in other frames it is different you know again I was telling that this is what was being thought earlier okay in fact when this aspect I will be discussing more in detail today then we have said in relativity we take frame s moving along plus axis can we take along y and z axis of course you can take it the equations will turn out to be different okay so long we have the frames of references are supposed to be you know inertial frame of reference you can always in principle choose your x y z z axis only thing I mean in fact x y z direction can also make arbitrary angle with respect to x y x prime y prime z prime directions only thing the mathematics will become much more complicated and you will realize that relativity already there is not a mathematics so you would like to choose certain system you know which at least makes your mathematics easier so as a convention we have always assumed that relative velocities are on the x direction but in fact there are some problems where you can even talk more relative velocity being in the y direction there is nothing wrong we can I mean things will I mean things are to only written in different fashion but that writes perfectly all right now what would be the consequence if there is no limit of velocity of light as special theoretical dimensions actually unfortunately I will not be able to talk about this particular limit in a great detail the first and most important consequence that you will get that all these distances etc will turn out to be imaginary and you know we do not know what you mean by imaginary time we do not know what is meant by imaginary distances so I mean as I said anything which is observable should be real and that is one of the reasons why we do not expect the velocity is greater than a velocity of light but there is also another thing which is we call you know the causality phenomenon of causality see we always feel that cause should precede the outcome so for example if I throw a ball to somebody then the person ball the person receives the ball the person ball receives the ball only because somebody has thrown the ball so throwing the ball was the cause and outcome was that another person receives the ball now obviously this particular outcome should come later it should not happen that in a particular frame of reference one can observe that the ball is received first but thrown later so this is what is called causality one can also show that if the speed is greater than speed of light we are allowed then causality will be violated it means it may be possible in that particular case that you could have some frame of references where cause appears you know later and its outcome appears first so you first know that the person has received the ball and then you will notice that the ball has been thrown so this is something which sort of is not acceptable so that is another reason why we do not allow speeds greater than the speed of light because as we say that this will violate causality then there is a question is that as you know what is at rest in the universe but it is mentioned so again I have said that rest is I mean this concept of absolute rest was being taught only in the sense of classical mechanics I am not classical mechanics but the classical ideas when we thought that ether exists but you know now we know that ether does not exist there is nothing like ether so those particular thing whatever I have mentioned in first lectures has to be taken as the things the thoughts which were before special theory of relativity as I had mentioned that my first way is to tell why relativity was needed where I had my weak points weak arguments that is what I had discussed in that particular lecture that the concept of ether was introduced but people were not very happy with that particular concept because in mechanics if you have a pure mechanical phenomenon you find that all inertial frames are equivalent but it seems to be showing that when you are coming to electromagnetic theory all the frames do not appear to be equivalent which is something is not very easily people are not happy with that particular situation and that is why and of course eventually as I said they have to perform an experiment and see that this concept of ether is really correct or not and that is what I am going to discuss that this does not turn out to be correct. In relativity we take two inertial frame can we take two non-inertial frame no not in special theory of relativity special theory of relativity it always means that the two frames it means applicable only when the two frames are inertial can we say that earth is inertial or non-inertial that also had mentioned last time that strictly speaking there is no really a 100% inertial frame exists okay in fact we these are all idealized concept and there is nothing like which is 100% ideal anywhere in life or universe okay but earth is actually everyone knows that it is a non-inertial frame of reference it is not only undergoing orbital motion across about the sun okay it means its velocity keeps on changing as a function of time it is also spinning about its own axis and that spinning is also fairly complex as all of you know okay so all these problems definitely makes the earth as a non-inertial frame of reference but as also I discussed yesterday that the effect of these non-inertial frame of references generally for I mean many of the phenomenon will be negligible and therefore most of the time when we are observing a phenomenon earth we could treat earth as an approximately inertial frame of reference then there is a question is that earth is assumed to be inertial based and what will be the impact on special theory of relativity as I said you know in general in any mechanics if you are having non-inertial frame of references then you may have to introduce some pseudo forces I mean I am not sure special theory of relativity whether it will cause certain specific changes because in a special theory of relativity sort of assumes them to be inertial frame of references then there is another question can frame S double prime move along wide z axis which I have just now explained that it is possible to do it so I think this takes care of all the questions so let us now come back to my second lecture on relativity okay so this is what we had discussed last time that it was felt that consistency of electromagnetic theory with relative velocity expression required the presence of a special inertial frame which was termed as ether okay in fact I had mentioned and today morning again I have mentioned that this concept is was sort of not very happily accepted but people thought that this is only way out under the classical mechanics but then we needed experiment to support whether really ether exist so this is what I am going to discuss whether there is an experimental support to the factor to the fact that ether really exist okay so this is what I am going to discuss today which we call as a Michelson-Morley experiment but let us first discuss before we start discussing actually the Michelson-Morley experiment what is the logic behind that experiment how this experiment was thought I mean what was the thought process behind this particular experiment and then we will talk about the actual experiment so basically what is our idea I mean this is what I want to prove that I want to measure I mean if I as I said that if the concept of ether is correct then if I take two different frames of reference okay then you should measure different speeds of light okay because we said that is only in ether that the speed of light is given by c is equal to 1 upon under root epsilon not mu not if you go to a different medium the velocity of light will be different as it is governed by the normal relative velocity formula okay which is standard as we said I have given example last time that if you are sitting in a train you find the speed of the train which is approaching to you or which is going away from you to be different okay same thing would normally you would expect to be for the velocity of light in the realm of classical mechanics okay so it means if you are going to two different frames of reference you should be able to measure two different speeds like a classical particle that is what I am able to do it okay so then that will confirm the confirmation of the existence of ether in which the speed of light is this so this basically I want to do if I am able to measure the speed of light to be different and two different inertial frame of references then I have actually at least partly I have explained the fact that I mean probably that ether exist and probably the velocity of light is a frame dependent quantity let us see whether this works out or not so this is the plan of the experiment you know how you thought about it so this blue thing you know I have put it here let us assume that this is somebody okay I will call it earth little later okay which is moving with a speed V and there is a light beam which is approaching it okay now when we are defining V let us be clear we are not biased about it that this V as well as this C is are both relative to earth ether so as we have said that I mean according to the classical ideas the speed of light C should be dependent on the fundamental constants 1 upon epsilon mu not only when the medium is ether this is what was classical thought okay let I am repeating unless people have confusion about this things as I said these are all classical ideas okay so let us assume that there is a ether medium at which exist and therefore this particular speed of light C will be given by under root 1 upon epsilon not may not in this particular medium and this particular body is travelling with a speed V again as measured in ether medium now question is that what will be the speed of light that will be measured by an observer sitting in this particular row okay and of course at this particular moment only thing which I have is the classical relative velocity formula so I will apply classical relative velocity formula to find out what will be the speed of this light when measured in this particular frame of reference as you can very easily see that if the observer is sitting here that particular person will not notice his or her own speed so actually the velocity of this particular light as seen in this particular medium will be vector C minus vector V so this is what I am trying to essentially assert on this particular thing so the measured velocity of light in the object frame would be given by this output M which means the measured velocity will be given by C minus V this V was in ether medium this C was also in ether medium alright but now I have changed by frame of reference I applied the classical relative