 So, now we know that quantum mechanics is very different from classical theories. It is a probabilistic interpretation of the microscopic world. You see in classical theories we are capable of predicting the exact position and momentum of a particle and its trajectory for all of time as long as we know the forces involved, the particles initial conditions involved. Quantum mechanics on the other hand is probabilistic, it is indeterministic, it is incapable of predicting exactly where the particle is going to be in the next moment. It is incapable of predicting what is the energy or the momentum of the particle at the very next moment. What quantum mechanics provides us, however, is a probabilistic interpretation. It can give us an idea of what the likelihood of the particle's position, the likelihood of a particle's energy and the likelihood of how a physical system is going to evolve. Now how is that probabilistic interpretation relevant to us in terms of the theory? You see it gives us an idea about what is known as the average of certain quantities and therefore today's video we are going to talk about the expectation values, the expectation values of position, the expectation values of momentum and in general the idea of operators in the mathematical theory of quantum mechanics. So hi, I am Divya Jyothidas. Welcome back to For the Love of Physics and as we have been discussing till now, the most important equation in quantum mechanics, the Schrodinger's equation. So whenever we try to study a given quantum mechanical system, what is it that we do? We start with this equation. We try to plug in the information about its potential, we try to solve it and when we solve it, we end up coming up with what is known as the wave function solution of the Schrodinger's equation. Now this wave function solution is most of the time a complex mathematical function and because it is a complex mathematical function, it doesn't have any physical meaning. It doesn't represent a physical quantity. However, just because it doesn't have any physical meaning or physical significance doesn't mean that it doesn't have any relevance. It does have some relevance in the context of the quantum mechanical theory because it contains information about the particle. You see, okay, fine. The quantum mechanical theory that we are talking about cannot exactly predict where the particle is going to be or exactly predict what the energy of the particle is going to be or the trajectory of the particle is going to be, but it does contain some information. And given all the limitations that we have seen in the context of quantum mechanical theory, let us try to figure out what information the theory actually reveals to us about the evolution of a physical system or the evolution of a microscopic particle. So this wave function solution contains some information and today we are going to look at what is that information that we can extract out of it. And one of these things are known as expectation values, which basically gives us an idea about the average of the position of a particle or the momentum of a particle if we make a large number of measurements on similar systems. But before we go into all that, I actually want to give you a rigorous sort of a discussion on the particular subject. So let us do some minuscule revision of probability theory. So let us go back and revise some of our understanding of what probability is, what average or mean values are because that is going to give us the right framework for coming back and analyzing the wave function. All right. A quick announcement for my students who are preparing for CSI and Net Physical Sciences or GATE Physics. We at Elevate Classes organize live classes for these examinations every six months. The next batch is starting January. So this is a full-fledged live batch with live interactive classes, recorded access, test series and everything that you need. It's a complete package. So if you're interested, you can check out further details at ElevateClasses.in. You can avail the early bird discount coupon, which is available for the first 50 students. And if you're not really very sure about these online classes, then the first week of lectures are free to attend upon registration. So Elevate your physics preparation with Elevate Classes. So let's just do a quick revision of the average values in the context of probability theory. And without taking too much time, let me give you a very simple straightforward example of what I want to discuss. Let's say that I have a class and in my class, I have a large number of students and I want to understand the distribution of, let's suppose, their age. And I could have taken any other variable, their height, their marks or anything else. But for the time being, I just want to do analysis of the age distribution of the students in my class. So let's suppose that I have around, I have a given data set. Let's suppose I have two students in my class who are of the age of 20 years. All right. And I have four students in my class who are of the age of 21 years. Okay. And I have three students in my class who are of the age of 22 years and two students in my class who are of the age of 23 years and one student in my class who is of 24 years old. This is a simple data set and I have given this data set just because I want to highlight a few key points that I'm going to use here in our quantum mechanical theory. So here I'm trying to study a variable and that variable is basically age. Okay. I'm trying to look at the distribution of age of the students in my class and I'm going to say that this variable is J. Okay. This variable is J. And on the left-hand side we have the number of students corresponding to that particular age. So I'm going to call that as capital N. This is the number of the students or the number of people having a specific age which is I'm representing as J. So if I calculate what is the total number of students then I essentially do a summation of N. Okay. So N is the number of students having a specific age J. I am using J to represent the age here. And if I do a summation for all the values of J then this basically gives us the total number of students which comes out to be what? 2 plus 4, 6 plus 3, 9 plus 2, 11 plus 1, 12. Okay. There are total of 12 students in my class. Now I want to understand a couple of things. First of all, I want to understand that if I pick a student at a random in my class then what is the probability that the student has a specific age? So for example, I'm sure that most of you are familiar with the basics of probability theory. So if I pick up a random student in my class and I say what is the probability that the student is of 20 years age? Now what is the answer to that? How many students are there that has 20 years of age? Two students. What is the total number of students in a class? 