 So, let us start by recapitulating what we did last time. We described the Young's double slit experiment and we discussed that we cannot determine from which slit the electron goes and simultaneously observe the interference pattern. So, if you really want to ensure that electrons really goes through slit 1 or slit 2, you can ensure that is correct but on the other hand then you will not see interference pattern. And in case you want to see the interference pattern, you have to have the uncertainty how the electrons are passing through this slit. Then we said that if you really want to observe the electrons, it will perturb the experiment that is probably the reason and we will always perturb it enough so that the interference pattern will get destroyed. Then we discussed if we see the interference pattern, we may not be able to see how electrons reach their screen. Particles tend to be mysteriously when they are as tiny as fundamental particles. Then we showed a video about the double slit experiment. Then we started looking for a new equation which could describe the behavior and we started with trying out this particular equation. And we give some logic why you want to look at this particular equation because we want that the constant which relates the time derivative and the position derivative should not have any dynamical variable. Then we started with this particular type of equation. We realized that a typical sine or cosine function will not be a solution to that equation. So, this particular function has to be in the complex form. So, we started with a complex function A e is power i k x minus omega t and using that eventually we obtain this particular equation is at the end the last equation this is what we obtained this is what we did last time. Now, let us go ahead ok let me just comment on this thing the value of gamma indeed turns out to be independent of dynamical variable but does depend on m that is what we also discussed last time. We said that see we have not seen particles of different m having interference with each other electron interference electron beam interferes only when there is electron beam ok electron beam does not seem to interfere with neutron beam. So, this particular constant being dependent on m does not bother us. The second thing which we have said that we could have also used a wave equation of this particular form where A e is power minus i k x minus omega t this is what we discussed last time ok nothing will change except that the sign of gamma will change in that case the value of gamma would change would have changed no physics would have changed. But conventionally we always use equation of this particular form where time dependent part has a negative sign. See in the earlier one time dependent thing would have a positive sign because there is a negative here there is a negative here. So, that would have changed it to a positive sign there is a negative here, so conventionally we always used a type of equation where time dependent part has a negative sign. So, we always write A e is power i k x minus omega t that is the way we start with. The wave function turns out to be complex but that shall not pose any problem so long the observables turn out to be real. But I never mentioned that this particular wave which is associated with the particles generally we call them as wave function probably all of you know it. We shall later see the physical interpretation of the wave function which we will discuss today. Now, let us discuss a fairly abstract quantity called operators. In fact you know when normally I teach this course to first year students for now the from last two years the course pattern has slightly changed. But before that you know we used to teach relativity and after that we used to immediately come to the quantum mechanics. And relativity was generally shocking enough for the students and at that time I used to always warn them that you will realize that quantum mechanics is much more abstract and many of the students used to laugh they thought that you know relativity is abstract enough. I said relativity whatever you say you can see the arguments through it you know the arguments are very clear and concise you know see those arguments may shock you that is a different question. But they are very very clear cut arguments on the basis of which you go ahead. While here sometimes you lose track of arguments you know you talk of really an abstract thing which does not seem to be having any bearing with the real space. But that is the way you know quantum mechanics has been developed. So in that sense we are using a lot of abstract concepts which does not immediately you know are digestible. So we introduce a concept of which we call as an operator which is a very very important concept in quantum mechanics. Of course as far as this course is concerned I mean especially the first year course we just give a flavor of operators. We do not talk about the operator you know mechanics or for example that way the matrix mechanics we do not talk about it. So these operators have to be defined such that eventually this particular quantum mechanics should have consistency with classical mechanics. We know that when we are talking about bigger particles to obey classical mechanics they do obey Newton's law of motion. So in the limit when the particles are bigger when the wave aspects are not very very clearly visible in that case it is reasonable to assume that they will follow Newton's law of motion they will obey classical mechanics. In order that this consistency has to be met using those things these operators are defined. Unfortunately the first course of this type of modern physics type of course we cannot give lot of physical sort of explanation of why the operator should be defined only in this way and not in any other way. But I mean in a probably better course in a higher course on quantum mechanics we do talk about that particular thing especially the consistency with classical mechanics. Here we do we are not able to talk so we just expect the students to just take this as just digest it. So we define what we call as a momentum operator. I always tell that in operator only the names sound very big it is nothing very very important only creating a different name. It essentially means that when there is some function which operates on something gives as a result of something different. For example it could be just a operator could be just a square operator whether that is a quantum mechanically used or not that is a different question. It means on any quantity which operates it makes its value squared. For example there could be just differential operator when it operates on a particular function the outcome is a differential of that particular thing. It could be a sign operator which can operate on something and the outcome is sign of that particular thing. So word operator in its own sense does not have a very very special or very important or very strange meaning. So it just operates on certain functions and eventually gives you some result. So if you say that momentum or rather px is defined by an operator minus ih cross del del x it means if there is a function on which this particular operator operates what we have to do we have to take partial derivative with respect to x of that function and multiply by minus ih cross whatever is the result that we will get that is the outcome of operating that operator on that particular function. So it is a very very simple thing only thing in the name looks very very big and probably strange. Of course the question is that why this operator should be defined in this particular fashion for which I do not normally give an argument I only say that this is squared in order that all our loss becomes persistence. Of course when we come to the eigenvalues at that time we also talk that the eigenvalues of these operators have to be real. So we do talk little bit about that but we do not want to talk in great amount of detail about why this operator has to be defined in this particular fashion. So which I say that now eventually it means that if this operator operates on any function of let us say x and t then outcome is minus ih cross del psi del t. So this is actually the meaning of this particular operator. Then we define what we call as the energy operator which is similar thing except that there is no minus sign here you have ih cross and the operator is just del del t. So again I give the explanation that if it operates on psi x t the outcome is ih cross del psi del t. So this is what we call as energy operator. Now once we have defined this operator this particular equation that we have derived just now or not derived but the equation that we have obtained just now can actually be represented in terms of an operator. I also mentioned that the square of an operator means that this operator operates twice. It does not mean that whatever you have got you square it but it means it operates twice. So suppose we say cube of an operator it means that operator operates three times. For example if I take px square operator it means first you operate px then whatever is outcome again you operate px on that particular thing. So that is the way these particular operators are defined. So then we looked at this particular equation which we have earlier obtained. This was the equation which we had obtained for a free particle which was a relationship between the second derivative of x with respect to the first derivative on time. Then we said that this particular operation equation can be written in this particular form here. So you can just verify that this minus h square by 2 m del 2 psi del t x square