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.  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.  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.  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?  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 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.  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.


CHAPTER XX. But never was a fight managed so heartily and in such a surprising manner as that which followed between Friday and the bear, which gave us all, though at first we were surprised and afraid for him, the greatest diversion imaginable. As the bear is a heavy, clumsy creature, and does not gallop as the wolf does, who is swift in light, so he has two particular qualities, which generally are the rule of his actions. First as to men who are not his proper prey. He does not usually attempt them, except they first attack him, unless he be excessively hungry, which it is probable might now be the case, the ground being covered with snow. If you do not meddle with him, he will not meddle with you. But then you must take care to be very civil to him, and give him the road, for he is a very nice gentleman. He will not go a step out of his way for a prince. Nay, if you are really afraid, your best way is to look another way and keep going on. For sometimes if you stop and stand still, and look steadfastly at him, he takes it for an affront. But if you throw or toss anything at him, though it were but a bit of stick as big as your finger, he thinks himself abused, and sets all other business aside to pursue his revenge, and will have satisfaction and point of honour. That is his first quality. The next is, if he be once affronted, he will never leave you night or day till he has his revenge, but follows at a good round rate till he overtakes you. My man Friday had delivered our guide, and when we came up to him he was helping him off his horse, for the man was both hurt and frightened, when on a sudden we aspired the bear come out of the wood, and a monstrous one it was, the biggest by far that ever I saw. We were all a little surprised when we saw him, but when Friday saw him it was easy to see joy and courage in the fellow's countenance. Oh, oh, oh! says Friday, three times, pointing to him.  I was surprised to see the fellow so well pleased.  Eat a me up, eat a me up! says Friday, twice over again.  You all stay here, me show you good laugh.  The bear was walking softly on, and offered to meddle with nobody, till Friday coming pretty near calls to him, as if the bear could understand him. Hark ye, hark ye, says Friday, me speak ye with you. We followed at a distance, for now being down on the Gaskany side of the mountains, we were entered a vast forest, where the country was plain and pretty open, though it had many trees in it scattered here and there. Friday, who had, as we say, the heels of the bear, came up with him quickly, and took up a great stone and threw it at him, and hit him just on the head, but did him no more harm than if he had thrown it against a wall. But it answered Friday's end, for the rug was so void of fear that he did it purely to make the bear follow him, and show us some laugh, as he called it. As soon as the bear felt the blow, and saw him, he turns about and comes after him, taking very long strides, and shuffling on at a strange rate, so as would have put a horse to a middling gallop. Away rains Friday, and takes his course as if he ran towards us for help, so we all resolved to fire at once upon the bear, and deliver my man, though I was angry at him for bringing the bear back upon us, when he was going about his own business another way, and especially I was angry that he had turned the bear upon us, and then ran away, and I called out, "'You dog, is this your making us laugh? Come away and take your horse, that we may shoot the creature.'" He heard me, and cried out, "'No shoot! No shoot! Stand still, and you get much laugh!' And as the nimble creature ran two feet for the bear's one, he turned on a sudden on one side of us, and seeing a great oak tree fit for his purpose, he beckoned to us to follow. And, doubling his pace, he got nimbly up the tree, laying his gun down upon the ground at about five or six yards from the bottom of the tree. The bear soon came to the tree, and we followed at a distance. The first thing he did, he stopped at the gun, smelt at it, but let it lie, and up he scrambles into the tree, climbing like a cat, though so monstrous heavy. I was amazed at the folly, as I thought it, of my man, and could not for my life see anything to laugh at, till seeing the bear get up the tree, we all rode near to him. When we came to the tree, there was Friday, got out to the small end of a large branch, and the bear got about half-way to him. As soon as the bear got out to that part, where the limb of the tree was weaker, "'Ha!' says he to us, "'Now you see me teach you the bear dense!' So he began jumping and shaking the bow, at which the bear began to totter, but stood still, and began to look behind him to see how he should get back. Then indeed we did laugh heartily. But Friday had not done with him by a great deal. When seeing him stand still, he called out to him again, as if he had supposed the bear could speak English. "'What? You come no farther? Pray you come farther!' So he left jumping and shaking the tree, and the bear, just as if he understood what he said, did come a little farther. Then he began jumping again, and the bear stopped again. We thought now was a good time to knock him in the head, and called to Friday to stand still, and we should shoot the bear. But he cried out earnestly, "'Oh, pray! Oh, pray! No shoot! Me shoot by and then!' He would have said, by and by. However, to shorten the story, Friday danced so much, and the bear stood so ticklish, that we had laughing enough, but still could not imagine what the fellow would do. For first we thought he depended upon shaking the bear off, and we found the bear was too cunning for that, too. Before he would not go out far enough to be thrown down, but clung fast with his great broad claws and feet, so that we could not imagine what would be the end of it, and what the jest would be at last. But Friday put us out of doubt quickly, for seeing the bear cling fast to the bow, and that he would not be persuaded to come any farther, "'Well, well,' says Friday, "'You come no farther? Me go! You no come to me? Me come to you!' And upon this he went out to the smaller end, where it would bend with his weight, and gently let himself down by it, sliding down the bow till he came near enough to jump down on his feet, and away he ran to his gun, took it up, and stood still. "'Well,' said I to him, "'Friday, what will you do now? Why don't you shoot him?' "'No, shoot!' says Friday. "'No yet!  