12. So 2 upon 12 is equal to 1 by 6. Fine. The basic definitions of probability. Now if I say what is the probability that if I pick up a student, the student is of age, let's suppose 21. Again, that is simple, 4 upon 12, which comes out to be 1 by 3. If I now say what is the probability that the person is 22 years old, that is 3 upon 12, which comes out to be 1 upon 4. And then what is the probability that the person is 20, 21, 22? And then 23 years of age, that comes out to be 2 upon 12, which is 1 by 6. And the probability that the person is of 24 years of age, which is 1 upon 12. So these are the probabilities corresponding to if I pick a random student in my class, what is the probability that the student has that particular value of j or that specific age. Now notice here a few things. First of all, what happens if I add up all the probabilities? Again, if you're familiar with probability theory, then I should end up getting 1, right? So if I do a sum over of all the probabilities, that is p and j. So how many values do I have here? So I have all these values, 1 by 6, 1 by 3, 1 by 4, 1 by 6, and 1 by 12. So if I add up all these, so in the denominator, I have 12. And here I have 2 plus I have 4 plus I have 3 plus I have 2 plus I have 1. So this should give me 4 plus 3, 7, 8, 9, 10, 11, 12 upon 12. This comes out to be 1. So the sum of all the probabilities is equal to 1, fine. Now what if I want to find out the average age in my class? I have this many number of students. What is the average age of the class? So I want to find the average of j, all right? So j is age, essentially, j is age. So if I want to find out the average age, then the average age is, so there are two students, 20 years of age, two times 20, right? There are four students, 21 years of age, four times 21. There are three students, 22 years of age. And there are two students, 23 years of age. And there is one student, 24 years of age, all right? So I do this multiplication and then I divide by the total number of students, which is equal to 12. Now this actually ends up coming out to be around 260. Let me just give you the numbers, 260 upon 12, which comes out to be 21.67. 21.67 years of age. Now notice one thing, that this is the average age of all the students in my class, but there is no student who is exactly this age, all right? There are students of 20 years, 21, 22, 23, 24, but not exactly this. So this is just a mathematical value, which gives us an idea about the mean of the distribution of the age of all the students in my class, all right? Now let's look at this particular quantity. What have we done here? So if you look at this particular expression, the general expression, if you take a look at it, then what is this general expression? What I'm doing here? So on one hand, I'm multiplying the number of students corresponding to that age and I'm doing a sum over. So essentially if you look at the general formula, I am doing a summation of the age corresponding to that particular number of students, all right? So I have the age, which is 20 years times the number of students belonging to that age and then I am dividing it by total number of students, which is nj here, okay? Now what is basically the probability in this case? If you look at the probability, again, let me just give you the general idea of probability of finding this specific j or age. The probability is simply the number of students having a certain age divided by the total number of students, okay? So nj upon summation of nj is the probability of finding j in our data set. So this can be simplified as, I can write this as summation, you have j and then you have nj upon summation nj, which is just the probability of finding j here. So this is essentially speaking the average of j, all right, or finding age. So the summation over the entire data set of finding a variable j. Now this is what I wanted to obtain because this is where we will start with the quantum mechanical solution and as how we can interpret this in this particular context. So if I have a variable and I want to find out the average of the distribution of students corresponding to that variable in a given data set, then the most simplest definition is this for a discrete data set, okay? Now here I have taken j to be what age? Now I'm not just interested in age, I could as well have studied their height, okay? If I had a data set of somebody being 5 foot, somebody 5, 5, somebody 6, somebody 6, 5, I could as well have used the same formula to find out the average height of all the students in my class or I could have used to find out let's suppose the distribution of their marks. Let's suppose I have two students getting 40 out of 100, three students getting 50 out of 100, five students getting 60 out of 100. So I can find out the average marks. So whatever variable I'm interested in, whether it is age, height, marks, as long as I have a given data set of how many students correspond to that particular variable, I can find out the average of that particular variable using this particular formula. Now this is a formula for a discrete data set where the set of data is a very discrete, you know, the specific numbers. What