can be written as 1 upon 2 m which we take it out minus ih cross del del x then again minus ih cross del del x psi x t which is reasonably alright because you take minus ih cross because these are just purely constant. You square it because they are appearing twice. So you will get minus h square. This minus h square is lying here. Then once you operate the first operator you will get del psi del x then you operate second time del x then you will get del 2 psi del x square. So this is exactly the same thing as this particular equation and of course this right hand side is clearly obvious which means that left hand side can be written as 1 upon 2 m px operator operating twice psi x t and equal to energy operator operating just on psi. And as we have said px operator operating twice can be represented as px square operator. So this equation can be written in terms of an operator in this particular form which has been written at the bottom line in the red. So this equation can in principle if we agree on the definition of the operators which we have used then this equation can be converted into an operator equation and this operator equation is px square by 2 m psi x t is equal to e. Now I say that basically it says it is essentially an equation of the type p square upon 2 m is equal to energy which is actually the case for a free particle that we remind you again which I have mentioned e here is only the total energy total classical energy it does not use relativistic energy. Now we come to the Schrodinger equation. The prescription is rather simple see earlier what we had done we had taken p square upon 2 m is equal to e and all that we did because in that case of the free particle there was no potential energy there was no force and just replaced px square by its own operator and e by its own operator. Now if the particle is experiencing a force and there is a potential energy involved with that particular thing this equation gets changed it is no longer e is equal to p square by 2 m but the equation now becomes e is equal to p square by 2 m plus v. If that happens I can use the same prescription you write this energy equation in the traditional form which is e is equal to p square by 2 m plus v multiply this equation by psi on the right hand side put psi on the right hand side. Of course normal multiplication does not matter whether you multiply from the left hand side or right hand side but because I am going to convert this to an operator equation therefore I decide to put this psi on the right hand side and replace px square by its own operator and energy by its own operator. So this is what we do of course this is a fairly abstract step which is not very easily understandable but nevertheless you know we one has to do it multiply each side by psi and the other side is that we use a lot of abstract operators in quantum mechanics but you know in every branch of physics we use a lot of mathematics and mathematics is abstract all the operators in mathematics is abstract what is so special here I mean I have one point you know we are developing a new theory which is which should be self consistent and experimentally very fabulous. Exactly. So that is the reason behind these things so if we say to the students that this is not understandable this is it may create more confusion. It depends on how you look and how the students look when I am trying to say and I am making this statement okay let me sort of explain why I am making this statement see it is true that mathematics is fairly abstract but on the other hand see right from beginning you know you can always answer this question suppose somebody asks this one question which always as why 2 multiplied by 2 is equal to 4 okay well of course when we teach students you know at that time they are so young probably they are not really asking the question why 2 multiplied by 2 is 4 and not 5 okay of course this is the definition okay and when we define this is fairly abstract why when we say 2 multiplied by 2 is equal to 4 and you could have also said 2 multiplied by 2 is equal to 5 okay 2 multiplied by 2 is 4 because that is the way we have defined it but why we have defined it because I can always give them example that okay there are lot of physical examples where this multiplication would help okay so I can give a lot of examples for every mathematical step even if you are talking of differential calculus already for that matter in digital calculus I can always give immediately some physical reasoning see look at this particular thing if you know for example you know your acceleration is not constant or velocity is not constant I can always take a derivative at this particular point essentially time taken let me complete it okay so in that particular sense okay for most of these mathematical things at least those which are used in physics we can always give some sort of physical argument okay see I am defining this way because this helps in this particular way now when I am coming to this particular abstract especially at this particular point where I have not really shown them that these they to define the operators in this particular fashion is necessary for consistency with classical mechanics okay people get a shock you know why you are talking of suddenly something which is I mean I do not understand see so long for every step you have been giving me some reasoning I may not agree with that reasoning I mean disagree with that reasoning but nevertheless you are giving some reasoning here other than making one statement that this is necessary for having a you know sort of consistency with classical mechanics you are not giving any reasoning in that sense it makes things abstract and why I always tell see things that you know at least in my teaching career I have always found out that if there is a step which is likely to be difficult for people to understand it it is better to warn them okay so that people are little more vigilant about that particular thing if I am going to go through a sort of difficult step for somebody you know talk of mechanics you talk of coriolis force it is a fairly complex complicated thing I warned them see remember this particular thing is little complex okay it is little more difficult to understand okay generally I find that students become more vigilant okay because they start thinking okay there is something strange here and then they apply their mind more why they think that is sort of good thing yeah now my point is isn't it better to refer to the postulates I mean we have some difficulties in explaining certain experimental facts as we did in special theory of relativity we had two postulates here also we have some postulates so these are the basic things on which we should develop quantum mechanics okay again I avoid that particular thing if I am doing a pure course on quantum mechanics for physics students okay probably that is the way I would do it okay the reason I do not do it in fact generally in quantum mechanics course where that is what I am planning to do today I do the postulates of quantum mechanics last I do not do in the beginning I mean this again the way I teach the course and I say see every teacher has a different way of teaching the course I mean I have been taught by many great teachers you know I can never follow them because they have their own style I have my own style okay so everyone has different style I have found this particular thing little more for first of all I think you are talking of quantum mechanical postulates they are not so well defined and so clean as for example you just lost a motion for example you tell anybody in your second law motion everyone knows what is your second law motion okay when you say you just start law motion everyone knows what is you just start law motion you do not have to tell okay if a person has read those things these are sort of very well clear similarly special theory related postulates first postulates everyone understand what is first postulate if you say second postulate everyone understand second postulate but if you say quantum mechanics what is And as quantum mechanics keeps on growing and you are talking about matrix mechanics, our postulates also keep on changing. So if I start right from the postulates, my fear is always that a large number of students who are probably, you know, finding this subject anyway is quite abstract and probably somewhat useless, which I do not agree by the way. But nevertheless, at that time, if I start talking of fairly abstract things, okay, I think my impression is that they generally get a shock. So I try to introduce things slowly and slowly, you know, which normally people I find difficult to understand. So I introduce operator here at this particular moment. Then I do not talk about anything. Then I try to do little bit of mathematics. Then try to solve one or two problems. Then I introduce something else about the collapse of wave function. Then I try all those things. So one by one, I start introducing things so people can slowly and slowly digest and then eventually when everything is over, then I talk about the postulates. That's the way I have sort of, I mean, I prefer to teach our engineering students. And generally I have found that this generally goes well. But I agree with, I mean, I'm pretty sure some of my colleagues, especially those who are critical physicists, I mean, they will not agree with me. You know, they would say, I mean, we had arguments in the department that we would say that, you know, you should start right from the operator business and you should try to start with metals mechanics because that's the more general mechanics. Okay, don't talk about wave