Me no kill! Me stay! Give you one more laugh!' And indeed so he did, for when the bear saw his enemy gone, he came back from the bow, where he stood, but did it very cautiously, looking behind him every step, and coming backward till he got into the body of the tree. Then with the same hindren foremost, he came down the tree, grasping it with his claws, and moving one foot at a time very leisurely. At this juncture, and just before he could set his hind foot on the ground, Friday stepped up close to him, clapped the muzzle of his piece into his ear, and shot him dead. Then the rogue turned about to see if we did not laugh, and when he saw we were pleased by our looks, he began to laugh very loud. "'So we kill bear in my country!' says Friday.  "'Why, you have no guns!' "'No!' says he. "'No gun! But shoot great much long arrow!' This was a good diversion to us, but we were still in a wild place, and our guide very much hurt, and what to do we hardly knew. The howling of wolves ran much in my head, and indeed, except the noise I once heard on the shore of Africa, of which I have said something already, I never heard anything that filled me with so much horror. These things, and the approach of night, called us off, or else, as Friday would have had us, we should certainly have taken the skin of this monstrous creature off, which was worth saving, but we had near three leagues to go, and our guide hastened us, so we left him and went forward on our journey. The ground was still covered with snow, though not so deep and dangerous as on the mountains, and the ravenous creatures, as we heard afterwards, were come down into the forest in plain country, pressed by hunger to seek for food, and had done a great deal of mischief in the villages, where they surprised the country people, killed a great many of their sheep and horses, and some people, too. We had one dangerous place to pass, and our guide told us, if there were more wolves in the country, we should find them there, and this was a small plain, surrounded with woods on every side, and a long, narrow defile, or lane, which we were to pass to get through the wood, and then we should come to the village where we were to lodge. It was within half an hour of sunset when we entered the wood, and a little after sunset when we came into the plain. We met with nothing in the first wood, except that in a little plain within the wood, which was not above two furlongs over, we saw five great wolves crossed the road full speed, one after another, as if they had been in chase of some prey, and had it in view. They took no notice of us, and were gone out of sight in a few moments. On this, our guide, who by the way was but a faint-hearted fellow, bid us keep in a ready posture, for he believed there were more wolves incoming. We kept our arms ready, and our eyes about us, but we saw no more wolves till we came through that wood, which was near half a league, and entered the plain. As soon as we came into the plain we had occasion enough to look about us. The first object we met with was a dead horse, that is to say, a poor horse which the wolves had killed, and at least a dozen of them at work, we could not say eating him, but picking his bones rather, for they had eaten up all the flesh before. We did not think fit to disturb them at their feast, neither did they take much notice of us. Friday would have let fly at them, but I would not suffer him by any means, for I found we were like to have more business upon our hands than we were aware of. We had not gone half over the plain when we began to hear the wolves howl in the wood on our left in a frightful manner, and presently after we saw about a hundred coming on directly towards us, all in a body, and most of them in a line, as regularly as an army drawn up by experienced officers. I scarce knew in what manner to receive them, but found to draw ourselves in a close line was the only way, so we formed in a moment. But that we might not have too much interval. I ordered that only every other man should fire, and that the others, who had not fired, should stand ready to give them a second volley immediately, if they continued to advance upon us. And then that those that had fired at first should not pretend to load their Fusees again, but stand ready, every one with a pistol, for we were all armed with a Fusee and a pair of pistols each man. So we were by this method able to fire six volleys, half of us at a time. However, at present we had no necessity, for upon firing the first volley the enemy made a full stop, being terrified as well with the noise as with the fire. Four of them being shot in the head, dropped, several others were wounded, and went bleeding off as we could see by the snow. I found they stopped, but did not immediately retreat. Thereupon, remembering that I had been told that the fiercest creatures were terrified at the voice of a man, I caused all the company to halloo as loud as they could, and I found the notion not altogether mistaken, for upon our shout they began to retire and turn about. I then ordered a second volley to be fired in their rear, which put them to the gallop, and away they went to the woods. This gave us leisure to charge our pieces again, and that we might lose no time we kept going, but we had but little more than loaded our fusees, and put ourselves in readiness, when we heard a terrible noise in the same wood on our left, only that it was farther onward the same way we were to go. The night was coming on, and the light began to be dusky, which made it worse on our side, but the noise increasing we could easily perceive that it was the howling and yelling of those