velocity formula which essentially says that whatever is the velocity of this particular frame of reference of this particular object okay that must get subtracted vectorially subtracted from the velocity so if I measure the speed of light when I am sitting on this particular frame of reference or this particular object then I would measure a velocity C minus V now I have drawn two figures here in this particular case where I have used exactly the same length of C exactly same length of V I have even maintained the same direction of V I am only have only changed the direction of C so in this particular case this is value of C in another case this is the value of C okay now I take a vector difference of C minus V as you can see very easily in this case this will be the velocity of light which will be measured and in this particular case the velocity of light this will be the value velocity of light which will be measured so now you can see very easily if this would have been the direction of C remember the C and V are constants in magnitude okay dotted direction okay now C and V magnitude wise are constant but still the vector difference of them this Vm its magnitude could be different depending upon what is the direction of C okay so if C keeps on changing if the direction of C keeps on changing then in this particular object frame we will not only find that the direction of this particular light velocity will be different but we will also notice that its magnitude will be different even though the magnitude of C and magnitude of V are identical. So this is the basically the thing which I want to mention because this is key to understanding the Michelson-Moduli experiment so I have emphasized that velocity object velocity is same in both the figures in fact the magnitude of C is also same in both the figures this what I have said the magnitude and direction of the measured velocity would depend on the direction of velocity of light hence if we measure velocity of light in object frame in different direction they would in general be different if whatever I am saying is correct okay let me re-emphasize this is purely from the classical ideas whatever I am going to say I am going to contradict later once I talk about special theory of relativity so you should not have confusion that what I am trying to say is correct these are purely classical ideas which are going to be rejected okay. So if whatever I am saying classically would happen true then what I will do in object frame if I am sitting if I measure speed of light in different directions they will turn out to be different from magnitude point of view their magnitude will also turn out to be different because of the figure which I have shown here that these magnitudes will be different depending upon in which direction I measure the velocity of light all right. So if I choose a particular frame of reference o-frame of reference and I measure the speed of light in different directions and I find them to be different it means they are in one way I have said I am not directly prove that ether exists but I have at least shown that velocity of light is frame dependent though I am making the measurement exactly in the same frame okay. But at least it becomes consistent with the idea of ether that the velocity of light c is a has a particular value only in the ether medium. So let us consider a very special situation that is the situation which was actually utilized by Michael Sunday-Bordley. So let us assume two different situations one which the measured velocity remember its measured velocity Vm okay I do not know what is the direction of c because I do not know where I am sitting on object o I do not notice my own velocity okay. So I can only measure with respect to I mean I erect some axes along x, y, z direction. So what I will do I will measure in two different directions okay. So let us assume two different situations one in which the measured velocity in the object frame is along the velocity another in perpendicular to it. So let us visualize this particular type of situation that suppose I have this how actually it will be done I will show it later but let us suppose that we have situation in which I am trying to measure in two different directions okay and in one direction it is V is perpendicular I mean the measured velocity of light is perpendicular to the direction of V another it is in the same direction of course as I just now say if you are sitting on the earth frame of reference or that object frame of reference you do not know your velocity okay how they will solve this particular problem let us postpone it till the next at a later point. But let us assume at this particular moment suppose I am looking it from ether medium or from outside medium okay and then I know that this particular object person sitting on the object is actually performing an experiment in which the velocity of light is being measured in a direction perpendicular or parallel to the velocity direction. So let me just make this particular situation clear. So let us suppose this is being observed from ether medium this is moving the velocity V okay and I feel sitting here that this particular person is observing the velocity in a direction perpendicular. So he is measuring velocity in a direction perpendicular to V this person would be able to measure velocity in a direction perpendicular to this particular V only if C is moving in this particular direction because the person what he would notice the velocity or she will notice the velocity directional velocity will be C minus V okay. So according to the person sitting on this particular object frame of reference the velocity will be measured perpendicular to V only when this Vm is in perpendicular direction to V and not when it is perpendicular to this C direction. So this is what has to be very clear that if this particular person has to measure in a direction perpendicular to V then this has to be value of C so that what this person is going to measure is only C minus V. Similarly, if the person is going to measure the velocity the speed of light in this particular direction of course he has to be traveling only in this particular direction. Now my question is that can I calculate what would be the value of this particular velocity what to this particular velocity in case this particular object wants to measure velocity of light in a direction perpendicular to the direction of V and along the direction of V. So let us first look at the perpendicular situation. So in perpendicular situation this has to be direction of C as I just now discussed this is the direction of the object and this is Vm perpendicular this is measured in a perpendicular direction. So if he has to measure Vm and remember C is what is given by the fundamental constants and V is the velocity of the object relative to ether. So as we have said Vm must be equal to C minus V so I just calculate what will be the magnitude of this is the right angle triangle. So you can very easily see that the magnitude of this square must be equal to magnitude of this square minus magnitude of this square but if you are not very happy you can take the dot product of these two if you take dot product you will get C square plus V square minus 2 C dot V and you can again show that C dot V will actually equal to V square because you just take C dot V this will give you again a component along this particular direction this will be C multiplied by V multiplied by cos theta and C cos theta will because of this figure will again give you V. So this will become V square otherwise you can directly look at this triangle okay. So this particular length must be equal to C square this particular arm square minus this particular arm square. So this particular thing should be equal to C square minus V square because I am looking only at the magnitudes here. Now in a parallel situation things are much more simpler if you have parallel situation remember this C is only in the ether medium according to my earlier thinking and this is the V here. So actually V parallel will be smaller than this thing because Vm parallel must be I should have put Vm here Vm parallel must be equal to C minus V alright. So because this has to be the difference of these two as we have always been telling so this particular measured velocity will turn out to be smaller than this thing because you know this is what is actually the V alright. So this will be this will get they are in the same direction so they just get subtracted by the magnitude of V okay. Now we look actually at the Michelson-Modell experiment. So as we said that we do not know immediately the direction of V but you can perform by taking some directions you can perform at various seasons you can keep on rotating I mean there are various ways of doing it we will see that you know how this particular person I mean they perform this particular experiment but let us assume at the moment that I am able to measure a velocity of light in a direction perpendicular to V and parallel to V okay. Let us just assume that we can do it okay in case it does not work out we will always try to I mean we will always keep on making modification I will explain this particular thing I think let us first go about this particular small mathematics then we will discuss it about it more detail. So using above idea Michelson-Modell designed and performed experiment to test ether hypothesis. So let us first ignore because I am performing this experiment on earth so let us ignore all the motions of earth other than the orbital velocity and let us assume that this orbital velocity is also constant okay.  