if we had a continuous distribution? So this is for a discrete data set, okay? For a continuous distribution, it is slightly different, okay? So for a continuous distribution what we are going to do is we will look at the probability of a certain variable in a given limit of that variable. So what we do is we said, okay, j average is essentially equal to, we replace the summation with an integration, j, the probability of the particle corresponding to j and then we have dj here, okay? So this is the basics of probability theory that I'm going to talk about here because this is all that is relevant to us. I hope you have understood what we have discussed. We are just interested in a certain data set and the average corresponding to some kind of a variable. Here we have taken age as the example. Now, if I use this formula in the context of the Schrodinger's equation, what can we do? You see in the Schrodinger's equation, I just now told you that we have the fundamental equation and its solution, which is a wave function solution, which contains some information about the physical system. Now, we have done discussions previously where we talk about probability, right? I specifically talked about the bond's statistical interpretation. Remember the bond's statistical interpretation? The bond's statistical interpretation simply tells us that whenever we have a wave function solution, then that wave function solution contains information about the likelihood of where the particle is going to be in a given region. Now, because the Schrodinger's equation is for 1D, so we are essentially assuming that the particle's motion is restricted to 1D, which is the x-axis. So along the x-axis, what is the probability that at a given location the particle can be found? And the probability that the particle can be found at a given location is given by the wave function mod square, okay? Where the mod square is simply nothing but the complex conjugate of the wave function times the wave function itself. So at a given location, of course, psi is a function of x and time. So at a given location x, at a given time t, if I find psi mod square, that gives us the probability or the likelihood of the particle being found at that location, at that instant. So this is basically a probability corresponding to if I made a measurement of the system, the particle will be found at that location, at that point in time, is given by psi x t mod square. So this opens up the possibility of what is the average distance at which the particle is going to be found? Because I have given you an idea about the average of any quantity instead of age, height or marks. What if I want to find out the average of position? The average of position, if I made a large number of measurements on a given system. So that formula can actually help us in finding out that average. So let's suppose that we want to make a measurement in finding the position of a particle. So because I said that the particle is restricted to 1D axis, all right, so this is let's suppose the x-axis. And I make a measurement. And when I make a measurement, I find that the particle is let's suppose here. I'm going to call this as x1, all right. Now I again prepare the system and I repeat the measurement of the position again. This time I find that the particle is let's suppose here. Let's suppose x2, all right. Again, I repeat the preparation of the system and I repeat the measurement. And I find that the particle is this time somewhere here. All right. Remember what I said? Quantum mechanics cannot exactly predict where the particle is going to be at a given point in time. So therefore, whenever we make measurements, we find that the particle is anywhere in the x-axis. Now what quantum mechanics does provide us is a probability of the likelihood that if I keep on performing this kind of a measurement over and over and over again, what is the likelihood that the particle will be in one of these locations? So let's suppose the probability that the particle will be at x1 is given by px1, all right. The probability that the particle will be at x2 is given by px2. The probability that the particle is at x3 is given by px3. How can we know px1, px2, px3? We can know that by psi mod square. So the psi at x1, mod square will give us the probability or the likelihood that if I keep on repeating this process of measurement over and over and over again, then how many number of times the particle will be somewhere here, all right. So that is what is provided by the bond statistical interpretation that we have a likelihood of whether the particle is going to be based on psi mod square. Now, I want to find out the average of all the measurements. If I perform a large number of measurements, what is the average value that we will get or the average of the position? So based on this idea, based on this definition of the average of a given quantity, according to probability theory, now I am equipped in finding out what is the average of x. So if I make a large number of measurements on similar systems to find the position of the particle, then what is the average value I get? The average value I get, because let's suppose this is a continuous distribution of data points, the particle can be anywhere in the x-axis. Therefore, I use this expression, which is an integration of x. The probability of the particle being found at a given location x dx in the entire x-axis, which goes from minus infinity to plus infinity, all right. So this is the formula for figuring out the average of the position of all the measurements that we perform on a similar systems of a given quantum mechanical system. Now what is px? Px we know from the Born Statistical Interpretation is essentially nothing but the wave function solution of the Schrodinger's equation, mod square, all right. So if I a little bit, rightly in a different fashion, the expectation value or the average of the position of all the measurements comes out to be. So here I have psi star psi. So I'm just going to rearrange