hellish creatures, and on a sudden we perceived three troops of wolves, one on our left, one behind us, and one in our front, so that we seemed to be surrounded with them. However as they did not fall upon us we kept our way forward as fast as we could make our horses go, which, the way being very rough, was only a good hard trot. In this manner we came in view of the entrance of a wood through which we were to pass, at the farther side of the plain, but we were greatly surprised when coming nearer the lane or pass we saw a confused number of wolves standing just at the entrance. On a sudden, at another opening of the wood, we heard the noise of a gun, and looking that way out rushed a horse, with a saddle and a bridle on him, flying like the wind, and sixteen or seventeen wolves after him full speed. The horse had the advantage of them, but as we supposed that he could not hold it at that rate, we doubted not but they would get up with him at last, no question, but they did. But here we had a most horrible sight, for riding up to the entrance where the horse came out, we found the carcasses of another horse and of two men devoured by the ravenous creatures.  But as to the man, his head and the upper part of his body was eaten up. This filled us with horror, and we knew not what course to take, but the creatures resolved us soon, for they gathered about us presently in hopes of prey, and I verily believe there were three hundred of them. It happened, very much to our advantage, that at the entrance into the wood, but a little way from it, there lay some large timber-trees, which had been cut down the summer before, and I suppose lay there for carriage. I drew my little troop in among those trees, and placing ourselves in a line behind one long tree, I advised them all to alight, and keeping that tree before us for a breastwork, to stand in a triangle or three fronts, enclosing our horses in the center. We did so, and it was well we did, for never was a more furious charge than the creatures made upon us in this place. They came on with a growling kind of noise, and mounted the piece of timber, which, as I said, was our breastwork, as if they were only rushing upon their prey, and this fury of theirs, it seemed, was principally occasioned by their seeing our horses behind us. I ordered our men to fire as before, every other man, and they took their aim so sure that they killed several of the wolves at the first volley, but there was a necessity to keep a continual firing, for they came on like devils, those behind pushing on those before. When we had fired a second volley of our few Cs, we thought they stopped a little, and I hoped they would have gone off, but it was but a moment, for others came forward again. So we fired two volleys of our pistols, and I believe in these four firings we had killed, seventeen or eighteen of them, and blamed twice as many, yet they came on again. I was loath to spend our shot too hastily, so I called my servant, not my man Friday, for he was better employed, for with the greatest dexterity imaginable he had charged my few Cs and his own while we were engaged, but as I said, I called my other man, and giving him a horn of powder, I had him lay a train all along the piece of timber, and let it be a large train. He did so, and had but just time to get away, when the wolves came up to it, and some got upon it, when I, snapping an uncharged pistol close to the powder, set it on fire. Those that were upon the timber were scorched with it, and six or seven of them fell, or rather jumped in among us with the force and fright of the fire. We dispatched these in an instant, and the rest were so frightened with the light which the night, for it was now very near dark, made more terrible that they drew back a little, upon which I ordered our last pistols to be fired off in one volley, and after that we gave a shout. On this the wolves turned tail, and we sallied immediately upon near twenty lame ones that we found struggling on the ground, and fell to cutting them with our swords, which answered our expectation, for the crying and howling they made was better understood by their fellows, so that they all fled and left us. We had, first and last, killed about three score of them, and had it been daylight we had killed many more. The field of battle being thus cleared we made forward again, for we still had near a league to go. We heard the ravenous creatures howl and yell in the woods as we went several times, and sometimes we fancied we saw some of them, but the snow dazzling our eyes we were not certain. In about an hour more we came to the town where we were to lodge, which we found in a terrible fright in all in arms, for, it seems, the night before the wolves and some bears had broken into the village, and put them in such terror that they were obliged to keep guard night and day, but especially in the night to preserve their cattle, and indeed their people. The next morning our guide was so ill, and his limbs swelled so much with the rankling of his two wounds, that he could go no farther. So we were obliged to take a new guide here, and go to Toulouse, where we found a warm climate, a fruitful, pleasant country, and no snow, no wolves, nor anything like them. But when we told our story at Toulouse, they told us it was nothing but what was ordinary in the great forest at the foot of the mountains, especially when the snow lay on the ground. But they inquired much what kind of guide we had got who would venture to bring us that way in such a severe season, and told us it was surprising we were not all devoured. When we told them how we placed ourselves and the horses in the middle, they blamed us exceedingly, and told us it was fifty to one, that we had been all destroyed, for it was the sight of the horses which made the wolves so furious, seeing their prey, and that at other times they were really afraid of a gun. But being excessively hungry, and raging on that account, the eagerness to come at the horses had made them senseless of danger, and that if we had not by the continual fire, and at last by the stratagem of the train of powder, mastered them, it had been great odds but that we had been torn to pieces. Whereas had we been content to have sat still on