So let us assume that this is only a constant velocity V related to ether okay let us see what is the effect of this particular thing. So the experiment performed was something like this that you have a source of light okay then you have a half silvered mirror so light beam comes here and this is put at an angle of 45 degrees with respect to this particular beam so you find that half of the light is reflected here while half the light approximately half the light gets transmitted then you have a mirror here the light gets reflected from this mirror then again part of the light starts coming here of course part of the light will be reflected which I do not care at this moment then similarly the light which has been transmitted here with respect to this from this M1 they will get reflected when it gets reflected again part of the light will be reflected part of the light will be transmitted I do not bother about the transmitted light I bother about the reflected light. So once I see this particular beam that is what is of interest to me because that is where I have put my telescope to observe so what you will find that this particular light which is going this way and coming back will come here this light is coming back from here will also come back here these two lights will interfere and if we perform the experiment in a proper way by slightly tilting this these mirrors you will be able to get the interference fringes and using these interference fringes we can find out that was the idea we can find out that is there a path difference when we try to make these lengths more or less same so you look at the path difference and try to find out what happens to the interference fringes. So this is basically the experiment of course in this case I am assuming that this direction is parallel to me this direction is perpendicular to me which as I said may not be immediately obvious but on the other hand when this experiment can be performed at various different seasons this particular experiment can also be rotated if it Michaels and Moldy actually rotated the experiment because of some other reasons so all those things can be taken care of by in a proper way. So as of now for calculation let us assume that this direction is actually the direction of earth velocity in ether medium and this is the direction which is perpendicular to the velocity of earth in earth ether medium. So now I have just now shown that the velocity if this is direction which is perpendicular to V then the velocity in this particular media will be under root c square minus V square you can also show exactly in the same fashion which I am not showing you are doing exactly the same type of calculation that if it was going backwards then also its velocity will be c square minus V square. So according to observer sitting on earth the velocity of light will travel up with speed c square minus V square and will again travel downwards with the same velocity c square minus V square but if this is the direction of velocity of earth we have seen that in this particular direction the velocity of light will be c minus V but when I am going in this particular direction because now I am going opposite to the velocity of the earth my velocity will be c plus V exactly the type of thing that we have said in the case of now train compartments that if the train is approaching you the velocity will turn out to be larger than so this is applying basically the old standard classical relative velocity formula so this is what you will be getting. So now you can calculate what will be the time taken by the light to go here and come back here and what will be the time taken by the light to go here and come back here and therefore we can calculate what will be the path difference. So I will just sort of hurry through these particular transparencies calculation of the phase difference if T parallel was time taken for the light to travel to and fro from the parallel arm of the length which is this particular arm of the length in one direction it will be c minus V in another direction it will be c plus V okay. So if you take this particular thing just calculate this thing denominator will become c square minus V square then just take this particular thing you will get that this time will turn out to be 2L by c where L is the length of this particular arm multiplied by 1 divided by 1 minus V square by c square. So this will be the time taken by the light to travel in the parallel arm parallel means parallel to the velocity direction of earth in ether medium. Similarly I can calculate T perpendicular okay let us just this T parallel I have sort of approximated it because generally Vs are supposed to be much smaller than c. So this 1 upon 1 minus 1 V square by c square I can write this to power minus 1 and then expand this into binomial series and then you will get this approximately this will become 1 minus nx and will be equal to minus 1 so this becomes plus 1 so this is approximately will be equal to 2L upon c 1 plus V square by c square. Now let us calculate in the perpendicular direction in perpendicular direction velocity going up going upward and coming down is exactly the same. So the light will travel total distance of 2L L going up L coming down with the same velocity c square minus V square. So T perpendicular will be 2L upon under root c square minus V square so I calculate exactly the same thing then you calculate what is T perpendicular use exactly the same thing take this to power minus half then there will be minus sign will get changed because of this minus half power and because there is a half so this will turn out to be factor of 2 here. So you can see that because of this particular 2 square the time taken by the light travel in perpendicular direction okay will turn out to be slightly smaller. So therefore there will be small time difference between the light traveling on the right hand direction I mean on the parallel direction and the perpendicular direction which will cause a phase change okay once it causes a phase change we can calculate how much will be the phase change because first we calculate the time difference this T parallel minus T perpendicular this time difference will be given by this thing this amps sort of moving little faster because you can only these are simple calculation. Now we have to observe fringes of course you know if the path differences are there is small amount of difference or path difference that does not really give you immediately the fringes. So what you do that mirrors you make slightly tilted so that different portions of the light you know which are getting reflected from these 2 mirrors they also have small phase shift therefore you are able to see the interference fringes. So here you slightly tilt here so you have a beam of light which is coming here so because of slight tilting there will be small amount of phase differences which will be coming because the light is some light is getting reflected from here some light will be getting reflected slightly from a different place so that the distance table will be slightly more so there will be continuous phase differences over the entire beam and therefore what you will be seeing will be the interference fringes but in addition to this there will also be a path difference because of the distance different time which has been taken by the light to travel the 2 distances. So this is what I said if mirrors M1 and M2 are slightly inclined they would create a gradual phase shift and one would observe a fringe pattern in the telescope the time difference in arrival of the waves in the 2 arms would cause an additional phase difference but biggest problem is that you know it will cause phase difference so you will probably find a fringe I mean if this particular experiment would have really given you that particular phase difference you will find a fringe shift but how do you notice this fringe shift because all you will see is the fringe pattern when you shift the fringe pattern you will not be able to really know you do not know where is the 0 if you know that this was the fringe pattern and because of this additional phase difference fringe pattern has shifted this much then you could have noticed it but this you do not know the way the experiment has been done. So then Michelson modally had this idea that you know why do not you keep on rotating this particular setup so you keep on rotating setup so here they have developed a huge platform if at this experiment was a very very interesting experiment if you look at historical aspects of this particular experiment this particular entire setup was being rotated.  Now when I have rotated it by let us say 90 degrees then this particular arm 1 has become in direction perpendicular to it and this particular arm 2 has now become in a parallel direction. So whatever was the time taken by the light in this particular position in going this way from this particular way in arm 1 now will become different. Now what was the time taken by arm 2 in this particular configuration will become again different here. So what was time taken by 1 now will be taken will be the time taken by in the second arm okay whatever was the time okay this was original situation. So whatever was the time taken by the light to go from this way to this way in this arm same time will now be taken by this particular arm 2 here and whatever was the time being taken by light to go up here and come down okay the same time difference will be taken by now in arm 1 because now it is arm 1 which has become perpendicular. So essentially what you have happened your phase difference has just shifted. So originally parallel was larger and this t perpendicular was smaller. So this was you can write t parallel as t1 and this is t2. So originally t1 minus t2 was positive now t2 minus t1 will become positive okay. So as it we are gradually changing you will find that the phase difference will change from this particular quantity to minus of this particular quantity all right because initially it was t1 minus t2 now it becomes t2 minus t1. So if I take the total phase shift it will be total time delay upon rotating by 90 degree will be delta t minus delta t I hope this particular point is clear because you know now the time difference takes negative sign. So in originally in arm 1 it was taking more time and originally arm 2 was taking less time now in arm 1 it is taking less time originally time 2 it will be taking more time. So when you are taking the difference now the difference has just become negative of that. So overall thing as I am keeping on rotating you will find out that the time delay will change from delta t to minus delta t. So total time delay has been this particular thing this you must divide by the time period which is 1 upon frequency 1 upon frequency is actually you divide this becomes c by lambda okay. So you can actually calculate what will be the overall phase shift overall phase shift will turn out to be this much. So this is divided by as we said time period when you divide by time period essentially means multiplied by frequency multiplying by frequency means multiplying by c by lambda okay. So this will be overall phase shift. Now these parameters have taken from some book so in one of the original experiments was of this length was not very large it is very interesting 11 meters this was the lambda which was used and we assume the velocity from whatever we we have some idea about what mean with the orbital velocities okay this I think this again unit problem so I will change this particular thing okay no sorry is v is 10 to the power minus 4 c multiplied v is equal to 10 to the power minus 4 multiplied by the velocity of light I am sorry 10 to the power minus 4 times the velocity of light. If you take this particular thing substitute in