the terms a little bit. I'm going to write psi star first here, okay. And then I'm going to write x here because it's essentially a multiplication, nothing else is happening. And then I'm going to write psi here and then you have dx between minus infinity to plus infinity because the particle can be anywhere between minus infinity to plus infinity. This gives us a formula for finding out the average of the position of the particle of that given quantum mechanical system if we performed a large number of measurements and this quantity is known as the expectation value of x or expectation value of position. So this is known as the expectation value. So the expectation value is essentially speaking the average, the mean of a large number of data points. All right, now please be careful about the word expectation. The word expectation feels like if I make a measurement I should expect this result but technically speaking that's not what it is. Technically speaking expectation value is simply the average of all the measurements. Now do you remember when we found out the average for the age of all the students in my class? I got a number which is not exactly corresponding to anybody's age because I had students in my class of 20 years, 21, 22, 23, 24 years but nobody having the age of 21.67. So this is just a number that gives us an idea about how a sort of a data set is distributed. It doesn't necessarily mean that somebody has this specific age. So when we get the expectation value it doesn't mean that that is where the particle will be found, no. That is just the average data point, the average position of all the measurements that we make on a given system. Now here again I have to be a bit careful about the measurement process. You see in quantum mechanics, I've tried to elaborate earlier that the measurement is something that is very unique. When we measure something, the waviness of the particle becomes the particle. Okay, so the particle is now concrete. In quantum mechanics, whenever we make a measurement we say that the wave function collapse into one of the specific eigenvalues. We are going too much into the detail. The act of measurement disturbs the system. So what we do is we do not really make repeated measurements on the same system. What we do is we make repeated measurements on similar systems. So if I have a large number of systems prepared in a similar fashion and I perform a large number of measurements on them, then the average of the measurements of all the positions on all the systems will give us this. So for example, if I have a large number of hydrogen atoms in the ground state, in that situation if I make a measurement of the position of the electron in a given hydrogen atom, I might find the electron to be at any given length from the nucleus. Now if I look at the specific radius for all the hydrogen atoms prepared in a similar fashion and I find out the average, now the average gives us an idea about where the distribution of the probability of the position of the particle in the hydrogen atom actually is. Now that is what the quantum mechanics as a theory can predict. It cannot predict that if I have a hydrogen atom where the particle is actually going to be if I make a measurement, no. What it predicts is if we do have a hydrogen atom in a certain given wave function state and if I make repeated measurements on similar systems, then what is the average value of the radius of the electron from the nucleus? That is what the quantum mechanical theory can actually predict and therefore, the expectation value of position becomes so very important because the quantum mechanical theory is not really giving us too much window of figuring out how the microscopic world behaves. They are just giving us some little bit of an understanding of what we can predict. So this is something that the theory can actually predict. What is the average of a large number of measurements on similar systems for the position of a particle that is the expectation value of position? Now just like I found out the average for the position, we can also find out the average for momentum, energy, et cetera, et cetera. So let me move forward the discussion to momentum. So we have found out the expectation value of position. Now let's find out the expectation value of momentum. All right, so next we want to find out the expectation value of momentum. But before we do that, one small addition I want to make that this is the expectation value of position if the wave function is normalized. Now, if you remember in my last lecture we talked about the various conditions that we impose on a wave function, right? One of the conditions was normalization. This is because I assumed in my definition of the average of a quantity to be such where the sum over of the probabilities was equal to one. Remember the previous slide, the summation of the probabilities was equal to one. So many times what happens is that whenever we are dealing with wave functions we are sometimes dealing with wave functions which are not necessarily normalized. In that case what we do? So this is the formula for a normalized wave function. If we have a wave function which is not normalized we can either normalize it first. So by normalization what we do is we multiply with a normalization constant. Or if we don't do that then the most general formula for the expectation value of position is integration psi star x psi dx, okay? Upon integration psi star psi dx. Now for a normalized wave function the denominator is actually equal to one. So for a normalized wave function we end up getting the same expression. For an unnormalized wave function if we include this then this gives us the actual expectation value. So this is the most general sort of an expression. Now let's move on to expectation value of momentum. You see in the similar fashion we can also find out the expectation value of momentum of the particle. Now to find out the expectation value of momentum what I'm going to do is I'm going to take a time