horseback, and fired as horsemen, they would not have taken the horses so much for their own when men were on their backs, as otherwise. And with all, they told us that at last, if we had stood all together and left our horses, they would have been so eager to have devoured them that we might have come off safe, especially having our firearms in our hands being so many a number. For my part, I was never so sensible of danger in my life. For seeing above three hundred devils come roaring an open mouth to devour us, and having nothing to shelter us or retreat to, I gave myself over for lost. And as it was, I believe I shall never care to cross those mountains again. I think I would much rather go a thousand leagues by sea, though I was sure to meet with a storm once a week. I have nothing uncommon to take notice of in my passage through France, nothing but what other travellers have given an account of with much more advantage than I can. I travelled from Toulouse to Paris, and without any considerable stay came to Calais, and landed safe at Dover the fourteenth of January, after having had a severe cold season to travel in. I was now come to the centre of my travels, and had in a little time all my new discovered estate safe about me, the bills of exchange which I brought with me having been currently paid. My principal guide and privy counsellor was my good ancient widow, who in gratitude for the money I had sent her, thought no pains too much nor care too great to employ for me, and I trusted her so entirely that I was perfectly easy as to the security of my effects. And indeed, I was very happy from the beginning, and now to the end, in the unspotted integrity of this good gentlewoman. And now, having resolved to dispose of my plantation in the Brazils, I wrote to my old friend at Lisbon, who, having offered it to the two merchants, the survivors of my trustees, who lived in the Brazils, they accepted the offer, and remitted thirty-three thousand pieces of eight to a correspondent of theirs at Lisbon to pay for it. In return I signed the instrument of sale in the form which they sent from Lisbon, and sent it to my old man, who sent me the bills of exchange for thirty-two thousand eight hundred pieces of eight for the estate, reserving the payment of one hundred moydories a year to him, the old man, during his life, and fifteen moydories afterwards to his son for his life, which I had promised them, and which the plantation was to make good as a rent charge. And thus I have given the first part of a life of fortune and adventure, a life of Providence's checker work, and of a variety which the world will sell them be able to show the like of, beginning foolishly, but closing much more happily than any part of it ever gave me leave so much as to hope for. Anyone would think that in this state of complicated good fortune I was past running any more hazards, and so indeed I had been if other circumstances had concurred, but I was inured to a wandering life, had no family, nor many relations, nor, however rich, had I contracted fresh acquaintance, and though I had sold my estate in the Brazils, yet I could not keep that country out of my head, and had a great mind to be upon the wing again, especially I could not resist the strong inclination I had to see my island, and to know if the poor Spaniards were in being there. My true friend, the widow, earnestly dissuaded me from it, and so far prevailed with me that for almost seven years she prevented my running abroad, during which time I took my two nephews, the children of one of my brothers, into my care. The eldest, having something of his own, I bred up as a gentleman, and gave him a settlement of some addition to his estate after my deceased. The other I placed with the captain of a ship, and after five years finding him a sensible, bold, enterprising young fellow, I put him into a good ship, and sent him to sea, and this young fellow afterwards drew me in, as old as I was, to further adventures myself. In the meantime I in part settled myself here, for first of all I married, and that not either to my disadvantage or dissatisfaction, and had three children, two sons and one daughter, but my wife dying, and my nephew coming home with good success from a voyage to Spain, my inclination to go abroad, and his importunity prevailed, and engaged me to go in his ship as a private trader to the East Indies. This was in the year 1694. In this voyage I visited my new colony in the island, saw my successors the Spaniards, had the old story of their lives and of the villains I left there, how at first they insulted the poor Spaniards, how they afterwards agreed, disagreed, united, liberated, and how at last the Spaniards were obliged to use violence with them, how they were subjected to the Spaniards, how honestly the Spaniards used them, a history, if it were entered into, as full of variety and wonderful accidents as my own part, particularly also as to their battles with the Caribbean's, who landed several times upon the island, and as to the improvement they made upon the island itself, and how five of them made an attempt upon the mainland and brought away eleven men and five women prisoners by which at my coming I found about twenty young children on the island. Here I stayed about twenty days, left them supplies of all necessary things, and particularly of arms, powder, shot, clothes, tools, and two workmen which I had brought from England with me, that is a carpenter and a smith. Besides this I shared the lands into parts with them, reserved to myself the property of the whole, but gave them such parts respectively as they agreed on, and having settled all things with them and engaged them not to leave the place, I left them there. From thence I touched at the Brazils, from whence I sent a bark, which I bought there, with more people to the island, and in it besides other supplies I sent seven women, being such as I found proper for service, or for wives to such as would take them. As to the Englishmen I promised to send them some women from England with a good cargo of necessaries if they would apply themselves to planting, which I afterwards could not perform. The fellows proved very honest and diligent after they were mastered and had their properties set apart for them. I sent them also, from the Brazils, five cows, three of them being big with calf, some sheep, and some hawks, which when I came again were considerably increased. But all these things, with an account how three hundred caribis came and invaded them, and ruined their plantations, and how they fought with that whole number twice, and were at first defeated, and one of them killed, but at last a storm destroying their enemy's canoes they famished or destroyed almost all the rest, and renewed and recovered the possession of their plantation, and still lived upon the island. All these things, with some very surprising incidents and some new adventures of my own, for ten years more I shall give a farther account of, in the second part of my story. This is the end of the Life and Adventures of Robinson Crusoe. Thank you for listening.