this particular expression then the overall phase shift that you will be getting is approximately 0. Now in fact they had done a very good error analysis Michael St. Molday and they are very very sure that this 0.4 phase shift would be observable by them but unfortunately they could not observe they perform this experiment in various seasons to be very sure okay and of course we are always rotating so my my original arm was not really parallel to v once I have rotated at one particular angle it will at least become parallel to it and you are continuously rotating so at least one direction it will always be parallel to v okay and other arm will be in perpendicular direction to v so even if in my original I mean when I start this particular experiment it is my arm may not really parallel to it but when as I keep on rotating at one time it will become parallel to it so let us start that my as my t is equal to 0 position okay then you are not very sure you try to do it again a different season when you know that earth's motion about the sun has changed its direction of v so the velocity of v has changed its direction with respect to ether okay so this experiment was performed in number of ways and we never found any fringe shift okay of course these days this experiment you know much much later than this experiments are performed with much better and better accuracies and the final result was exactly identical you never observed any fringe shift it appeared as if the path difference that we have calculated okay by assuming the velocity of light to be different in two different direction okay assuming that velocity of light is c only in ether medium and because earth is moving in ether medium therefore you will find the velocity of direction in two velocity of light in two different directions to be different okay that particular hypothesis does not work. This is what I call as a failure of experiment experiments were performed in various seasons but never gave positive results various unsatisfactory reasons were also floated to explain the negative results there are lot of people try to explain various reasons that why this particular experiment was not successful because people were always bound with that particular idea people always were bound with that particular thought process there are also some other series of experiments where people try to calculate the force on a charged particle in the magnetic field because you know as we say that it depends on v cross v and of course thought at that time was that v was velocity in ether medium okay so if you for example move to a particular in a different frame of reference and you measure the force on that particular charge okay that should turn out to be different if this hypothesis of ether medium is correct so there have been some other attempts based on the use of electromagnetic theory a moving charge in earth may produce a magnetic field all such or for example you know if earth is moving in let us say ether medium okay then v even if this particular charge is stationary on earth okay then actually it is moving in ether medium okay therefore there should be a magnetic force on this particular thing because it should be v cross v okay so if I apply a magnetic field okay even if the charge is stationary in principle it should experience a force because actually this charge is moving in ether medium okay none of these experiments were found successful then came the postulates of special theory of relativity in 1905 as all of you know these are the two postulates of Einstein okay they are very very bold postulates and what was outcome of these particular things were equally bolder and as I have told you last time it was not very easily acceptable even the citation of Nobel Prize for Einstein did not mention special theory of relativity even though this particular thing was given to him more than 10 years later after the advent of special theory of relativity okay so these are the postulates of special theory of relativity the first postulate is laws of physics are same in all inertial frame of reference no preferred inertial frame exist okay Einstein was a very very non-orthodox thinker he could think out of the blues you know he can really think something very very different and he said that there is no reason why phenomenon mechanics should be different from the phenomenon in electromagnetic theory okay if from pure mechanics we find all inertial frames to be equivalent they should also be equivalent even from the point of a electromagnetic theory so very very clear that all laws of physics are same in inertial frame of references so long you are talking of an inertial frame of reference okay you always have exactly you can apply all laws whether they are based on pure mechanical phenomenon or based also on electromagnetic theory they should already be identical they will not change there is no preferred inertial frame there is nothing like ether okay you do not talk of any frame of reference which is special okay and talk that you are talking of velocity of light only in that particular medium but then you have a few issue the issue is that as I have just now discussed if you believe that all inertial frames are equivalent which as I have even told you yesterday you would have like to believe it okay because I have given an argument that all mechanicals I mean from mechanics point of view that all frames turned out to be equivalent okay how is it so that the nature has made a bias towards a mechanical phenomenon that from the point of a mechanical thing no you find all the inertial frames of equivalent but one it comes to electromagnetic theory you can distinguish between the various inertial frames of references so this he did not found Einstein did not find it acceptable so he made a bold postulate saying that no preferred inertial frame exists but then become this becomes inconsistent with the velocity addition formula because then what will happen C will turn out to be frame dependent quantity and C turns out to be frame dependent quantity then my fundamental constants also become frame dependent so he makes a second postulate which is also very very important postulate the speed of light C is same in all inertial frame of reference it is not very certain whether Einstein knew about the Michael Snell-Money experiments outcome see this particular paper special relativity I was told does not contain any reference it was really in original work I was also told that there is some reference that there had been some experimental evidences but the fact is was that in its own self it was exceedingly bold statement which says that speed of light cannot be frame dependent in all inertial frame of references you will measure the same speed of light what does it mean this postulate implies that there is no absolute rest first postulate no absolute velocities I can never talk about the absolute velocity there is no ether a space is filled with nothing it is nothing like ether okay electromagnetic wave travels it actually travels in space okay without being occupied that space is not occupied in something okay we require something real to which we can attach frame of reference then only with respect to that frame we can talk of velocities so whenever I am sitting at some place with respect to that particular place I can talk about the velocities okay but I cannot talk about the absolute velocities okay so there is nothing like ether with respect to which I can always talk about the velocities all right for example I mean when I am sitting on earth I can always talk about the velocities of let us say train plane or you know car okay assuming that this is related to earth okay but such type of liberty does not exist in overall nature okay you have to always talk with respect to something real okay when you are talking of the velocities of galaxies or a particular star okay you have to specify when you are talking velocities with respect to what what does the second postulate imply that this is a figure which I had gone earlier yesterday okay and I had mentioned that the velocity of light measured by the two persons here will be different now the second postulate means that this person this person this person all three of them will measure the same speed of light c so whether this person approaches this particular source of light whether this person recedes from this particular source of light this person this person person sitting on earth all three will measure exactly the same speed of light c this will not obey c minus v or c plus v all the observers will measure the same speed of light so the second postulate if you believe it to be correct which we now believe that it is really correct then in that particular case this light will not be a frame dependent quantity so basically means that the relative formula velocity formula that we have earlier talked about it is not correct okay I cannot add at least it is not correct for velocity of light all right so definitely requires certain modification so the classical formula formula of velocity addition is definitely not working here if postulates of Einstein are correct a new formula is needed which will ensure that the speed of light is same in all the inertial frame of reference so I did a particular formula which ensures that the light is same in all the inertial frame if this is correct and also not only that this new formula must be consistent with the first postulate and that is what Einstein started working on it and that is what we are going to discuss in today's lectures and tomorrow lectures or tomorrow's lecture first will evolve a new formulation a new relative velocity formula which is consistent with the fact that speed of light should be same in all the frame of references okay second thing that I will test whether this really works with the first postulate whether this makes I mean with this particular formula is consistent with the first postulate of special relativity and whether any modifications are needed in our concept then we will introduce those modifications and then we will finally concept come to the most famous expression Einstein is equal to mc square so before I do that let us just discuss certain concept the concept of transformation but this is something which we use very very often in special theory of relativity so what I mean by transformation it basically means that a set of dynamical variables are known or measured for a particle in a given frame for example if you are I mean I have just now mentioned that there can be many inertial frame of references you have one inertial frame of reference in which you have measured let us say the coordinate of a particular particle or you have measured the