function, don't talk about wave particle duality, okay. While I find it a little more convenient, because if you are going through wave particle duality concept, students slowly are able to understand that, I mean, why we are doing this thing is because of this particular thing. And wave concept, probably people can get a little easily, rather than talking of very upset concept like, you know, a matrix mechanics and saying that there are operators and let's talk whether they are Hermitian operators or not. And then start doing about that thing, which I find especially at the first year students. As I said, if I am doing a course on quantum mechanics for my MSc student, okay, probably that's the way one would do it. But first year student just coming out of high school, I think it's better to gradually increase the effectiveness in the subject. That's the reason I do that way. Again, as I say, it depends on the individual, you can always do it in a different fashion, if you feel that's the way it should be done. Okay, so this is what I said. So my prescription is that I start with this particular equation, multiply psi on the right hand side, then replace energy by its own operator and px square by its own operator. We get this particular equation. This is what we call as time dependent Schrodinger equation. In three dimension, this is the way this equation is written. So all that has happened, that del psi del x square or del 2 psi del x square is replaced by what we call as a del square operator. And I just define this particular operator. See, unfortunately we do this particular course. This is supposed to be the first course on physics in IIT Bombay, when they have not even done the electrodynamics course, which is actually the second course. So in the electrodynamics course, they know about these del square operators very well. They teach a lot about it. But in this particular course, because they are not familiar with del square operator, I just mentioned this particular thing and I do not talk about this much, until I do a particle in three dimension also. Then we try to say that many times it happens in many of the problems which. Sir, excuse me. Previously we were having operators, suddenly we are introducing, just converting the operators to the real values. Instead if we assume psi is equal to e power i omega t minus kx, if we assume. So omega can be replaced with e by h line and k can be replaced with p square by 2m. So without losing the operator, just we get the eigenvalues and it says the next equation, what you write immediately. I am not sure whether I understood your query. Here we are having p square and e both are in operator form. Next to next slide immediately you draw p square. See normally I start when we talk the free particle. We say that we know for a free particle the wave is off to form e raise power i kx minus omega t. Using that particular thing, we got, we use this equation e is equal to p square by 2m to get rid of that particular constant and make this, not get rid, but make that constant independent of dynamical variables. So I use that time equation e is equal to p square by 2m. Total energy government. Then I define operators and using those operators, using e is equal to p square by 2m, I obtain this equation. Yeah, that is all right. So if I forget about all those things, remember we are not deriving Schrodinger equation as he himself said that we are not really deriving it. Okay, we are just, I mean eventual proof is always the experimental proof because there are we accept this particular thing or not. So my prescription now, I mean holding the ear like this, I hold it in a different fashion. I say now you look at the equation energy conservation equation which is e is equal to p square by 2m, replace e by its operator, p by its operator. Then you will get into this equation. Suddenly we are changing the operator to the real values. I am not, I am not, I am never changing to the real values. No, no, no, no, no, I probably, I think I have not made myself clear enough. See what I said, I started with e is equal to p square by 2m. Okay, then land it up into that equation, a equation. Then I reinterpreted this equation in terms of operators. Yeah, instead what I am suggesting is if psi can be e per the previous definition of equation. Yeah, here itself, if you replace k by, using duality, k can be replaced with p square by 2m echelon. Yeah. And same way omega can be replaced with e by t, e by echelon t. But that's the way you operate. Yeah, but if you operate again, the operators form cancels around, you get the eigenvalues. eigenvalues as p square by 2m and e. That's certainly what I did last time. So eventually what I did, this k was replaced in terms of momentum and omega was replaced in terms of energy. That's what we did. Okay. Yeah, exactly. That gives an operator directly give the eigenvalue. So that gives an immediate. Yeah, I don't want to talk about eigenvalue at this particular moment because that's a little too much at this moment because I say eigenvalue, I want to reduce much later. So what I said that at least I know for a free wave, a wave should be looking like this. Okay. Using some sort of arguments, I obtained a differential equation. All right. See, let's accept the fact that this is not the only way that one can arrive at Schrodinger equation. If you look at different textbooks, every textbook is a different way of arriving at Schrodinger equation. Okay. I mean, many people, as I said, many of the great theoreticians will not even talk about Schrodinger equation. They will start from matrix mechanics and when necessary, they will give Schrodinger picture. So there are multiple ways of arriving at that equation. What I am trying to say, the way I introduce it is that I use this particular equation about which people have no arguments because they agree that this is a wave equation corresponding to a free particle, which is not under the influence of a force. Using this, I obtained an equation. Now this equation, I reinterpret it in terms of operators. And the equation that I obtained is E is equal to p square by 2m because that is the equation which I have used. Now all I am saying is that I want to generalize, I want to go one step further. Now instead of E is equal to p square by 2m, I am going to use the equation E is equal to p square by 2m plus v. That is the way I put the argument. I mean, I will fully agree with you that there are other ways of doing it. So initially what I had done, I had used only E is equal to p square by 2m and obtained that equation. That equation was not the Schrodinger equation because that still assumed that the particle is free. Sir, if such a particle is a function of value, it creates a function because v is a function of x. Of course. And p is there. So if you write some value for x, you will not be able to write any value for p because of uncertainly principle. So this equation is not valid. No, no, no, no. This equation, see let us put it like that. This equation is valid even in the quantum mechanics provided they are replaced by their average quantities. Okay? So about that particular thing, I mean, see, again we are coming because we are, no, you are doing about the advance things and then you are coming back to the thing. I am saying that let us start, I am looking at an equation, I am again repeat, I am not deriving it. Okay? If you find that this is not convincing, you can always use the argument which you find more convincing. I have absolutely not, I mean, I am only telling that this is the way I introduce it. Okay? So in this particular fashion, because I use e is equal to p square by 2 m and obtain that from that differential equation, which I reinterpret in terms of operators, then I generalize an equation e is equal to p square by 2 m plus v, okay? And I mean, I do not talk about the expected values, I have not even started talking about the expected values. I have not even talked about the probability interpretation of the wave function. I have not even talked about what is that wave means. None of those things I have talked. I have just introduced uncertainly principle in a funny way earlier, all right? So I am obtaining equation how I obtain, I am not deriving it. I mean, I mean, it is almost like similar arguments if I have to give for example, for a lot of this transformation. Okay? There are a lot of arguments about which you can have doubt, you can have worries. Okay? I am somehow I am landing at an equation. Okay? Some of the arguments may not be all that, in that sense, a perfectly valid arguments. Otherwise, I would have said I am deriving the equation. I am not deriving the equation. I am just somehow looking at some of the arguments and coming to that equation. So all that I am saying is that I started with this equation and now I am adding a v term and using that particular thing, I am going the other way. Now I start with this equation and rather than giving any other arguments, replacing these dynamical quantities by their operator. Sir, why I am worrying because if we interpret the operator directly to the eigenvalue or rather just p operator to p immediately. So if you ask to later on, if you ask the student to calculate the expectation value of p on some cases. So many times they, instead of replacing p with operator, the p operator with i h lentil. So many times the student replace just p square or p immediately. So that leads to miscalculation or some wrong results. I am not sure in what concept they are probably likely to be confused. But because see, at the moment I have not talked about any of the outcomes which I am going to get from floating equation. I am, I mean