derivative of the expectation value of the position. So I have figured out what is the expectation value of the position? What if I do a time derivative of that, okay? If I do a time derivative of that that will give us an expectation value of the velocity of the particle which is not the velocity of the particle. Again going back to our general interpretation of the quantum mechanics we make a large number of measurements on the system. And out of that the average value of a certain quantity is what the theory can predict and that's what we are discussing. So this is the average velocity of a given system which is equal to this. So if I plug in all these values here so let's say I want to do a time derivative of average x. So this comes out to be minus infinity plus infinity you have psi star x psi dx. Now this is a time derivative because it's a time derivative I can take the x out of it I can take the derivative inside and this simply gives us an integration x I take out and now because it's a time derivative I take it as a partial derivative, okay? Because psi is a function of x and t both. And here in the bracket I have psi star psi dx, all right? So this is what we have. So let's say that this is point number one, all right? Now let's move forward. Let me just rub this portion a little bit just to create some space, okay? So now I want to analyze this particular term first, okay? Let's analyze this term, okay? So now the time derivative of psi star psi we can write this as psi star del psi upon del t plus psi del psi star upon del t. So again let's say this is point number two. Now I am going to invoke the time dependent Schrodinger's equation, okay? The time dependent Schrodinger equation. So which is iota h-cut del psi upon del t is equal to minus h-cut square upon 2 m del 2 psi upon del x2 plus v psi. If I take the iota term to the right hand side in the Schrodinger's equation, I end up getting and the h-cut term into the right hand side. I end up getting del psi upon del t is equal to, so 1 upon iota is minus iota, which is equal to plus iota. h-cut gets cancelled, h-cut upon 2 m del 2 psi upon del x square. Here I'll have 1 upon iota, which is minus iota and v upon h-cut psi, okay? So this is del psi upon del t. What is its complex conjugate? Okay, if I want to find out its complex conjugate, then the complex conjugate is del psi star upon del t. So this comes out to be, I just replace the iota term with minus iota. So this is a way to find out the complex conjugate of a complex function. As you replace the iota, the imaginary number, root over minus one with minus iota and then you get this complex conjugate. So this is minus iota h-cut upon 2 m del 2 psi upon del x square plus iota v upon h-cut psi. Now if I plug both these two expressions, okay? Into point number two. In point number two, if I plug both these two expressions, this one and this one, what do I get? So partial derivative of psi star psi is equal to psi star d psi upon dt. So this thing I'm interested in, okay? So I'm interested in this and I'm interested in this, okay? So I have psi star and then d psi upon dt. So this becomes iota h-cut upon 2 m psi star psi star del 2 psi upon del x is to minus. Again, I have iota v upon h-cut psi star psi. So this is the first term, okay? This is the first term, this one. Now let's write the second term, psi times d psi star upon dt. So this becomes minus iota h-cut upon 2 m psi and then you have del 2, okay? Sorry, I'm sorry. I should have said this is psi star, complex conjugate, right? Complex conjugate, just a small correction. When I take the complex conjugate, I replace iota with minus iota, which I have done. But the wave function, because we don't really know what the function actually is, it could as well be a complex function. So wave function is also substituted with its complex conjugate. So therefore this should become del 2 psi star upon del x square. And then we have plus iota v upon h-cut psi star because it's a multiplication. I can rearrange the terms and I say psi star psi. So here you can say that these two terms cancel each other out. And then we are only left with these two terms, okay? It's just a mathematical calculations. Let's be a little patient, okay? Don't lose patience because these are mathematical calculations. And this here, the iota h-cut upon 2m is common, fine? If you look at these two terms, psi star del 2 psi upon del x2 and psi del 2 psi star upon del x2, these are second order derivatives. I can reduce this further by taking a derivative of a series of terms. So I can say that del upon del x and I can say psi star del psi upon del x and then you have minus psi del psi star upon del x. So let's say that this is point number three. Now let's substitute point number three in point number one and see what we get, okay? Let's just create a boundary here so that we don't confuse the equations and substitute three in one. So if I do that, the velocity actually becomes, velocity expectation value becomes d x average upon dt which is equal to this x del upon del t of psi star psi d x which is, again I'll substitute this term here, del upon del t of psi star psi. So I end up actually getting iota h-cut upon 2m is common and here I have x and then I have del upon del x of psi star del psi upon del x minus psi del psi star upon del x d x. All right, now I can invoke integration by parts at this point. So if I invoke integration by parts, again this comes out i h-cut upon 2m is common and then I have x here and then I have integration of this term, right? So basically del upon del x of psi star del psi upon del x minus psi del psi star upon del x dx. This is one term and then minus I have integration of dx upon dx which is just one and then again integration of del upon del x psi star del psi upon del x minus psi del psi star upon del x dx dx, okay? Now the integration is happening between minus infinity and plus infinity. So if I just, let's just look at the first term, iota h-cut upon 2m. The first term, what is the first