velocity of a particular particle at a given time or you have measured let us say acceleration at a given time for example acceleration could also be time dependent or for that matter you could have measured angular momentum or whatever it is any dynamical variable is measured in a given frame of reference at a given time okay if I want to know what will be exactly the same equivalent component in a different inertial frame of reference the equations that I am going to use to obtain one from another are called transformation so let me first read that I will explain little more a set of dynamical variables are known or are measured for a particle in a given frame we are interested in finding out the values of the same set of variables in a different frame the equations that we have to use are called transformation equations so for example I may know vx vy vz which are the x y and z component of velocity of a particle in a given frame I want to find out what will be the velocity of the same particle as being seen in a different frame as prime okay then the equations that I will use to convert this vx vy vz to the velocities in a different frame of reference will be called velocity transformation equation or if I know what is the value of coordinate at a given time of a particular particle in a given frame and I want to find out what will be the coordinate of the same particle at the same time in a different frame of reference then the equation that I am going to use will be called the coordinate transformation equation so this is actually a simple concept but we always use this for transformation so let us just I have to spend a few minutes to explain what is meant by transformation so for example there are examples of transformation like coordinate transformation velocity transformation so first what we will do we will formalize the classical coordinate transformation which probably all of us are knowing may not be knowing in that formal language but you know unknowingly we have always been using it so let us just take the classical coordinate transformation and then I will look what is required and what has to be done to change it to a consistent transformation with respect to the second postulate of Einstein. So let us first talk of problem formulation this I have mentioned in the last lecture also okay while in the transparency which I had mentioned what I have given last time the set of axes were not very important because I was talking only position vectors okay but here let us be very very clear because I am talking about the coordinates so let us be very clear what are our choices of axes and we are taking some sort of special choices of axes but the transformation equations will be derived using both set of axes but you can show very easily that okay if you want to use any general thing I mean you can always derive an equivalent type of equations nothing of physics will change only thing it makes our equation slightly simpler so I take two inertial frame one is s and another s prime choose x and x prime x is along the direction of relative velocity of the two frames so these two frames are both are inertial so their relative velocity has to be constant so whatever is the relative velocity direction that I choose as my x direction and also x prime direction remember x y z are the directions x y z directions in s frame x prime y prime z prime directions are the directions in s prime frame so once I know what is the relative velocity direction between two frames I can always decide that my axis axis will be in that particular direction alright and the other person also decides person sitting in s frame of reference that whatever is the relative velocity direction okay corresponding to that that person chooses x prime axis so x and x prime axis of both these frames are collinear they are in the same line now take two z axis directions y and z which are of course have to be perpendicular to x and also have to be mutually perpendicular but then assume that s prime also has y prime and z prime axis parallel to them as I said this looks like some special case but this is general enough okay and all the equations it only makes your equations somewhat simpler so this is what I mean I have two frames of references s and s prime so s has axis x y and z this is axis x prime y prime and z prime okay and I am assuming that this x and x prime are collinear and y and y prime are parallel z and z prime are parallel we make two conditions again these conditions are sort of general enough frame s prime moves with respect to s along the x axis with a velocity v which I have always mentioned remember here I am using the word v okay we I am emphasizing this particular aspect because now I have to be little more careful the velocity v the symbol v is always reserved for relative velocities between the frame of reference so v is always supposed to be constant as a function of time whenever I am talking of a particle velocity I will be using the symbol u so this one has to be very clear v is the relative velocity between the frame while both the observers could be observing a given particle okay and they could measure their velocity and what velocity of that particular particle they will be measuring will be called u or u prime depending upon whether it is being observed by s or s prime the second condition which I impose is that I assume that both the observers have a clock and obviously when they were moving like this relative to here there would have been some time when o and o prime would have been coincident okay when they were coincident both the observers decided to start their stopwatch from zero time so time zero from for both the observers s and s prime started from the time when their origins were coincident the clock of each observer is set to zero time when their origins coincide or rather coincided all right so this is what we call as original transformation classical transformation which is called Galilean transformation when I say direct transformation essentially means that I know x y z in s frame and I want to find out what are x y z in s prime frame of reference when I say inverse transformation it means I know what are the x y z coordinates in s prime frame of reference I have to find out what are the coordinates in x y z directions remember here I am assuming that s prime moves along this particular direction with the logarithm v now according to an observer sitting here this particular s frame goes away recedes in this particular direction minus x direction if I change v to minus v I should get exactly the same we can transform from direct transformation to an inverse transformation you can very easily see the Galilean transformation will be this because if you are looking at one particular position here something y coordinate will be same because they are parallel similarly the z coordinate will also be same x coordinate will be different because the origin of this is differing from this particular thing by distance of v t because they were coincident at t is equal to 0 so at a time t they will be far away from this particular thing from each other by distance of v t so they are x coordinates these are with x coordinates these x coordinates will differ by an amount v times t so this is my direct transformation which says x prime is equal to x minus v t pi prime is equal to y and z prime is equal to z here there is something which is implicit in the classical mechanics which we have not been never been talking very very loudly is that we also assume that the time remains same in both the frames of reference so once they have started this top watch whenever I mean an event happens both when both of them start observe an event for example could be observation of the location of a particular particle somehow we use always the concept of event in special theory of relativity so it could be for example observing an object when we try to observe their object then the time this particular event time for both of them will always remain same so t prime is equal to t we never talk about it very clearly but this is sort of implicit in all the classical mechanics that t prime is equal to t so these are what we call as Galilean transformation now I can always calculate the classical velocity transformation okay what you do you differentiate with respect to time if you differentiate with respect to time it is as simple you just differentiate with respect to time so here you will get vx prime you differentiate this with respect to time here you will get vx because rate of change of x with respect to time will be v when you differentiate this with respect to time you will get v similarly if I differentiate this with respect to time I will just get vy prime here I will get vy vz prime vz so if I differentiate with respect to time this is what I will get vx prime is equal to vx minus v uy prime is equal to y uz prime is equal to z which is the same as the classical velocity addition formula so what I have found out is just velocity classical velocity addition formula which is the one which I had used when I was discussing the classical mechanics the original idea that light will turn out to be frame dependent quantity obviously this formula is not consistent with the second postulate of spatial theory of relativity because if I keep on using this formula I will always get velocity of light to be frame dependent these are my comments it is the same as the old relative velocity formula which is inconsistent with the second postulate we can always write inverse velocity transformation okay which will be ux is equal to ux prime plus v so remember one can always obtain that inverse transformation from direct transformation by interchanging primed and unprimed quantities and replacing v by minus v see we generally try to write I mean there is nothing new about inverse transformation but because relativity things can become somewhat complex so people find it easy to think in terms of direct transformation and inverse transformation but even if you do not remember inverse transformation all those things as it can always be obtained from direct transformation so direct transformation is much more interesting now we have to look for a transformation which is consistent with the second postulate as we have said that relatively rejects this particular transformation because this is not consistent with the second postulate of spatial theory of relativity and therefore we have to look for a new transformation which is consistent in the second postulate so that we get velocity of light frame independent so that is what I am going to discuss in my next lecture so I will stop here we can have some questions because we have today we have some time for