I repeat the same sentence. I am somehow obtaining an equation. Okay? So what I am going to say is the most general equation. These equations. There are many ways to repeat. This equation is not being derived. Let me complete it. Now using that particular equation, I am going to repostulate and postulate my many of those things. Okay? I mean it is almost like Bord's Atom, you know when you did, he used totally wrong arguments probably to arrive at any energy equation which turns out to be correct. Okay? I am giving some arguments. I am not saying that these arguments are 100 percent correct arguments. Okay? I am somehow lending up into an equation and once I come to that particular equation, okay, I use that particular equation. In fact, there was one of my third-edition friend, he said that he will not like to talk how I have obtained an opportunity equation. Okay? In fact, there was a plan, once we are planning to write a book and he was very, very clear that we will not talk about how the floating equation is arrived. I will just give the first step. Here is the equation which you have to use. How? Don't ask me questions. Okay? That is one of the ways of doing it. I mean as I say, I find it a little simpler to give some arguments. Okay? Those arguments may not be 100 percent correct. I mean I always imagine myself to be in that particular time when I have no idea of what is happening. Okay, that is what we should have done. And therefore, I am looking for something which can lead me somewhere and therefore using some arguments, some funny arguments, some abstract arguments, I will end up in an equation. Then I develop mechanics from that. I mean, remember, you know, this particular thing took almost 30 years to develop. Yeah, which is not agreeable for me or whether I have a difficulty is directly replacing quantum operator with a classical value. Yeah. So, that is what I am really not acceptable as worrying leaders. I would say that, you know, don't do this way. I mean, you can always say that, you know, this is showing an equation, don't ask question why it has been arrived. That is a perfectly valid way of putting the argument. All I am doing as I am saying, I find it a little more convincing for my students to tell them that this is the way I landed up this equation. Okay? These arguments, the path that I have followed, may be not 100 percent correct, may be full of arguments. Okay? Maybe some difficulties in understanding, but this is only to convince you that somehow if I am sitting in totally dark and finding out my ways, okay, I use something, you know, some small things to arrive at an equation. You may not agree with those ways of arriving, okay, that is the way I would like to say. Okay, so this is the way I put it. So I start with this equation and replace this by their operators. And then we arrive at this particular equation. And then finally we say that many times it happens that, and most of the problems that we do at least at the first year level and for that matter, even in the first course on quantum mechanics, the potential energy is only an explicit function of x, it is not a function of time. In that particular case, it is possible to separate the variables and obtain what we call as a time independent Schrodinger equation. So for that particular thing, there is a standard mathematical way where I can write the way function as a product of phi x and f t, where phi is purely a function of x and t is purely a function of time. So phi does not contain any time term and f does not contain any extra. Then I substitute this particular equation in this particular form, which I will probably go through rather very fast. And then this particular equation splits into two different equations. The time equation becomes comparatively simpler to solve, which is this equation. And this becomes the energy equation, which is the x equation. So this equation is when it is solved, then it becomes a e raise to power minus i e t by h cross. This minus sign unfortunately is not coming out very clearly in this transparency, so I should have put a gap. And comparing with the standard value of equation, because I know that this particular equation would also be valid for a free particle. And when it is valid for a free particle, then this e must be equal to the energy in that case. Therefore, I interpret this particular constant, which I have used as energy. Then write this particular equation, which we call as time independent Schrodinger equation. But I always warn them that remember whatever we are going to get as a solution of this particular equation is only the special part, the complete way function, you have to multiply it by e raise to power minus i e t by h cross. So that is very, very important, because many times, especially in first course, we do not talk about it. So people think that is, I mean, when I say the wave function has to be complex, some of the people who have read a little bit more say particle in the box wave function is not a complex. It is a pure sign term. I say what you are forgetting about the time dependent term, which still contains an i. So, when we derive this particular equation, all I am getting is phi x, okay, and actual wave function has to be multiplied by e raise to power minus i e t by h cross. For many of these things, multiplication of these things may not be very important, but there are certain cases where multiplication by this thing is going to be important. That is what I will discuss later. So this becomes our time dependent Schrodinger equation in three-dimension, and this becomes time independent Schrodinger equation in three-dimension. So this is just generalized in three-dimension. Our next question is what we do with these cases. Now I introduce the first postulate of quantum mechanics at this particular moment after I have got this equation. I introduce postulates one by one. As I said, I mean I like to define all the postulates always at the end. That is my style of teaching, okay. As I said, had it been a pure course on quantum mechanics, probably I would have started directly with Schrodinger equation and started with postulates directly. But the way I introduce especially to the first year students is introducing them slowly to these things. So then I introduce this particular thing that a microscopic particle is described by a wave function. This is the first postulate or this is one of the postulates which contains all the information that we can have about the physical property of the particle. The Schrodinger equation gives the position and time dependence of the wave function. So this is the postulate of quantum mechanics that for every particle there is a wave function associated with it and this wave function contains all the information that can be obtained about that particular particle. And this particular wave function can be obtained by solving the Schrodinger equation. So first we define that there is a wave function and this wave function contains all the information and how to find the wave function, the wave function has to be found after the solution of Schrodinger equation. Now let us forget about how we obtained the Schrodinger equation. We obtained the Schrodinger equation. Now we go ahead assuming that this is the equation. This is going to follow. This is going to be correct as far as we know. Then use this equation to find out various things about the particle. So now I come from the, I mean I always say that in quantum mechanics there is a development part of the quantum mechanics and then the use of quantum mechanics. So we are ending with the development part of the quantum mechanics now we are saying that how to use this particular thing in quantum mechanics. Then we say the probability of finding the particle between x and x plus dx is given by psi x t square dx. This is what I mentioned at this particular point. That this particular wave function actually is a sort of probability wave ok. Therefore the amplitude square which has to be real because psi x t square has to be real. This particular quantity will give me the probability of finding particle between x and x plus dx. So now I am coming to that particular part. Now I take some time to explain what is this probability because this is what a large number of students are fairly confused with. This is what I mean by probability. We imagine a very large number of separate identical particles described by the same wave function. If a measurement of position is done on all of them at a time t the result will not be identical. All right. Let us be clear about it. Let us assume that we do not have one particle but you have very large number of such particles which are all described by the same wave function and I make a measurement at a given time on all of them of position. The unlike classical mechanics I would not find a unique answer. I will find the position of the particles different at different points. Now if I say the probability of finding the particle let us say between x 1 and x 2 is 10 percent. Then in means in 10 percent of measurements I will find the particle to between x 1 and x 2 and in 90 percent of the measurements I will find it outside x 1 and x 2. So what I want to emphasize that when I am talking of probability this probability does not refer of probability of finding the same particle at different time. Rather on a very large number of particles on which measurement is being done at the same time. So this is something which is very very important to tell which I think most of the students are confused. If the probability of finding particle between x naught and x 1 is 0.1 then in 10 percent of measurements the position