term? You have x here, okay? And then you have the first term here. Let me just put square brackets. dx and dx gets vanished and then integration of del this. So this is just nothing but psi star del psi upon del x minus psi del psi star upon del x, okay? Now the integration is happening between, as I said, from minus infinity to plus infinity and then I have the second term here minus integration of again dx dx vanishes. I am left with psi star del psi upon del x minus psi del psi upon del x star and then here I have dx close the bracket. Now again, we discussed it previously that whenever we are talking about a physical or a realistic scenario of a particle, then usually the wave function goes to zero at infinity because one of the criteria for wave function to be applicable for physical systems is that it has to be normalizable. So if a wave function should be normalizable, it means that its value as well as its derivative should go to zero at infinity. So if we are looking at the wave function, its complex conjugate or its derivatives at plus minus infinity, they should all go to zero. So this term therefore will vanish. And then finally we are just left with this particular term, which is this is equal to or I should say the expectation value of the velocity is equal to minus iota h cut upon 2 m and then you have integration of psi star del psi upon del x minus psi del psi star upon del x dx. Now let's say this is point number four and keep it here. I hope that you are not this hardened by the level of calculations. It's just a couple of more steps is left and then we'll come up with the expression. Let's first rub the board, okay? All right, so now let's use this term from point number four. This term psi star del psi del x, okay? Psi star del psi del x and then again do integration by parts, all right? So there is another, okay, with respect to dx and dx term is here. So if I do an integration by parts, what do I get? I have essentially what I get is I psi star comes out and then I have del psi upon del x dx minus, here I have del psi star upon del x integration of del psi upon del x dx and then dx, okay? So this becomes what? So this essentially becomes you have psi star and then del psi upon del x dx. This is just nothing but psi, okay? If I put up the integration limits from minus infinity to plus infinity and here I have minus del psi star upon del x, this becomes integration of del psi dx is just nothing but psi dx, okay? So again, what did I say? For any kind of a wave function to represent a physical system, we have certain conditions. One of them is that the wave function should go to zero at infinity, where plus minus infinity, this should go to zero. So therefore, this essentially just comes out to be minus. You have del psi star upon del x psi dx, okay? So we are about to reach the final conclusion. Psi star del psi upon del x dx is just equal to minus del psi star upon del x psi dx, okay? So now let's bring point number four. So if I bring point number four here, then what do I have? I have the average of the velocity. The average of the velocity is equal to minus iota h cut upon two m and then we have the first term, psi star del psi upon del x. And then I have another term psi del psi star upon del x. So because these are multiplicative terms, so I can re-enage a term. So this is essentially equal to, so this term is essentially equal to psi star. So the first term is del psi upon del x. And the second term is basically psi del psi star upon del x. This can be written as this. So I have a minus term, minus, and minus becomes plus. I end up getting psi star del psi upon del x. And here I have dx. So this is twice, two and two gets canceled. And this essentially comes out to be minus iota h cut upon m integration of psi star del psi upon del x dx. And if I take the m to the left hand side, if I take a m del v, sorry m expectation value of v, this simply comes out to be integration of psi star. Again integration from minus infinity to plus infinity, minus iota h cut, I'm taking it inside, minus iota h cut, the constants. And then you have del upon del x of psi dx. What is this? Mass times the average of the velocity. This is nothing but the expectation value of momentum. So the expectation value of momentum finally comes out to be this particular expression, integration of psi star minus iota h cut del upon del x of psi dx. So this is the average of the values of momentum of the particle, the expectation value of momentum. So just like we found out the expectation value of position, we have found out the expectation value of momentum. So if I make a large number of measurements on similar quantum mechanical systems which exist in a given state psi, then just like we have an average of the position of the particle, here we have an average of the momentum of the particle because the particle could be moving at different speeds with different momenta, but quantum mechanics predicts the average of that particular momenta distribution. Quantum mechanics cannot predict the exact momenta of the particle, but it can predict the average of the momenta of the particle if we make a large number of measurements on similar systems. And theoretically this is a prediction that we can come up with. This is the beauty or the uniqueness of quantum mechanics. It cannot predict where the particle is going to be exactly at the next moment or what its momentum is going to be exactly at the next moment. The only thing that it provides us is a probabilistic interpretation. If we have a particle or similar particles or similar systems and if we make repeated measurements, then the theory of quantum mechanics can predict the average of the position of the particle based on the wave function solution of the Schrodinger's equation and the average of the momentum of the particle based on the wave function solution of the Schrodinger's equation. So this is what the theory can predict. No matter how limited it is, this is what it is and this is what we have to live with. So