some questions the valchand college sangly question are related to the michelson more or less experiment yes my first questions is you have used the term phase difference instead of path difference what is the correct term there number two question is you could not talk in your or the compensating plate is not shown in your diagram please explain the concept of compensating plate there and last question is is it possible to show actual practical experimental arrangement because in the original experiment the michelson and more lay use the mercury tank for carrying out the experiment thank you sir yeah see thing is that you know of course I have not really talked about what exactly the experiment the way we have they were doing because that is not really of interest when my idea is to focus more on the physics aspect of it and that is what I was I had emphasized on this particular thing about the compensating plate I am probably they might have used but you know my idea was only to tell I mean I mean just given the general philosophy of the experiment okay now coming to the your path difference and phase difference see path difference and phase difference are two different things because path difference tells you exactly in the unit of distance well when I am talking the phase difference it gives in the unit of the angle okay so basically you have to look at the path difference and compare with the correspondingly convert into let us say a time difference and eventually look that I mean I hope you understand what is the phase angle difference when I am I am trying to draw there is one particular wave like this okay now this particular wave differs in the phase by let us say this particular wave I am sorry I am not drawing it very very good well so whatever is the angle difference okay I mean for example if I am writing this a cos theta or cos kx minus omega t another I write as k cos kx minus omega t plus phi so I am essentially interested in what is this phi so this is what I am calling as the phase angle difference so this is what I have calculated phase angle but if you are not very comfortable with that okay basically you know how much additional path difference it has created and you can also work out exactly the same thing using the path difference because I have been using this particular term right from quantum mechanics kx minus omega t so I am little more comfortable myself with the phase angle but if you are not very very comfortable you know one can work out purely with path differences is just like any other interference experiment you know you know Michelson was one of the great optics person who designed Michelson interferometer and things like that so for him to design and think about optics experiment was very I mean straightforward so he had designed this particular experiment by essentially taking the same concept of path differences which is also being used in Michelson interferometer so it is exactly the same thing so whatever is comfortable that is alright but basically the idea what I am trying to you know focus on that generally if I am finding out I am basically there are two things which I want to say that if velocity of light is not same in all the inertial frame of reference and is constant or in and is given only in the ether frame of medium in terms of fundamental constants in Earth's frame of reference the velocity will be direction dependent velocity of light measured in different directions will turn out to be different in general so this is first aspect and if this is so then if I make light travel into two different directions which are perpendicular to each other I will in general find out that there is going to be path difference and this path difference would cause a shift in the fringes the thing is that whether I am able to measure this fringe shift or not and that is what he was trying to measure he designed an experiment in a fashion where he could was really able to whether he was sure that he could measure the fringe shift in case there was a path difference okay which he did not find sir I am wondering whether I am allowed to ask a question on crystallography a topic that you did yesterday yeah please you can you can yeah sir we discussed about Bragg's law and the diffraction of acrylic and we had arrived at a condition of Bragg's law that's my question is pertaining to that sir and I request you to elaborate on that so we always consider this Bragg's law as a Bragg's reflection also maybe to certain extent Bragg's scattering also and we say that whatever is the angle of incidence the angle which is certain accident makes with the plane by the same angle it gives a emerge also yeah and on that hypothesis we have been the relation so sir can you elaborate on to the reason why do you take that exactly the way it should be see let's not talk about name because people use different names depending upon I say people are not very exact when we speak we always feel that we understand so somebody may call Bragg's scattering somebody may call Bragg's reflection somebody may call Bragg's diffraction okay all the three things are same as far as I am concerned as far as 2D synthetize and lambda is concerned all right of course there could be also scattering which are called non-elastic scattering which are very very different about which I have not talked it's also possible that the part of the energy of the photon is used to excite some other other excitation in that case the wavelength of the outgoing photon turns out to be different which we call as inelastic scattering so in general when we are talking about Bragg's law we also call it elastic scattering in which we always assume that the energy of the photon has not changed all right so now what you can do you can always take for example what you consider is scattering from different one particular plane if you consider from various planes one particular plane okay if you take let's say this particular light which is X-ray which is coming in and let's suppose it's not satisfying the condition of Bragg's reflection okay it's going like this so the angle let's say I am looking at different planes okay then you have another thing which is coming here and coming here going like this then you can show that these between these two beams you know if you are going this particular beam and this particular beam the path difference between this beam and this beam will be different okay so only when this angle and this angle turns out to be same from these particular planes then only you will find that for scattering with all the atoms within this particular plane there is no path difference okay then only there is a possibility of a constructive interference so first we consider one particular plane and we try to argue out that the Theta's must be same okay then we starting in the second plane and we say that in addition the light which is coming or X-ray which is coming from here to here the path difference additional path difference which has come which is which is because of this particular plane which turns out to be 2 to sin Theta should also be equal to n lambda so first we consider take scattering only from one particular plane and if you take this particular direction and let's say another beam which is going exactly in angle like this which is not this particular angle then you can draw a perpendicular here let me draw a much more clearer figure let's let's take only these two points and let's suppose my X-ray is coming like this and it's coming like this okay and let's suppose getting transmitted in a angle which is different from here so let's say going like this okay then you can calculate what will be the path difference this path differences will be this okay and between this this will be the path difference so this particular portion of the beam travels this particular distance larger while this particular part of the beam travels this distance larger this distance and this distance are not equal so therefore there will be additional path differences which will be created by these lines when you are taking scattering from different points all right now in order they are constructively interfering then it is necessary that this path difference and this path difference should be exactly identical which is only possible when this theta and this theta turns out to be same so this is the first argument we do when we derive 2D sin Theta is equal to n lambda okay unfortunately I did not do the derivation of 2D sin Theta is equal to n lambda because we thought that this is generally I mean done even in high schools so this is the first step of that saying that condition of speculo reflection must get satisfied in order that I get you know a constructive interference from one particular plane then I take additional second plane and then I put additional condition 2D sin Theta is equal to n lambda I hope I have answered the question the last class we told yeah in the bus is a field of Ithar big part of the initial stage of universe what is the nature of Ithar and what are the components of Ithar yeah see you are asking what I mean of course Ithar does not exist so I cannot answer the question see at that time they thought you know I mean I mean that was also one of the problems about Ithar because Ithar should be very very special medium because you know if it all was present because it should have excellent good transmittivity because light keeps on coming from millions of light year away and they are still being observed on earth okay it was very very difficult to envisage that particular medium these are all the problems associated with Ithar so that's what I was telling that people were not happy with Ithar idea but I mean today I cannot talk about Ithar medium that I because I know that Ithar does not exist it was only thought which was 110 years back well actually light can travel in space with medium of Ithar but it is a hypothetical medium that's correct but we can also assume that on the basis of particle light can travel without medium I mean that's what we believe now that light can travel without medium it doesn't require a medium to travel that's precisely what I am trying to say that you know the transparency which I put today essentially says that there is no Ithar medium if we believe in postulates of special capability which we do now okay in that case Ithar does not exist and light does not require a medium to travel one more question sir yeah that's the relativity frame s prime moves along a quadratic x axis can we assume in y axis or z axis sir that's what I am trying to answer today morning of course you can always do it the equations that I am going to get will become different the Lorentz transformation which I am going to not derive but I am going to tell about Lorentz transformation in the aftermountain lecture okay that will turn out to be different instead of x there'll be y instead of y there'll be z that's alright you know it makes no difference I mean see we have to choose some convention when we give the formula in the textbook okay we we've chosen this particular convention physics will never change if you change they are your x, y, z and axis because x, y, z axis okay is have been chosen by you it has not been chosen by nature nature has not biased any particular direction to call it x direction okay so whatever you are calling x direction you could have called y direction the equations will change I mean even if you would not have made y direction parallel to y prime direction okay you'll get probably much more set of complicated equations okay but physics will not change the outcome has to be same alright okay thank you have a nice lunch