Hello and welcome to the session. My name is Mansi and I am going to help you with the following question. The question says draw an angle of measure 135 degree and bisect it. Let us see the solution to this question. First thing that we do is we make an angle of 135 degree. We can make an angle of 135 degree with the help of a protractor or with the help of rural and compasses as we have seen in the question number 5. So this is an angle of 135 degree. Now what we need to do is we need to construct the bisector of angle A. So the first step would be with A as centre and using compasses draw an arc that cuts both rays of angle A label the point of intersection as B and C. So with the pointer of the compass at the point A we draw an arc BC. The step 2 is with B as centre we draw in the interior of angle A and arc whose radius is more than half the length BC. So with the pointer at the point B we draw an arc like this. Step 3 is with the same radius and with C as centre draw another arc in the interior of angle A. Let the 2 arcs intersect at D then line segment AD is the required bisector of angle A. So like this with B as centre then C as centre we draw these 2 arcs. Let the point of intersection of these 2 arcs be D. Now we have to join A to D then we see that line segment AD becomes the bisector of angle A because when we measure angle DAP it comes out to be 67.5 degree that is half of 135 degree. Therefore we can say that line segment DA is the required bisector of angle A. So I hope that you understood the question and enjoyed the session. Have a good day.


Welcome everyone to another live video do this to forget about the narcissist I know many of us it's hard to take our minds off of them sometimes especially when they're hoovering us especially when they always find a way to pop up in our lives again and while it is good to process the trauma and the feelings that come along with dealing with these narcissists it's also good to take time just to enjoy your life to forget about them and that's why I decided to do this live today to talk about that do this to forget about the narcissist do this what I'm doing right now you see this I'm out here walking through a forest I'm out in nature I'm on the way to check up this monument I'm in Scotland right now so what am I doing what am I doing that's helped me to forget about the narcissist well as I said I'm out in nature lots of high vibes out here with the trees and the squirrels the birds just makes you feel good so that doesn't mean that you have to travel to Scotland it doesn't mean that you have to go hours and hours away from where you live even if you've just got a local park forest a lake whatever it may be just go for a walk there just go for a walk and take in the scenery observe so many times we just move through life without actually noticing things we're always so distracted we're always thinking about something else I know many of us even if we do get the chance to just go for a walk in a forest by the lake when we get there we're just thinking about something else we're thinking about what we have to do when we get home what we're gonna have for dinner when we need to set aside some time well we don't have to think about anything other than what's in this moment right now just observing things and that is very healing in itself but traveling does help too if you can go far away that can be very healing too because these narcissists they'll often enforce fly monkeys starts me again against you they want to take away all of your family friends and resources they want to ruin your reputation destroy your career and they just make your world feel so small they make you feel like everything you had they took away from you all the potential things you could have developed they've taken ownership of that now so you can't partake in their world which they have conquered or hijacked from you they want you to think that there's nowhere how you belong you're just an outsider you're an outcast you've got nowhere to go now well I'm here to tell you that that is not true there is a big world out there huge world billions of people they want you to think that none of that exists anymore they want to think that everything you you thought you had and everything you thought you were going to develop that doesn't even exist and all there is is what you've got now what they've taken from you which you can no longer have but once you leave the narcissist world you will find that there is so much more out there more than you could ever imagine just remember all of those good times you had before you got involved with these narcissists before they got involved in your life that stuff still exists I can tell you I've seen the lights I've seen the fun I've seen the good times the people the good people the kind generous people because yes there are still good people out here and there's just so many opportunities for you so don't let them feed you these limiting beliefs you are so much more than what they have led you to believe and there is so much more out there waiting for you and for you to worry about what they've taken from you what they've sabotaged what they're trying to prevent what they're trying to limit and restricting your life yeah they want to keep you in that scarcity mindset we're then left for you you think you don't deserve anything but you need to be in a mindset of