now I have written the expectation value or the average or the mean value of position and momentum theoretically speaking in terms of the wave function in this particular expression. We have obtained this based on the basic definitions of probability theory and then by taking a time derivative of this we have obtained this going through all those steps. So finally we have obtained two very important expressions the expectation value of position and the expectation value of momentum. Now, this again introduces us to a even more bizarre and murky world of quantum mechanics. See, now here we are going to introduce the concept of operators. You see if you look at the expectation value of position and the expectation value of momentum I have written them in a very peculiar fashion. You see in the expectation value of position I have psi star which is the complex conjugate of the wave function solution of the Schrodinger's equation and psi which is the wave function solution of the Schrodinger's equation. And I've integrated it with respect to dx but in between I have written x. Okay, I have written x. Basically I'm trying to find out the expectation value of position. Here again between psi star psi and integration with respect to dx I have written this particular term. These terms that I've written in bracket which I have sandwiched between psi star and psi. These terms are basically known as operators. So basically the operator of position is x and the operator of momentum is this. Now what is the operator in quantum mechanics? Well, I'm coming to that. Essentially every physical quantity that we have in quantum mechanics has an operator associated with it. You see the problem with quantum mechanics is that solving the Schrodinger's equation doesn't really give us a physical quantity. Like unlike classical mechanics where solving the Newton's laws gives us position, time derivatives gives us velocity, time derivatives gives us acceleration, et cetera, et cetera. All the physical quantities are connected to the theory of the Newtonian mechanics. But in Schrodinger's picture we don't really get the physical quantities. What we get is a very weird sort of a mathematical system in which we have to perform certain kinds of mathematical operations onto the wave function solution to get some information about the system. And that is the role that operators perform. So operator is an instruction to do something to the wave function that follows it. And it does something in such a manner that it opens up the possibility of providing us with some information about the physical system, the actual physical system. So the operator of position and we usually specify operators with a hat or a cap, okay? The operator of position which is known as a position operator is nothing but x itself which is basically we multiply x to the wave function that follows it. And the operator of momentum is nothing but this particular quantity minus iota h cut. And then I have the derivative del upon del x. So if I perform this operator onto a wave function then all I have to do is take the derivative of the wave function with respect to x multiply with minus iota h cut. So these are known as operators of position and operators of momentum. Again, what is an operator? Because quantum mechanical theory or the Schrodinger's equation doesn't give us direct physical quantities, what it gives us is the wave function. And the wave function contains some information about the system. And we need to be able to extract that information out of the wave function. So the operator provides us a possibility of extracting that information. The operator is essentially some sort of an instruction, some kind of magic we perform onto the wave function that reveals something about the actual physical system. So when we want to find out the expectation value of position, I simply sandwich the operator of position in between psi star and psi that gives us the average value of position. If I want to find out the expectation value of momentum, I simply substitute the operator corresponding to momentum between psi star and psi and that gives us the average value of momentum. So operator, therefore, at least in this context, is a mathematical operation corresponding to some physical quantity that which when we substitute in between psi star and psi and then integrate with respect to x, gives us the average value for that physical quantity corresponding to the system whose solution is psi as a wave function. So this technique opens up the possibility of finding out the operators for other physical quantities also. For example, if I'm interested in the operator corresponding to let's suppose kinetic energy. Now I have obtained the operator for momentum and position. So how is kinetic energy related to any one of them? So let's suppose I want to find out the operator corresponding to kinetic energy and I'm just going to specify that with a hat symbol here. So what is kinetic energy? Kinetic energy is nothing but p square upon 2m, right? We know that kinetic energy is half MV square which can be written as p square upon 2m p being the momentum. So here I again use the momentum operator. All right, so if I use the momentum operator, what happens? This simply becomes one upon 2m. What is the momentum operator? It is minus iota h cut del upon del x and then you have square which essentially means operator operating on to the same term basically. So which comes out to be, so minus minus square becomes one, iota into iota becomes minus h cut square upon 2m del 2 upon del x2. So therefore the operator corresponding to kinetic energy for the system is minus h cut square upon 2m del 2 upon del x2. So this is the operator corresponding to kinetic energy. So what can we say from here? We can say that if I have a physical system, I make repeated measurements onto the physical system and I'm interested in finding out the average value of the kinetic energy of the particle in the system. Then what I can do is theoretically speaking, this is