everyone, and welcome to Lillian's Vegan World. I'm your host, Lillian Cumick, with ThinkTech Hawaii, coming to you live from gorgeous downtown Honolulu. Today's episode of the show is 2021 A Plant Forward Revolution. I was almost going to say getting down and dirty, but it's getting down to earth. One of my favorite stores in Hawaii, and I have the beautiful Amber Brennan, who is part of the community outreach team over at Down to Earth, and she's going to be my guest on the show. Welcome, Amber Aloha. Aloha. Hi, Lillian. Hi, Amber. It is so, oh my gosh, I was just going to say, so great to see you. I used to work over with Amber and the team at Down to Earth, did some cooking classes there back in the good old days. Good old days. No, they're always good old days. We just have to adjust and keep moving forward. Amber, so great to have you on the show. How's everything going over at Down to Earth since COVID and the changing times? Yeah, we're adapting, I like to say, because everyone's changing how they do business nowadays. So once COVID hit and we were like, first of all, we're like, oh my gosh, we all had to take a breath and be like, what are we going to do? But then we thought, actually it could be a blessing because now we have all these new ways and new technologies to get plant-based foods out to our community. So we offer a lot of online shopping. We have delivery. We have in-store contactless pickup. And I think it's actually been really cool for us to find different ways to do business and keep everyone still happy and healthy. That is the main thing. And I did notice that during the lockdown and a lot of stores had to close last year, Down to Earth was one of the ones that stayed open. Thank you so much for doing that. You can't imagine what a great help it really was to the community. Not to mention that it's everyone's favorite store, especially if you happen to be into taking care of your health. You don't have to be vegan or vegetarian to be able to enjoy anything at Down to Earth. And it's also a fantastic place to start if in fact you are interested in starting a plant-based diet or a journey. So Amber, what's going on? What's happening in 2021 for you personally? I named the show A Plant Forward Revolution because I think that people are honestly starting to really reconsider their health and what they've been eating. And I'm hoping, I'm predicting that this year is going to be the year that sees the biggest increase in people going plant-based. What are your thoughts? I think so. I think a lot of people are re-evaluating their whole life honestly, myself included. I've been vegan personally for six years now, but even I'm thinking of like, oh, I should try to experiment with new superfoods or incorporate more fruit into my diet. So honestly, and then this is not just product placement, but with the help of being connected with Down to Earth, I can do that. And we have all these different tracks of if you're trying to keto or you're trying to be fruitarian or whatever it is. There's a lot of different avenues you can take. So yeah, we'd like to support the journey of everyone. Absolutely. Amber, before we continue on just for people outside of Hawaii, can you explain exactly what kind of store Down to Earth is? Sure. So Down to Earth is a natural foods retailer, kind of like a health food store. So we like to be not just a place to get plant-based groceries because we are 100% vegetarian. We do have a lot of vegan options and we do sell dairy and some honey, but that's the only animal products that we sell. And so we have groceries, we have produce, a wellness department with knowledgeable staff. You can ask them about all kinds of natural wellness options. But my favorite part is the deli because we have a huge spread of a hot bar, a salad bar. Some of our delis have sushi, Asian wok bowls. Our smoothie bowls are amazing. Just pretty much anything that you want that you thought couldn't be plant-based, we definitely have it for you. And for our friends in Hawaii, we definitely have the local flavors. We're constantly updating our menu to make it more local friendly, if you will. So we have a lot of local favorites which includes a lot of Asian fusion. We always experiment with a little bit of Hawaiian food and all kind of cultures in between. That is interesting that you pointed that out because Amber, you have actually prepared some slides for us of some of the food available down to earth. And this first one, speaking of fruit, just I'm sure that the viewers looking at this are going to be confused as to what it is because it literally looks like ahi poke. What is it? In fact, it's not. So the slides that I sent you are recipes that we've created. They're not available at our stores, but they're on our website. We found a recipe and we kind of tweaked it to make it like just we use our poke, a standard poke recipe that probably we got from grandma or auntie or uncle or what have you. But we just roasted watermelon and then we just substituted that instead of the fish and it was that easy. And just like ahi poke, it's best when you let it marinate over for a day. Yes, if you can't wait. That looks amazing. I do want to mention, this is a great tip for anyone who enjoys the Hawaiian flavors of poke. Watermelon actually you should roast it in the oven, as Amber said, or you can just dry bake it off in the fry pan. No oil. Just let it release its liquids and then marinate it. You can also marinate it first, then roast it or dry it off in a in a skillet or a fry pan. And I tell you, this stuff is crazy because it really does take on that not fishy texture, but it doesn't. It loses its watermelon nest. That's eating fruit.  I love watermelon poke. I actually tried to poke a apple the other day, which was interesting. And it kind of came out almost like a a daikon or a radish consistency, which was interesting. Yeah, you should give it a try the apple. I'm considering, I'm thinking pear might work as well. But anyway, poke is just one of those, you know, just a loved recipe or dish in Hawaii and something you should definitely try. Amber, you did mention that the recipes are all available on your website, which is sitdowntoearth.org. And then our recipes section on the top will and then you can search all the recipes or I think you have the links that will be available down below. Yes, they will be. Let's have a look at another slide. This is the tofu musubi. Yeah, these are so this is my version of the tofu musubi. You know, back when we had gatherings those good old days, I would have to prepare a lot of musubi for my family and friends. So what I did was I just put a baking sheet with rice, and then I would just get my musubi, not musubi, but the tofu slices and marinate them and fry them up and then just place them onto the baking sheet and cut them so that you can make a lot of musubis that way. So such a great idea. I was just looking at your photo and how beautiful, beautifully fitted the rice is to the musubi. You can actually, you know, buy the musubi containers or, yeah, that you can make them with, but that's a great idea if you are making a lot. Amber, you do a lot of cooking. Is your husband vegan as well? He is, yes. Oh, do you want to swap husbands with mine?  How long has your husband been vegan? You mentioned you have been for six years or so. I think he may be coming up on four years. Okay, so you were first. Yeah. Okay. And was it his choice to go vegan? I'm sure because of the cooking like that would have definitely helped push him in that direction because you cook so well. Was it hard for him? Was it hard for him to give up the regular American diet or Hawaiian diet? So I actually feel like it was really easy for him because when I went vegan, he was very supportive and he said, you know, I'm okay with not having any animal products in the house. So I was like, yes, great.  So he was like, yeah, this is easy. I can do this. So I kind of set him up real good. Yeah, I was going to say, I think he could be spoiled in a very wonderful way. My husband is not vegan, but our home is 100% vegan. We don't have any leather or anything in our home. And he loves it. He actually enjoys being vegan here. And you know, when he needs whatever the fix is, he'll go out and do that. But who knows, I always keep my fingers crossed that maybe someday he'll wake up and just realize that he doesn't need it anymore. We shall see. We shall. And with a lovely cook like you, I'm sure he'll see the light someday. Well, I know you would think by now he'd be, you know, totally on the vegan wagon, but he's getting there. He's very good though. Let's take a look at another. I know that this is true. We, you know, crazier things have happened. Are you able to see this slide? Yeah. Awesome. Tell us a little bit about it. My mouth is watering right now. I know. It's about that time too. Yeah. So this is our garlic shrimp or my gochum recipe that I kind of adapted. It's inspired by the North Shore shrimp trucks. I've never had shrimp from the trucks before, but I've seen a lot of recipes and my friends have shared their recipes with me. So I kind of frankenstein them all together. And then I use this beautiful vegan shrimp that we sell in our frozen department to create this dish. So it's pretty much just fried garlic with a little bit of vegan butter. And you just put the shrimp in there. It's a konnyaku base, which is kind of like a Japanese potato kind of starchy kind of situation.  