abundance and that's what this video is about it's about reminding you that there is so much in this great big world I don't know what it is 350 400 countries there's just so much to see out there so many different people so many different different cultures opportunities that there's just the point for you to waste your life worrying about these narcissists who never brought anything of value to you anyway here we are the early monument in Scotland this is what life is all about getting out here traveling around the world exploring things life these narcissists want to filter everything that goes into our minds they want us to limit the things that we think and feel and experience they want to change our perspectives so that all we can think about is what they want us to think about but you need to expand you need to grow you need to realize that there is so much more you need to realize how big the world is it's like they put us in this box it's like these there's just these four walls around us and they want us to think that that's all there is it's only when you get out of the box that you realize that there really is so much more to life don't let these narcissists take away your freedom it's sad you know like what so many people have to go through with these people they can trick you for your entire life where you think that the box is all there is where you think that there is nothing else when it's just not true there is so much more it's almost like they do some kind of hypnosis on you it's like they put you to sleep and then everything you were doing everything you were interested in everything you were so passionate about all of the dreams and aspirations you had you just forget about it all then you're left just focusing on whatever they want you to think about then you just leave all of that behind you it's really sad put you in a cage I think I just got off there I was just going to say no just how valuable you are fully aware of that if something's no good why would you ever try to control it if something's worthless and they want to make you think that you're no good but really they know your value and that's why they just can't leave things alone but yeah just take this in taking this view on the monument you can see all the ice and the mountains over there someone's saying it's my birthday it's not my birthday today it was about two months ago I was 32 so yeah do this to forget about the narcissist come out in nature breathing that fresh air observe the trees the grass so are there any questions or Elizabeth is asking where are you I'm in Scotland in Kirimu Joe Victor how do you stop wanting their attention you have to think about why do you want their attention you'll find that it's very similar to why people overindulge in food or why people smoke or do drugs or drink alcohol why do they do those things and if you look into it you'll find that the reason is because they feel like they're not enough and that's why they're searching for something outside of them to complete them because if you knew just how great you are and if you were in a mindset of abundance and you realize just how many greater opportunities there are for you out there you wouldn't care about getting that attention it just wouldn't matter to you anymore and that's why we need to focus on feeling enough because if you've been around these narcissists long enough I'm just gonna say bluntly you're gonna feel like shit you're gonna feel like crap like you're nothing I'm just gonna tell it to you straight up because you know I've been there I know exactly how it feels you don't feel good and you are around these people they just make you feel worthless of course that doesn't mean that we are worthless I mean we all know narcissists want people who are attractive successful powerful intelligent they target the best the best so as I said they know our value but they don't bring anything to the table so they have to put the focus on us to make us feel like we're not doing enough or we're not enough when it's just not true we are more than enough and we are more than capable of achieving great things in this world healing through inspiration asks tell us how to make money online like you do to further get away from the narcissist environment okay so all starts with your belief but one thing you need because no matter how capable you are if you don't believe in yourself you're not going to take action I often say to people when I first started this channel I was making videos I wasn't so sure if I really wanted to post these videos I was to think into myself who cares about my thoughts and opinions on this who wants to listen to my voice I didn't even like my voice but just imagine if I listen to that voice inside my head that voice telling me not to do it just imagine if I didn't believe in myself enough to post those videos I mean we've only got to look at the numbers to see the impact this has made 100,000 subscribers soon over 20 million views just imagine if I didn't believe myself if I didn't believe that I was worth the investment none of this would even exist there would be no knock survivor many people we never have found out about this they may have struggled to heal they may have thought that they were the problem and I'm not saying this to boast I'm just saying it because you need to believe in yourself too if you don't believe in yourself you're not going to take action or if you do take action and it fails the first two or three times you're not going to try again when it could be that third or fourth time that makes you succeed so yes you can make money online you most definitely can I am able to travel the world and you know I went from I went from living off 500 pounds a month that's how bad it was for me 500 pounds to look after myself like cat and now I made over 40,000 pounds just in a three month period I easily make over five digits a month and I only have to work a few hours a day I don't even need to do coaching sessions for the money I do it to help people I could easily spend that time making videos instead and I would make far more money but this is my purpose to provide value to people to support them through the difficult times that they may be enduring so yeah I went from living