the tool we have that the average value of the kinetic energy of the particle in the system is equal to, again I sandwich the operator of t in between psi star and psi. So I write minus h cut square upon 2m del 2 upon del x2 psi dx. This is the expectation value of the kinetic energy. Of course I can simplify it further by taking psi onto here so that we're actually doing the operation of the differentiation. So you can write it as psi star. Okay, let's take the minus h cut square 2m outside the integration. So if I do that, minus h cut square upon 2m integration. Again, we are integrating between the limits of whether particle can be. So usually it's from minus infinity to plus infinity along the x axis. So this is psi star and you have del 2 psi upon del x2 dx. This is the expectation value of the kinetic energy of the particle in the physical system whose Schrodinger's equation solution is the wave function here psi. We can also find out other things like for example, total energy. Now for total energy what I'm going to do is for total energy I'm going to use the time dependent Schrodinger's equation. Now we know what the time dependent Schrodinger's equation is. It is minus h cut square upon 2m and del 2 psi upon del x2 plus v psi is equal to iota h cut del psi upon del t. So if I just take the psi out, I can write this as minus h cut square upon 2m del 2 upon del x2 plus v and then let's put this in a bracket and psi is equal to iota h cut del psi upon del t. Now notice something here. This is a Schrodinger's equation. What is this? This is the operator for kinetic energy. What is this? This is nothing but the potential energy expression which we don't really know what the expression actually is because depending upon the system it could be anything. I'm going to call this as the operator corresponding to v. So essentially what is happening is the kinetic energy operator is acting on psi. Potential energy operator is acting on psi which gives us the right hand side. Now what is kinetic energy plus potential energy? Kinetic energy plus potential energy is basically nothing but total energy, right? So if I call this as a total energy operator then by making a comparison with this what is the operator for the total energy of the system? The operator for the total energy of the system is so this acting on psi gives us this acting on psi. So if I remove the psi I am left with minus, oh sorry, iota h cut del upon del t. This is the operator corresponding to the total energy. So essentially you see that the Schrodinger's equation is nothing but an operator equation where on one hand you have kinetic energy plus potential energy operator acting on the wave function psi is equal to this instruction acting on psi. So which is the same thing as the total amount of energy acting on psi because kinetic energy plus potential energy is equal to total energy, right? So essentially the Schrodinger's equation is a energy operator equation at the end of the day. So here we can write the energy operator as this. So now that gives us a possibility of figuring out the average energy, total energy of a given physical system in the context of quantum mechanics. It is just what you do, you take this operator and you sandwich that between psi star psi and then integrate with respect to dx. So you integrate with respect to dx. So iota h cut, let me take it out, you have psi star and then you have del psi upon del t dx. This is again between minus infinity plus infinity is the expectation value for total energy of a system. So you see that this provides us with a very beautiful theoretical framework of what we can expect from the theory of quantum mechanics. We cannot expect the Schrodinger's equation to actually give us a physical quantity for a given system but it can give us something about the system. And in this lecture we are discussing that something to be the average of a repeated measurements of a physical quantity on similar systems which we call as expectation value. So if we are interested in expectation value of position momentum, kinetic energy, total energy, we can always do that by sandwiching their corresponding operators in between psi star psi and integrating with respect to dx. If we can do that, we can then predict what is the average or the mean or the expectation value of position, momentum, kinetic energy, total energy, et cetera. Keeping in mind that all these physical quantities have operators corresponding to them. These operators are very important in the context of quantum mechanics. We will see their relevance even later on in our coming lectures. But this is something that you should probably memorize, operator for position, operator for momentum, operator for kinetic energy, operator for total energy, et cetera, et cetera. In general, if I want to find out the expectation value of any kind of a physical quantity, let's suppose I have a physical quantity q, all right? And the q could be anything, maybe angular momentum, maybe something else. And if I want to find out the expectation value of that physical quantity q, all I do is I sandwich q operator between psi star and psi and then do integration with respect to x between minus infinity to plus infinity. Now what is q? I'm just saying that it is a function of maybe p, maybe x, maybe time. So I try to represent them in those familiar terms and then sandwich that between psi stars. I integrate with respect to dx. I get the expectation value of that physical quantity. That is the capacity of our quantum mechanical theory. Yeah, so I hope that you have understood what we discussed till now and we will use these expressions and these values later on in our lectures when we are studying specific systems. So please keep these things in mind. Note this down. It's very, very important. Especially these formulas and these expressions are very important. So I'm Divya Jyothidas. This is for the love of physics. We will discuss something very important in the next lecture. But till then, goodbye. Take care. Bye bye.