Hello, everyone. Welcome back to Exploring Quantum Physics. I'm Charles Clark. In this lecture we're going to use the variational method to make estimates of the properties of quantum mechanical systems. Now there's been a lot of material that you've been exposed to in this course. And I'd just like to make one recommendation. And that is, if you can remember one thing, one thing to sort of commit to memory. Which we're going to give you some encouragement to do so by the practice in this part of the lecture, it's this simple Gaussian function. Its shape is it's a little lump, with a width that's proportional to d, and it's easy to integrate. Numerically it's a very practical thing to use, it's easy to calculate, and it happens to be, here where g is just a parameter. It has the units of length. It expresses the characteristic length scale of the function and it enters the normalization coefficient here. You see the wave function has the units of one over the square root of length. And now although d is just a parameter in this wave function, when we interpret this as the ground state of a harmonic oscillator. Then d the characteristic length scale, is the square root of h bar, Plank's constant, divided by the mass and the frequency of the oscillator. So in other words, you can use this function. You can remember it as the ground state of the harmonic oscillator, but then you can use it in a very arbitrary way to represent wave functions of complicated systems. So now, we're going to do a calculation and it's something, we're just going to do it once. Because when you do it once, I'm going to give you a numonic, I hope you can will be useful for you to remember the result. And that is to calculate the expectation value of the kinetic energy operator applied to this wave function, which is in the one-dimensional kinetic energy, operator mass is bar square root 2m dx squared. Now, this is a straightforward integral if you just differentiate this function twice. But I want you to think about how you can do it without actually doing explicit integral and be able to remember how the procedure works and to reproduce it when necessary. And the trick is the Virial Theorem, which you encountered in a previous homework, and I think it was mentioned in an earlier lecture. And that is for the harmonic oscillator, here's the Hamiltonian with the usual form. Kinetic energy and the potential energy that's indicated there. The Virial Theorem states that the expectation value of the kinetic energy over the wave function is equal to the expectation value of the potential energy. So these two terms make on the average an equal contribution to the total energy. So now I'd like you to think about what that implies for their actual values. So I hope that you remembered how to get that result. The Virial Theorem states that the two contributions are equal. Well then the, we must have T. +<v> =E, the total energy on the average. So in other words, the expectation value of the kinetic energy operator is half that of the total energy, which is equal to,</v> One-half times one-half h bar omega equal in the ground state. Now, I want to make two comments about this, the Virial Theorem very powerful. So, first of all it actually applies to any state E, the most arbitrary state you can make of the harmonic oscillator, including a time dependent wave packet. Now, in that case, this expectation value has to be generalized, meaning it's time averaged. But that makes it very useful and the next to the last point to make on this for the moment. It's valid in any number of the dimensions. So, if you have an n dimensional harmonic oscillator, then you can just count the contributions to the expectation value of the kinetic energy from each individual coordinate. A last comment is that virial theorems exist for other potentials, but this case of the harmonic oscillator is very special when we have this so called equipartition of energy. Between the kinetic and potential terms, as we'll see later, the Virial Theorem for the cool coolum problem, or the hydrogen atom. Also very useful, is an entirely different form. So to recapitulate, if we take this Gaussian as our trial function. And just compute the expectations out of the kinetic energy, is half the total energy. And it takes this form. I've written it this way just to emphasize there's a factor of a half out front, which is due to the levothym. And this is the ground state energy of the harmonic oscillator but written in terms of the characteristic length. So you'll see expressions of this frequently in quantum mechanics. H bar squared over the squared Planck's constant divided by mass and the square of a distance. That's the, this is the energy of localized function. In other words, if you have a function that's localized on link scale D, then the characteristic motion, the uncertainty principal based motion, the average value of the kinetic energy. Of that wave function is of the order of h bar squared over 2md². So if you remember the correspondence of the Gaussian with the harmonic oscillator ground state, which I've just been emphasizing over and over again in this part, you don't have to do this interval again. Having done it once, you can use it forever. And keep in mind that it always has a contribution that goes in the inverse square of the characteristic length scale. That means that as you try to compress a wave packet, you raise its kinetic energy. It's a manifestation of the uncertainty principle again. So, now let's see how this works in a simple application. Now, in one of the homework problems for last week, there's a variational estimate of the Attractive Dirac Delta function potential. Now, I can see from responses on the student forums. That you know some of you have a lack of familiarity with the direct delta function which is fine. Now I think there's a DLMF chapter. A DLMF chapter. D-L-M-F Section 1.17 for the Dirac Delta. I think that's very clearly and accessibly written. And the delta function is the limit of a sequence. You can think of a sequence of functions. Let's say localized on the line. Let's take this square well function on the line of width a, and depth v naught. And what you do is take a sequence of such functions, where a gets smaller and smaller. And v naught gets larger and larger. So that the product is preserved. So, in fact, you can, if your having trouble with the concept of the Dirac Delta function. Just think of it in terms of a very specific implementation like this, and think of this as an approximation of the delta function. And if we use this as the approximation of the delta function then, by its definition, this potential here, is equal to -V0a delta of x. Because as you can see, when we perform this integral of the potential, what we get is the only contribution is the product V0a with the negative sign. Over that finite interval. Now we use the delta function, the delta function is handy for sample applications. But it's widely used in practice as sort of a pseudopotential for representing information about complex interactions between particle. In terms of a single parameter. And this is useful when the de Broglie wavelength of the system, which is the characteristic wave length of the quantum mechanic wave function. That's h over p, is much, much greater than the range, the range of interactions that affect the function. Let me give you just a simple concrete example of that before proceeding. Here we see a schematic representation of a wavefunction for an ultra cold atom system, two atoms colliding. And here is the quantum mechanical wavefunction for two cases of bound in a and a free state and then this is the intermolecular potential. And you see in this system, there's a lot going on in the region of close approach of the two atoms, and many, many, many wiggles in the wave function. But in the application that's relevant, the application that's important, in the discussion of this paper, is the behavior of the wavefunction at large distances from the atom. And so, really the details of what happens in this inner region are only important insofar as it affects the long range functionality behavior of the wave function out here. And so that behavior can be built in by the use of a Delta function in an internal region that just sets the scale for the evolution of the wave function further out. So lets go and complete the problem. We're going to calculate the expectation value of age as a function of d. Using our trial function. Okay, so we have, we have this expression for the kinetic energy, it's h bar squared over 2m b squared. So and now let's just recall what it must be for the contribution for the potential energy. Well all we need to do there. Is when we integrate over the delta function we just get minus v0a Times psi squared evaluated x equals 0. Right? Because that, the delta function, this takes a measure of the weight on top of it at the origin. And so psi squared x equal to zero, is just d over the square root of pi. So this gives us, at small d a very repulsive, in fact let's just plot it out schematically. This would be T. Now let me see if I can change the color here. Yeah so now what about V? Well, V is minus one over D. So that is negative here, and it falls off inversely proportional to the potential. So this is v. So now let's look at the sum of the two. So the sum of the two, it's positive at short distances, but then it's got to become negative at long distances so it turns over. And so this, this defines a point of minimum. So you just have to find the minimum as a function of d, find the minimum value of this, the sum of these two terms. And that gives you the best choice of the scale parameter. To estimate the energy of the attractive delta function. I want like to say that this is the same minimization that one encounters for the hydrogen atom in three dimensions. That is first of all, obviously, the kinetic energy, the contribution of the kinetic energy is always the same. And then if you think of it, well the hydrogen atom has you know v of r goes as 1 over r. So it stands to reason that you'd have an inverse length in the expectation value for the potential energy. And we'll see in the subsequent part that that's the case.