off 500 pounds a month to making over five digits a month in just over a year it didn't take long at all and I believe that everyone in this live chat right now is fully capable of that and probably the only reason why someone would not have achieved that already is as I said the belief once you believe in yourself you can achieve anything and I do offer a YouTube coaching program now as well which will teach you what I did to get to where I am now to be able to travel the world and work for my laptop all I have to do is just take my laptop on my microphone with me wherever I go and I can just explore the world and make money on the go and I have this program which can teach you to do the same thing so you too can travel the world and in plenty of money I know people say money isn't everything but let's be honest you know we do need money in our lives you know if you want a house you want a car just to pay the bills food if we want to look after our families of course we need money money isn't bad it can help us and if we have money we can give to charity we can save people's lives money is an amazing thing as long as it's used wisely so if you are interested in the YouTube coaching program you can book a session through the website it's www.knocksurviver.uk and when you book the session in the notes just say that you're requesting YouTube coaching and who knows you know maybe one day you know when you build your channel and YouTube becomes successful on YouTube and you start raking in thousands millions of views maybe we could collaborate as well maybe we could work together on this purpose of helping and supporting people and just changing people's lives I am all about collaboration as you've probably noticed with my new series of episodes posting people stories and they also have partnerships with better help and being verified and have also collaborated with surviving narcissism as well I'm all about collaboration I'm all about working together I know times are hard now we are in a pandemic and you know it can be quite a lot to afford you know for these these coaching sessions but I believe that the value that you will get from them is far far greater than the cost but as I said you know I know that we are in a pandemic I know many people may be in difficult financial situations right now and that is why for the viewers in this chat I am offering a 25% discount on all of the YouTube coaching sessions and that doesn't matter if you book a 30-minute session 60 minutes 90 minutes the discount still applies so if you get on to the website and you book a session now all you have to do is type in the discount code in capital letters NS welcome 25 and you will get 25% off the price but that is only available for a limited time so make the most of that discount while it lasts anyway that's all for this video we do have two premieres later today so I hope to see you there thank you all for joining this life


What's up everybody? Jason here for Jasper's case breaks calm We're giving away five Zenith first off the line football spots away for random teams number one What does 2021 prestige football hangar one box break number 31 random teams first? We do the break randomized because we're names and teams to shoot the hits out and at the end of the break We'll take all the 32 cousin names that bought in and the top five will win a $240 Zenith first off the line spot Here we go. Good luck. Let's do the break and it's a six and a three nine times one two three four five six seven eight nine Ryan down to Russell nine times one two three four five six seven eight nine Dallas Cowboys out of the Minnesota Vikings All right, so there you go nine times nine times Now let's uh alphabetize this And I'll quickly show your teams right there remember no trades just a quick little hanger All right, so this is all just gonna be based up until about right here Sorry, Matthew We got a Najee Harris Kyle Pitts rookie All right, there you go guys nothing crazy there now let's switch teams and give away spots and zenith Good luck guys top five get in roll it It's a four and a six ten times. Good luck top five get in one two three four five six seven eight nine Ten down to Nico. Unfortunately not in but thanks for giving it a shot Congrats to top five is Steve Evan Derek there you go Tim And Derek wow look at that. Did you get two spots? I think you only had two spots Two out of the top five not too shabby. There you go, man Congratulations Now let's uh Fill you guys in here boom Appreciate it guys. Jasper's case breaks.com


Excessive Gossip Have you ever met someone who seems to beam with joy when speaking negatively about others? Gossip is a predictable and inevitable aspect of life and it's not abnormal for someone to gossip occasionally. However, gossiping becomes very dangerous and toxic when taken to the extreme such as starting vicious rumors and using others to obtain personal information that could be damaging to their reputation. Highly competitive. Possessing a strong sense of ambition is generally something positive and admirable, which it is. There's nothing wrong with being competitive. However, when a person is so competitive that they're willing to compromise their own values and hurt other people to achieve their goals, that's when it becomes toxic. Insincere Flattery Another prevalent behavior of a Machiavellian individual is the use of insincere and excessive flattery, especially to those in positions of power and influence. Take it from the Queen B.R.S.L.F., Regina George. She used


Hi, I'm Alex Dorgan. I'm a program manager here at Universal Hydrogen. I began my educational journey at Skidmore College where I studied jewelry and metals or art, and I minor in business, and from there I had a very crooked career path towards, you know, managing aviation programs. I had a short standard research center where I developed some of the technical know-how of the industry and I worked on some user interface design, and from there I moved over to UH2 and that's where I'm now working on program management for the projects that we have here. I'm originally from Alston, Massachusetts, which is a neighborhood of Boston, and I grew up playing tennis, spending a lot of times outdoors, and doing a lot of art.