That is incredible. That dish, that would really fool anyone, wouldn't it? And using the konnyaku is such a brilliant idea because it does take on that kind of, not rubbery, but it does have that consistency that you see with shrimp. In Japan, we call it putty-putty. Ah, yes. And that is sold in the frozen section down to earth. Yeah. Awesome. We're going to take a quick break, Amber, so I can just wipe the dribble from my mouth and listen to a few messages and be back after the break. Stay tuned. Welcome back, everyone, to Lillian's Vegan World. I'm your host, Lillian Kumi, a vegan chef and author of my newly released vegan cookbook, Hawaii, a Vegan Paradise. If I may ask to pull up the slide of my book, it's available in stores all over Hawaii. Also available on Amazon, Barnes & Noble, with free shipping for members. It's also available at the Kakako Down to Earth store. Hawaii Vegan Paradise, over 120 plant-based recipes from the islands. So definitely pick up a copy. I'll also be at the Kakako Farmers Market here in Honolulu. I'm there about two or three times every month on a Saturday from 8 a.m. to 12 p.m. So I do book signings there if you would like to stop by and say hi and get your book signed. Please come on over. The Kakako Farmers Market is amazing. And Amber also has news of the pop-up that Down to Earth will be also participating on, which is Saturday, February 13th. I'll let you explain that, Amber. Sure, thank you. So, yes, our Kakako store is having a pop-up market, which we've been, the goal for these events are to really lift up local artisans that may have been impacted by the 2020-ness of the world. So we have once again succulents, which I don't know if you've seen them, Lillian, but they do a beautiful succulence grown locally in Waimanalo. And then we also have Ada Swim, and she is amazing. Her bathing suits are made of 100% sustainable and recycled materials. So those are two local vendors that we really align with, and we're excited to be able to have them in our Kakako dining area. Like Lillian mentioned, it's going to be on February 13th, so the day before Valentine's Day. You can get all your gifts in. So we hope to see you there from 2 to 6 p.m. That's awesome. There's the slide of the succulent. That is really pretty. Are they easy to take care of, Amber? I'm a bad person to ask this because I'm really bad at keeping plants alive, although I shouldn't talk because they're all going to hear me now. But once again, staff actually provides plant support, so you can always reach out to them if you have issues. If your succulents are looking a little wonky, they're very friendly and they're willing to help you with any issues that arise. Wonderful. Definitely stop by and check that out and pick up some other fun stuff. And what a great day to have this pop up on the day before Valentine's Day. I know there's going to be a lot of people like myself who are leaving it till the last minute, so definitely do stop on by. Amber, let's have a look at another one of the slides that you've provided. Ooh. A gorgeous pataya bowl. A lot of people don't realize that pataya is actually dragon fruit. Yes. And it's a superfood too, so we try to make healthy options available in the form of things that people actually want to eat. So we have a lot of different smoothie bowls and the pataya bowl is just one of our most popular ones. That is gorgeous. Look at the color. One thing about the smoothies and the smoothie bowls down to earth, they are so gorgeous to look at. Thank you. You know, they're not only super healthy for you, but this is another one. Let's take another look at it, that vibrant color. Are you using the frozen pataya for this or is it fresh? I believe that it's frozen. I'm not 100% sure, but yeah, our deli is very, very picky about their ingredients, so I trust them 100% that it's very good quality. No, absolutely. That looks amazing. What kind of toppings are the toppings or vegan or is it you have a vegan, you have vegan options for the toppings? How does that work? We do have vegan options. So we have the granola and the banana and I think there's some different options you can choose if you're ordering online. You can add different kind of, I think you can do coconut cream, you can do different kind of nut butters, maybe some hemp seeds. You can get fancy if you like. Definitely. That's the whole point of the smoothie bowl. You just concoct your own version of it and go as crazy as possible. In my opinion, that is the point of the smoothie bowl. When I make it here, I might end up being big enough to feed four people, but I somehow managed to get it down just by myself. So that's another gorgeous one. Do you still have the blue-colored smoothies over it down to earth? I used to love them last year with the smoothie bowls. Yeah, we do. Those are the ones that are inspired by Hawaiian landscapes. So we have our lehua bowl which is made with goji berries, the mauka bowl which is inspired by the mountains. So we have green spirulina and then we have maukaibou which is inspired by the ocean, like this beautiful ocean behind me. So that's blue spirulina base. Gorgeous. Don't rub in the island thing Amber, you know, not everyone is able to live here in paradise. So okay, I actually rub it in all the time. Are you originally from Hawaii? Yeah, I was born and raised in Kaneohe on the east side. So when you think of Hawaii, this might be a difficult question given that you were born and raised here. Do you find Hawaii to be a vegan paradise? Is this somewhere you could say to people come to Hawaii if you're vegan? I think absolutely. Well, I think a vegan paradise actually is like a state of mind. It's kind of like if you have something that you really love to eat, you can find a way to make it vegan. But speaking of Hawaii, there's so much fruit and different kind of produce and I might be biased but I feel like our land is just so nice and fertile that the food, the produce just tastes better. So yeah, I think it is a paradise for sure. I love what you said just now that vegan paradise is a state of mind. It truly is because you can live in any country on planet Earth and if you are aligning with your morals and beliefs and truly, truly want to live a compassionate, healthy, positive lifestyle, then you can do it from any country in the world. But Hawaii just happens to be one of those places where, as you pointed out, we are lucky to have fresh produce all year round. So that's one thing I love about Hawaii. I've only been here for two years and I wrote a book and named it Hawaii, a Vegan Paradise. It seems like you've been here longer. That point across. I know it does. It's gone very quickly but I think magical things happen when you find that right place for you. Definitely. Let's have a look at another one of your pictures, Amber. Amber, please talk about your staff because I do want to say every time I go to Down to Earth, I always end up having a conversation with at least someone just standing by. The people that work at Down to Earth are so friendly and so willing to help. So please tell us about this slide. Yeah, absolutely. I'm glad you brought that up because our staff loves to share their knowledge, their wealth of information. They're all very passionate about health and wellness. This slide here is our produce manager from Kahului. This is Alex and he is here giving samples.  He will talk your ear off about the different farms. They all have really great relationships with local farms, because we try to source local when possible and support our local economy. So he'll tell you, oh yeah, brother man just brought in some really good bananas. Here you should try some. It's just a really great community and our staff will love to talk. So definitely ask them questions. They're there for you and they love to talk stories. So yeah. Absolutely. It looks like all of you are having a good time working there and I think that's what makes it so much more enjoyable as a shopper going there. It's just a really good vibe. I have to say I'm very biased when it comes to down to earth and if I can find all of the ingredients that I'm looking for, I will buy from there. I'm also a huge fan of the farmers market. The farmers markets I think are incredible as well. And what new products have you got in down to earth that you'd like to mention before we close the show? Oh boy, we have a lot of new products coming in every day. I would say the most exciting thing recently is we're going to roll out a value menu on our deli app. So kind of lower it's I think it's $6 and below. So it's it's like a veggie value menu. So that's coming soon. We're really excited to roll that out because we again, we want to make it accessible for everyone to eat, you know, plant based but still have that awesome flavor. So just a little teaser, but it will be coming out soon. What a great idea. That's awesome. I must say, though, down to earth is priced very, very reasonably for what it is. It's good quality produce that you use in all of your dishes. So I've never thought that down to earth was pricey in any way. There's another slide I do want to show Amber. There's this one. This mega, oh my god, burger. I showed this to my husband and he agrees like this is something that anyone could get excited about. Oh, definitely. So what we're looking at is our exclusive to Kapolei deli barbecue bacon burger. So we make all of our buns in house. So that's a housemade bun, fresh produce, lettuce, tomato, onion, sprouts. And then we have the crispy vegan bacon on top with some melted vegan cheese, your choice of one of our either housemade earthenburger or black bean patties or you can use beyond burger, which is kind of more widely known.  You just have to try it for yourself. So that burger is available at our Kapolei deli exclusively. Okay. Do you sell burgers at any of the other stores? We do. Yeah. Some stores just have, you know, made their own kind of special burgers. So at Kailua we have our double mushroom Swiss burger. Kakaako has a different version of their bacon burger. So yeah, we got a lot of things going on. I think the moral of the story is go to down to earth and check out their deli because you're going to find some really amazing things. I'm going to, I'm going to go as far as say and say, I think down to earth sells or some of the food that they're banging out in the deli is better than food in some restaurants. Thank you so much. Yeah, I truly mean it. It's amazing. Amber, it's been lovely talking and chatting with you again. Good luck with the 13th, the pop up. I'll definitely try and stop by and pick up some prezzies for Valentine's Day. Stay safe and before you go, a quick message to anyone, to the viewers. Oh, just stay safe out there. Come visit us and stay healthy. Perfectly said. Thank you so much, everyone, for watching another episode of Billions Vegan World. Thank you to Amber, our gorgeous host today, and I look forward to seeing you at the store. Stay safe, everyone, and until next time, aloha.