would be found between x naught and x 1 in 90 percent of the measurements is likely to be found that particle is not between x naught and x 1 but somewhere else. So when we have multiple system all this being described by the CMBA function the outcome of measurements is going to be different that is what is important. Probability is not defined over a measurement of a single particle. Suppose the measurement was made at a time t and the position is found to be at x naught and you make immediately a second measurement it does not mean that the particle will not be found at x naught it will be found somewhere else actually found at x naught. Once you have made the measurement otherwise there is no sense in having any experimental measurement. There is no sense in having an experiment. I have made an experiment and found the particle to be x naught immediately second time I make the measurement it will always be at x naught. It should be very clear that this measurement does not refer to this particular point that measurement is being done on the same particle multiple times. I make measurement now I find the particle here immediately after that I make the measurement I find particle there. No that is not correct. Once I have made a measurement and found the particle there next time the particle will be there of course if the time has elapsed the particle might have moved that is a different question. But immediately after that if I make a measurement the particle has to be there. So what is this probability? Probability is that I assume that there are large number of similar particles and I make measurement on all of them then I will find that this particular particle is here other particle is somewhere here third particle somewhere there. Okay that is what the probability refers to and not over the measurement of a single particle. Let me read it again. Defined probability is not defined over a measurement on a single particle. Suppose a measurement was made at a time t and the particle was found to be in the infinitesimal vicinity of x naught. If we take a measurement immediately after that it will still be found there. Experimental reproducibility. Otherwise there is no sense in experiment. At a time much later the wave function may again evolve understanding an equation. Measurements on different particles at the same time may however result yield different results. All one point which I want to emphasize very much. See whenever I am talking of probabilities I am talking of measurement of different particles. All being described by the same wave function. For a single particle it is not localized. So there is uncertainty in its position. So how can you say that if I measure now and immediately after that they will be definitely found at the same position there must be some uncertainty. I am not sure whether I have made myself clear. See when I am talking of the uncertainties when I am talking I am talking of a very large number of particles. I am talking about single particle. If they are in the universe if you have a single particle isn't it possible? It is possible to have a single particle in the universe. It is allowed by quantum mechanics. Let us say perform multiple measurements at different time. What is the problem? There is no problem. There is no problem. See it depends on what is the wave function of the particle. Because I have not come to that particular point. Basically the idea here is that once you have made the measurement you have disturbed the system. It is no longer being represented by the same wave function. And we use the word what we say the wave function has collapsed. Now the particle is no longer being represented by the same wave function. It is being represented by different wave function. Now it is going to follow the characteristic of a different wave function not of the first wave function. So whatever we are saying about the first particle particle is disturbed system has been disturbed now. Now depending on I mean remember see I mean unfortunately the way we have introduced uncertainty. We are talking in a very very generic fashion. But now once we are coming to the formal theory, formal theory has to be described in a somewhat different fashion. Now when we talk about uncertainty principle, uncertainty principle is that we make a measurement on a very large number of particles. I find what is the mean value of x. I find what is the mean value of x square. Use standard deviation and then eventually find out what is uncertainty in x. That is the way I am going to define it. It is true that this particular particle if for example it evolves, its wave function evolves as a function of time, at a later time it will be having different wave function. Then again I make a measurement. Then now its probability of finding that particular particle at a given position will be dependent on what is the wave function now and not what it was earlier. Now there are various approaches. These are somewhat philosophical which I generally find that students are excited about it. These were when the physics was being developed. Suppose I ask the question, if a measurement was done at the and particle was found at x is equal to x naught, what do you think was the particle there just before the measurement? If I ask the question that I made a measurement and I found a particular particle to be at x is equal to x naught. What happens? Can we answer some question just before measurement? So there is one particular idea of what we people call an idea of hidden variable. Let me just first read it. The measurement was done at the particle and the particle was found at x is equal to x naught. Where was it just before the measurement was done? This is one of the views which we call as a very realistic view or we call a hidden variable view. Indeterminacy could be different from ignorance. The particle was there only but we were not knowing. Some additional information was needed to provide the complete description. This is what is called the postulate of hidden variables. The example for example which we give. For example we always toss a coin and we ask head or tail and we assume that this particular thing is actually random. The chances of head coming is same as chances of tail coming. But strictly speaking if you know all the forces, the way I have tossed the coin, if I know all the air forces and all other forces, it is a 100% predictable motion. If I knew all the variables, if I could actually calculate the way I will do, I am sure that I am going to put either head or tail because perfectly we will define the dynamical quantity. Only thing the variables are so large and I do not have, I am not putting those variables all into place. I am not knowing all these variables. Therefore the overall answer appears to be random. So there are lot of hidden variables about which you do not know how you are throwing, what is the air resistance, how you are putting your hand there. So these are so many variables about which we do not have information so the information appears to be random, which probably is random in that sense. So probably the same thing applies for this particular particle also. We did not know, there are so many variables which define the way a particle is moving about which I had no information. Particle was there only. But we did not know it until I made the measurement because I did not have information about those hidden variables. Just like I was sure that it would be either head or tail. Only thing I did not know all those variables which make it either head or tail because I did not have information about those variables. So to me it appeared that was by chance that it came head or tail. So something similar could be happening that particle was there but particle was there because so many other variables about which I had no information. Therefore I mean I thought it is random but it is probably not random. So this is what is called the postulate of hidden variable. Now there is a second postulate which is called orthodox postulate. Which is also called Copenhagen interpretation. This is the one which you normally expect. I mean which is now believed. The particle was not really there. Our method of measurement forced the particle to take a stand to be present. In fact I perturbed the system and I actually made the measurement. At that time particle decided to be present there because I have changed the system. So at that time this measurement process of measurement made this particular particle come to that particular location. So it is a process of measurement which changed things. So this is what is called Copenhagen interpretation. And there is always a third thing which is generally called agnostic. Agnostic view is that okay why should I answer the question about which I don't have any idea. So forget about it. I don't want to answer. So Copenhagen interpretation is the one which is accepted these days. If you want to read more about this thing, probably one of the best books by Kirchhoff's introduction to quantum mechanics. Very, very latent book especially about all these arguments. Very beautifully explained in this particular book. Okay. Now we come to the normalization. I talk about that this particular thing wave function by the nature of the equation. If psi is a solution of this particular equation, A times psi will also be a solution to this equation. So this A you cannot determine by using any boundary conditions. This has to be obtained by normalization. Saying that the probability of finding particle anywhere in the universe has to be 1. So this is the equation which has to be governed. By which you can determine. This is what we talk about