So we're checking out the MQ boot here a bootloader for IoT. Hello, who are you? My name is David Brown. I'm with Lenaro and what are you? What is this? So this is a NXP Freedom K64 board. This has a ARM Cortex M4 based CPU on it. We're gonna be showing the bootloader that we've come up with it works both with Zephyr and Minute currently. There's interest in other OSes as well for this. So it's a shared bootloader between... what is Minute? So Minute is another real-time operating system similar to Zephyr and they originally developed the bootloader and we worked with them to port it to Zephyr as well. How are you able to share a bootloader between different platforms? So there's a set of common code that is used by all of them and then a small amount of shims that interface directly to the underlying operating system. The bootloader doesn't have a large number of requirements. It basically needs to operate the flash driver and print some messages out. All right, so what are you showing here? So what we're showing here is we have the bootloader that I can use a JTag debugger to flash into the device and then looking at the serial port we can see that there is a small JavaScript interpreter running. It's called jerry script. And if we reset, you can see that we're still in that. So that's pretty quick or? Well, so the thing that we can do now is an over-the-air upgrade. So there's a new version of the firmware. So this firmware will be sent and placed in a second partition on the flash device and then the bootloader will look for that, detect that there is a new image there, verify its signature to make sure that this is supposed to be running and then we'll swap the two images so that the new version won't runs. And once that's done, we now have a small Python interpreter running on the target. And as a one particular use case, this could be a test build and if the test fails, the bootloader would then swap the old image back that the upgraded image didn't work and the device is not running a useless image. So is that a fully secure, stable way of updating IoT? That's the idea. Fully secures a pretty strong term to use. It does require some hardware support to protect the bootloader itself from being replaced. As I began, I wrote the bootloader with a JTAG debugger. So it will be better on the Cortex M33? Yeah, well what's really required is the ability to lock the flash and prevent it from being written to once the bootloader is placed into it. Lock the flash, prevent it to be written to. Correct. So that's not something in Cortex M4 and M3? It's generally a feature of a specific SOC and in this particular case, for kind of clear reasons, you don't want to do that during development because you go through a lot of boards if you lock it each time you change your software. And then so you're going to be able to verify signatures and safely upgrade from the verified image. Yes. So what does mean the signatures verified? I have another demo where we can load in an image where I've corrupted one byte of it and the upgrade is just rejected entirely. Is there any way that people will be able to fake that or trick that somehow? Depends on which people you refer to. I mean, this is the same kind of cryptography that's used to secure websites. There's a ECDSA signature over a SHA-256 sum. And as long as the public key is kept, or the private key is kept private, it shouldn't be available. And this whole Zephyr stuff is kind of new, no? Yes. And it's been moving very fast? Definitely, yes. So that means you have a lot of stuff to do? Is that the... Oh, there's definitely a lot to do, yes. This is what you work on? Yes. So I've been working on this bootloader for about six months now. All right. Is it fun? I enjoy it, yes. It's challenging. Because this is very important. There's all these lights and all these things and if they're connected, they're not secure, but you want to make them secure. And really the bootloader is kind of the first important stage to any kind of security you have. If you don't trust the image that you're running, then any security you have after that could be thwarted by simply loading rogue images. So it's kind of the underlying thing behind any of the security we have on these targets. Is there other parts of the security that's important, or is just security bootloader stuff? Oh, no, the other parts are definitely important. Well, you need ways of protecting memory, secure cryptography between the device and other things it may be communicating with. Goodness, there's all kinds of things. So everything has to go through the trust zone all the time? Yeah, and obviously there's different kinds of levels of security. These devices are very small. The ARM V8 M's aren't even yet available and people want to build devices. So there's different levels of security you get depending on the hardware. And the narrow has taken a big role in all this solving all these issues. We're trying to. Yeah. And the team is spread around the world? Yes. Yeah, I'm based in Denver. I'm the only one on my team there. And you're chatting with some people in Sweden? Yes, that's pretty regular, yes. Okay, cool.