normalization that you know you can represent this d tau, various type of variable systems. Cartesian, spherical polar, cylindrical polar, whatever you want to do it. In fact, then I said that this particular square integrability that you know you can integrate this square, puts a condition where any arbitrary function cannot be actually a wave function. Okay. Even if it solves to take an equation in that sense. Okay. Square integrability has to be a proper condition. A wave function is what we call as has to be well behaved, essentially one of the condition of being well behaved is that it should be square integrable. The normalization is still not, I mean I talk about free wave, when we talk about free wave that you cannot strictly spring normalize it in that particular sense, because the probability of finding the particle is equal everywhere. Then we talk about the expected value that you know when you are talking of the mean value, when you are talking measurement on large number of system. So it is what I say in quantum mechanics, normally there is a probability associated with the result obtained after measurement. Expected value, the name is somewhat misleading, is the average value that one would get after a very large number of measurements made on identical system with the same wave function. This is the expected value defined. Then we say then many times you have to use something like mean value of momentum. So the general definition is that you use an operator. So instead of this, you use operator and it has to be written in this particular fashion, psi star g psi, which I am sort of going a little faster. So for example, mean value of momentum it has to be found out, then g will be replaced by its operator. If I have to find out mean value of px square, then I have to put px square operator which means px operating twice and this is the way I am going to find out px square. It can be shown that the expected value of momentum would always be real, which in fact I give you the problem generally. The operators of the dynamical variables are always such that their expected value turns out to be real. See these operators have to be chosen in such a fashion that their values are real. I do not introduce a Hermitian effect. Generally I avoid use of Hermitian in the first course on the quantum mechanics. See as far as possible I do not want to go and making too mathematically heavy, try to concentrate more on the physical aspect. Then for time independent vx, consider a solution of time independent variable and this is what I want to call about talk about stationary state that if I found the solution of Schrodinger equation and find it out to be fx, we realize that the actual solution will be psi into e raise power minus iet by h cross, but once you take complex conjugate of that, this time dependent term will cancel it out and therefore you need not take time dependent term into consideration if you are just calculating the mean value of x. So this is what I say that these are called stationary states where the probability of finding the mean values do not depend on time. Though in principle wave function has a time dependent term which is here, but when you take complex conjugate and multiply it and this will cancel out, so if a particular particle is in a stationary state if you have found the solution of Schrodinger time independent Schrodinger equation and take phi x, we realize that the solution is not really phi x, but phi x into e raise power minus iet by h cross, but when I calculate the probability I can ignore this particular term because complex conjugate of this will just make it 1. Then I give a particle in a box which I will skip as we discussed that we will not talk about the part. We often talk about time independent Schrodinger equation and time dependent Schrodinger equation, but in time independent Schrodinger wave equation we always have a term energy term and we know energy is always dependent on time. So how it becomes time independent? No see as far as the solution Schrodinger equation is concerned time independent Schrodinger equation comes it does never talks about time dependence because it will talk it assumes that V is only a function of x is does not have implicit time dependence. See there is always I mean I mean it does not have explicit time dependence there could always be implicit time dependence, but let us look at hydrogen atom or What does it mean implicit and explicit? Let me try to explain you know there is something explicit time dependence would mean that let us assume that wave potential energy is here something some value let us say V naught. After one second from V naught it has become V 1 this is explicit time dependence because at this particular location I am fixing the value of x but still the potential energy is changing as a function of time so there is explicit time dependence there is implicit time dependence that the potential energy here is V naught it always remains V naught but at this particular point potential energy is V 1 now particle moves from here to here therefore its potential energy keeps on changing ok so that is what I mean by it has a sort of implicit time dependence in that sense that as its keep on keeping on moving and because V is a function of x therefore the particle will experience different potential energy at different time. You see there is no difference in the formula time independent time dependent we can also derive time independent wave equation has a formula A plus B whole square that might be written A plus B square plus 2 AB and we can also write A plus B we manage B whole square plus 4 AB same thing so no difference so it is dependent upon Agentson ok thanks sir see the thing is that you know one more question sir psi in this case psi is time independent so that itself represents that that equation is for time independent case E is dependent on time but that doesn't make it time dependent Schrodinger wave equation because the particle is represented by the wave function and the wave function is time independent only psi acts so this is time independent Schrodinger wave equation no see I am going to also give an example little later when it is possible you to have a different type of wave function it is a linear combination which is not a stationary state in which you actually get a time dependence ok about that also I am going to talk about it therefore things are little more involved but what I want to tell you is that you know see if you find a particular particle in stationary state in principle there is no time dependence but there is a possibility of creating time dependence if you have overlap of wave functions about which I will be talking little later ok now let us come to this particular part which is about the superposition of wave function see when we started our Schrodinger equation at that time we always talk that wave functions can always superimpose I mean that is the way we started we rejected the original wave equation only by that psi 1 and psi 2 should be able to superimpose ok now question is that I have solved let us say particle in the box problem and I have found out different set of phi n's ok phi 1 is the ground state wave function phi 2 is the first excited state wave function phi 3 is like blah blah blah all those things you also realize that actual wave function will not be just phi n but one has to multiply by e raise power minus i e n because this phi depending on which n you are talking the energy of that will be different if I am talking phi 1 the wave function phi 1 will be phi 1 e raise power minus i e 1 t on h cross if I am talking phi 2 the wave function will be phi 2 into e raise power minus i e 2 t 1 h cross so therefore there is subscript n here there is subscript n there there is subscript n there now in principle this is also a valid state of the wave function like that can exist because this will solve time dependent Schrodinger equation what I want to say is that superimposition of wave function this is a valid state of the system where you have ground state wave function mixed with first state wave function mixed with second state wave function mixed with third state wave function this is a valid solution to this particular problem let me read wave function must be able to superimpose therefore the following is a solution of time dependent Schrodinger equation and is a general solution in fact this is the most general solution of particle in a box alright here phi n x are the solutions of the time independent Schrodinger equation corresponding to energies E n for the student show that if I have a state like this this will not be stationary state if you take psi star psi the probability will depend on time if you no longer be a stationary state but this is a valid state now let us put time t is equal to 0 in this particular equation if I put time t is equal to 0 this is what I am going to get because e raise power i e t minus i e t by h cross will become 1 now this will not be a solution of Schrodinger time independent equation this also you can verify that if I take phi 1 plus phi 2 you take under 2 by L okay sine phi x by L under 2 by L sine 2 by x by L okay this 2 will not be a solution of Schrodinger equation because one corresponds to energy even another corresponds to energy E2 okay when you mix them up there will not be solution of this thing but this is a valid state of the equation it means a particle could be found in such a state okay how do we interpret it further following is also a valid solution at time t is equal to 0 but is this the solution of time independent Schrodinger equation what will be the value of energy we shall get if a particle is found in the above state if there is a particle in the above state what will be the value of the state alright starting let us assume that this Cn2 summation of Cn2 is R1 Cn are the coefficient remember this Cn are the coefficient here so let us assume that summation of Cn2 is 1 then another postulate of quantum mechanics says that in that case if we make a measurement let us suppose we are talking of only 2 states mixture of 2 states phi 1 and phi 2 I have given the example okay this is the ground state wave function and this is n is equal to 4 wave function let us just take this example let us suppose we mix them up what I have talked about the authority of the wave function it is not bothered by the orthogonal so let us assume that wave wave function like this we agree that this is a valid state of the system at time t is equal to 0 alright this is what I have normalized it then if I make a measurement on this particular system then because this is a mixture of 2 states one corresponding to energy E1 another corresponding to energy E4 okay I may either get E1 or I may get E4 okay I will not get anything in between because the values are only E1 and E4 because this particular state is a mixture of 2 states corresponding to energy E1 and E4 so either I will get E1 or I will get E4 okay probability that I will get E1 is proportional to C1 square probability that I will get E4 is proportional to C2 square or C4 square okay so this is the way we talk about this particular thing if this is the one the physical interpretation implies imagining a large number of boxes where the wave function of the particle is given by the above if a measurement of energy is made in half of them we shall find particle to be 9 is equal to 1 state and half of them I will find the particle to be 9 is equal to 4 state because they are equally mixed C1 is equal to 1 upon under root 2 C2 is equal to 1 upon under root 2 so probability that is found in C1 state the ground state is proportional to 1 upon under root 2 square okay probability that is found in C4 state is equal to 1 upon under root 2 square and because it is already further normalized okay then in that particular case there is half probability of finding the particle 9 is equal to 1 state and half the probability of finding particle in 9 is equal to 4 state this is what we call as mixed states they are no longer stationary states now this is again a question which I normally give there is certain interesting implications let us just go through this thing first I say is this wave function is a normalized wave function it does turn out to be normalized wave function because this can be written as 1 upon under root 5 5 1x plus 2 upon under root 5 5 4x okay this square plus this square is 1 so this is C1 this is C4 C1 square plus C4 square turns out to be equal to 1 because this is 1 upon 5 this is 4 upon 5 which happens to be adding to 1 okay it means that if I make a measurement on system which is described by wave function like this okay and we have multiple boxes in 20% of the boxes you will find that the particle will be in n is equal to 1 state and 80% of the case the probability the particle will be in n is equal to 4 state this is what is the interpretation what will be the expected value of energy expected value of energy is simple that you just take what is the probability 0.8 times E4 expected value of energy is just a mean energy which can be different from E1 and E4 which will be some sort of average alright so the mean value of energy need not be equal to actually one of the eigenstates you know eigenvalues now this is the question if no measurement was done what would be the wave function at time t this was the wave function at time t is equal to 0 if we do not make any measurement what will be the wave function at time at later time t see each term will evolve with its own time dependence first term will evolve with e raise power minus E1 t upon h cross the second term will evolve with e raise power minus IE4 t upon h cross because each term will have a different time evolution so this will be the wave function at a lesser time if no measurement was done if a measurement is done at one of the boxes at time t is equal to 0 and the energy is found to be E4 what will be the wave function at a later time t this is interesting question if I made a measurement and found the particle to be at E4 at that instant of time the wave function collapses the wave function is just becoming 5 4 because you have disturbed the system ok this is what we call is a very standard term the collapse of the wave function wave function has collapsed you have created a state initially ok now you made a measurement particle is no longer in the original state state has changed alright so if I make a measurement now ok my wave function has become there system is disturbed wave function has collapsed alright this will be if I make a measurement then at a later time t the wave function the first term is disappearing now now the next question what will be the measurement of energy yield on this particular box on which a measurement has already been done ok it will always be E4 there is no chance that it will come back to E1 for the wave function has collapsed no longer representing the same I think with this example probably some of the arguments that we had earlier can be understood better that once we have made a measurement system is disturbed now I find I will always find out the energy to be E4 remember even with this particular wave function because this is always basically this particular term is always in the original state ok now this is in the stationary state ok the other missed state has gone so once it is in the stationary state at all time because time dependence has no meaning as far as energy is concerned you will always find the energy to be E4 so once in a box you have made a measurement and found the energy to be E4 it will always remain E4 now there but if you have not had made a measurement then you do not alright this of course gives to the famous you know as I am avoiding talking about paradoxes may I pardon if I determine the position of the particle so the basic thing is that if you are talking about the position measurement eventually whatever the system has to be always represented in terms of the Eigen value of the position operator ok then you look at those coefficients and find out what is the coefficient square and that will give you the probability ok or let us not go into that complication let us go to simple case because I already know that this is a wave function ok whatever is the wave function I mean then I have to compare ok let me complete it then I have to compare with those boxes which all of which have the same wave function my system is different now ok now I have a box ok in which the state of the particle is given by E4 ok I know what is the wave function I can find out what is the probability of finding the particle let us say between x is equal to 0 and let us say L by 4 alright I can find out the probability knowing this size square once I find the probability and I say the probability is let us say 10 percent ok then this 10 percent probability has to compare with identical boxes which are all being represented by the same wave function not the original wave function because the original wave function no longer exists sir we need to have we collapsed it has collapsed now wave function is different we will always get this wave function in this box unless you create another perturbation and because of reason you recreate your wave function that is a different question it means the wave function is changed that is what I mean the word which people use in quantum mechanics is collapse of wave function ok but effectively it means that wave function is changed now that you are saying that energy has that wave function is collapsed so it has taken a one definite value so now energy is completely determined so it has infinite precision that you can talk about so what about this so in fact that means in the energy time uncertainty relation you will have infinite uncertainty in the time can I make a second measurement of I mean what is going to what does it mean by having an infinite uncertainty when we are talking about time, energy, uncertainty we are talking about basically a wave state which is created with that particular type of energy first of all if it makes a transition which is totally different type of quantum mechanics when you are talking about time dependent time dependent production then you talk about time dependent ok since there is ok because that is not existing here so that is what we do not talk about I think as a teacher which I have one philosophy which I have always found you know just want to share with you know see let us all accept that we do not know everything ok so I mean there is no harm in accepting I do not know ok this is something which I have found very very useful throughout my career I do not know everything let us accept it I will do my best I expect others to do their best ok and the second thing that no question is useless every question has to be welcome I may not be able to answer it that is perfectly alright ok that is part of the game ok but on the other hand questions are always welcome in fact I mean I always tell my best teachers have been my students because many times they ask me questions which I have never been able to answer ok and that is what has forced me to learn from those questions otherwise I would have never sort of thought that whatever I have learnt is so this is what so that is the way every question is welcome in fact I will sometimes if we meet informally I will talk about some of the questions which students have asked me and some of them are very very intriguing questions excuse me sir when we are dealing with quantum mechanics we are always considering a particle in a box and we are applying some Schrodinger equations now what is the real time problem that we can apply that equations I mean see a particle in a one dimensional box of course people do apply for the case of you know emission of the alpha particle emissions and things like that ok see let us see to make a problem really realistic in quantum mechanics is much more difficult there are very very few problem in quantum mechanics which can be solved exactly most of the time you have to have some sort of perturbation you have to have some sort of approximation by which you solve those problems so those problems tend to become actual problems become much more difficult at least in this case if you go to a little more realistic situation a particle in a three dimensional box ok that entire solid state of physics that is what we apply entire semiconductor physics we apply the same result so the standard answer is quantum well in which we are confining our carriers charge carriers and we are studying the behavior but that is not in finite box that is actually finite box people can give a lot of examples but you know the thing is that generally what students generally ask that these are highly idealized situation which I agree that these are highly idealized situation ok because again my problem again I say the same thing that you know if you are preparing for examination you do not want to start with the most difficult problem first you want to start with the simplest problem first you understand the simplest problem then you go to more and more difficult problem so that is what we do I mean even physics that we start with the simplest problem which is easier to understand easier to solve it rather than trying to make it more realistic because in order to make realistic the problem may become so complicated that I may not be able to solve it so first let me solve a simple problem and then go slowly ahead and then try to find out go to more and more difficult problem ok I think we have another 10 minutes so let me just quickly blast through I have only few transfers left so what would be the measurement of energy yield on this box at a later time the particle is now in the stationary state hence the measurement would always lead to e4 that these are the value of x mean value of x in fact I told you about the uncertainty principle so I can for particle in a box I can calculate x mean value of x I can calculate mean value of x square I can calculate the mean value of px which turns out to be 0 I can calculate the mean value of px square then you can calculate delta px and delta x delta x and this is what happens to the uncertainty principle for n is equal to 1 it turns out to 0.57 h cross for n is equal to 2 it turns out to 1.67 h cross it is only for Gaussian wave packet that it will turn out to be exactly 0.5 h cross now this is the last thing that we eventually do on quantum mechanics the postulates of quantum mechanics after that of course I try to solve some other problems but you know this is generally in the formal quantum mechanics this is the last thing as I said you know I do the other way number one system description and time evolution a particle under a potential vx is described by wave function which contains the information about the all physical properties of the particle the time evolution of psi x is governed by the time dependent Schrodinger equation the wave function psi x is a single valued finite and continuous function of x this is what we call as a well behaved wave function the position derivative del psi d lx is also continuous unless vx shows infinite jump this is a statement which I have taken from Crane's book which personally I liked very much so this is a statement in which talk about classical and the quantum mechanics so in bracket I have written something which is classical so first I will read only the classical statement then I will read the quantum mechanical statement when an object moves across the boundary between two regions in which it is subjected to different forces the basic behavior of the object is found by solving Newton's second law the Schrodinger equation the position of the object is always continuous across the boundary and the velocity is also continuous as long as the force remains finite this is a classical statement which I think two will agree now same expression is replaced using quantum mechanics by using the red things when an object moves across the boundary between two regions in which it is subjected to different potential energies because in quantum mechanics we talk of potential energies the basic behavior of the object is found by solving the Schrodinger equation instead of Newton's law instead of position now it becomes wave function the wave function of the object is always continuous across the boundary and instead of velocity derivative d psi dx because that depends eventually on momentum the derivative d psi dx is also continuous as long as instead of force change in potential energy remains finite this is sort of convincing that you compare with the classical mechanics of quantum mechanics ok that is why we expect these things to be continuous then here I introduce operators little more formally each dynamical variable that relates to the motion of particle can be represented by an operator satisfying certain criteria the only possible results of measurement of the dynamical variable represented by an operator is one or the other Eigen values of the operator this is where I introduce Eigen values before that I do not talk about Eigen values the Eigen values are real numbers of the operator Eigen values are real numbers for the operators representing dynamical variables and for dynamical variables Eigen values have to be real then I talk about Hamiltonian operator so just saying that this particular Schrodinger equation which we are talking time independent Schrodinger equation that I am talking is actually an operator equation which is an Eigen value equation replacing thereby an operator then we talk about the completeness ok which again I mean rather hurriedly the Eigen states of an operator representing the dynamical variables are complete any admissible wave function can always be expressed in the following way in terms of Eigen functions of any operator these Eigen functions form the basis we had discussed this aspect in detail in Eigen function of the Hamiltonian operator that we did probability that an Eigen value g n would be observed as a result of measurement is proportional to the square of the magnitude of the coefficient g n in the expansion of psi the proportionality becomes equality if you have normalized wave function collapse of wave function if the measurement gives a particular value of Eigen value g n the wave function discontinuously collapses at phi n I think that is it so I mean these are some free state problems I will just upload the transparency if you want to have any questions you can always ask me tomorrow also I am available ok before that you know if you have time I will just mention about because somebody asked me this particular question that why we express all the wave functions in this particular thing in this particular case so boundary condition has to use the direction therefore see for a particle in a box instead of sine kx plus cos kx I could have used e raised to power ikx and b raised to power minus ikx nothing would have changed you will get the exact same solution ok but here it is necessary because I want to assign a direction so here we say that the particle is coming from the left hand side and there is a step potential and the energy of this is lower than this so classically I would expect that all these particles will cross over because classically if it has to cross over it has to pass through a region where kinetic energy is negative and if kinetic energy is negative it means the velocity is imaginary which does not make any sense so it is not possible in classical mechanics for such particle to cross over to this particular region however when you are trying in terms of the wave you can always think that even if the particle is material is opaque if you can make it thin enough there is always some light which will pass through it so this is that particular thing and realizing that the particles have a wave behavior ok we can write the wave function that three different regions these are the wave functions that we write we apply boundary conditions here this particular particle in region 1 can also travel backwards because that can get reflected from this particular thing so there is a, I mean see remember just by looking at this term we cannot say in which direction it moves but we should always remember there is minus i e raised to power ep by h cross time dependent term is negative so advantage of that way is that this particular term this positive sign means that this is a wave traveling in plus x direction and minus sign means that this is a wave traveling in minus x direction ok so here both these waves are possible but in this case this particular term will not be possible because once the particle has crossed over here then there is no way that the particle could come back so g has to become 0 so we put g is equal to 0 solve this equation and then eventually get the transmission coefficient so this is the way I do it I never actually worked it out to complete time the transmission you know there is a lot of mathematics for a word you see I think that is it