Hello everyone. Welcome back to Exploring Quantum Physics. I'm Charles Clark. In this final part of the lecture we're going to look at the polarizability of the hydrogen atom. A system in which electron and an ion move freely, according to their own equations of motion in three dimensional space, much richer system you might think than the diatomic polar molecule which can be effectively treated as a rigid rotator, but as we'll see, the concepts that were developed in of course the molecule, apply as well to the richer case of the atom. I might just mention, parenthetically, I eluded to some of my own scientific work in a previous lecture. I am very interested in the effects of electric and magnetic fields on matter. This type of approach, as applied to the hydrogen atom, is very similar to research grade work that is done in computing the black body radiation shifts of optical atomic clocks. So we're going to go through a calculation which has to do with finding the response of an atom to an electric field. It's convenient in these cases to the work in the CGS units, just because as you'll see, there's a lot of complicated factors floating around and that simplifies the book keeping. We're going to look at the general problem of a hydrogen-like ion, which might involve an electron and a nucleus, it might involve an antiproton and a nucleus, it might involve an electron and a positron. The nucleus has a charge of Z times Z, which Z is conventionally the atomic number. The electron nucleus distance is given by r. We retain the reduced mass so that this formula can be used to describe species other than electrons found in nuclei. And we're going to do perturbation theory, just of the very same type that we applied in the diatomic polar molecule. Slight difference here for notational convenience, I'm writing this, the typical conventionally, is H naught plus lambda V for the Hamiltonian, it's what's being solved. So I'm just going to use F, as the field strength, is equivalent to lambda, and that means that the perturbation here is just given in terms of the unit vector, the scale of the product of the unit vector with the radius times the charge. That half means unit vector now. So here is the Hamiltonian for the hydrogen atom that we've talked about before. Here is an expression for the ground state energy, E naught which is just the Rydberg formula, but with finite mass and the square of a nuclear charge. And then here is a wave function that was introduced in a previous lecture. This time it's correctly normalized. I don't know why I had the wrong expression before, but this is the normalized ground state wave function of a hydrogenic ion. Where this length scale, this, it's very close. If mu were the electron mass and Z would equal the one, this would be the vor radius, a naught. But now, we'd retain it as general form so that we can deal with particles of arbitrary mass and of arbitrary nuclear charge. So there's a lot of symbols floating around here and the mathematics does get fairly involved. But we're going to take a fairly high level view, and I'll show you what you need to do to actually perform the calculations in detail, should you so choose. Now as usual, the very first thing we do, because all of the ingredients are laid out, so we calculate the first order energy. I always love doing this because usually it's the easiest step in the problem and leads to useful results. So here, I've written it out in its full glory, the first order energy which is just the expectation value of the perturbing potential over the square density of the wave function. Now I would like you to look at that and see if you come up with any easy ways of performing that integral. Okay, I hope that you saw that. Again, this is an interval that vanishes by symmetry. If you started to write this out and try to evaluate it by long hand without noticing the essential symmetry property, it could be laborious. And I'd just emphasize that many of them, basically, all of the electrical field problems that one encounters in quantum mechanics, tend to have this generic feature. In other words, if you have this is potential, this interaction potential, v, is odd under parity. If you invert the coordinates, the potential changes sign, and so that if you're integrating that over a symmetric density, the integral necessarily vanishes. Okay, well, we got that easy part of the calculation out of the way, the determination of the first order energy. So now we come to what is really the most difficult and that is the solution of this equation, the first order equation for the wave function. This is more complicated than the one that we had for the diatomic molecule. Because, see here is the usual left-hand side, which has a Hamiltonian operator that's got Laplacian and coordinate function in it. And then on the right-hand side, I guess you can see the crux of the issue now, is that we have two variables, the radius and the angle. In the diatomic molecule, we didn't have the degree of freedom and radial coordinate. This makes the equation much more complex, but there is a saving factor, again, associated with the angular momentum. So recall that one again, the H naught is the Hamiltonian for the hydrogen atom. It's completely, spherically symmetric and it is being applied to thus psi 1, and the result is something that's asymmetric, so I emphasize that means that the asymmetry is present in psi 1. It's revealed by the action of h naught, but it is not created by it. So that means we know that psi 1 has to have this particular, again this is a y 1, 0 harmonic embedded in it, and if you go back and look at the Hamiltonian, you'll see that there's a del squared function of which the cosine theta is an Eigenvalue, and then also this part here. The fact that the cosine theta is accompanied by a single power of r, guarantees that the behavior of this function near the origin will be consistent with this powerful property, the angular momentum operator in that the angular momentum completely dominates the motion of the particle at very small distances from the coordinate of origin. So whenever you see the cosine theta at small r, this, what's called characteristic value of the radius, must also be present. Now this is more like a matter of experience because of the long range profile of an exponential decay, you need to have that present in the envelope of psi 1, and then the remaining part is a radial function in order to keep the angular character consistent with the cosine. So this is the description of the analytical steps that are needed to get an appropriate form of the solution. Now the determination of f is another matter, which we'll discuss on the next slide. So again, here's the analytic supposition that we wrote down for the first order wave function and with this in hand, it's not difficult to show that f must satisfy, but is proportional to this rather simple linear function. Again, I think those of you who like working out real quantum mechanics problems, this is perhaps the most involved one that has been offered during this course. And perhaps you can get a glimpse of that from this crazy looking solution. Yes, that is a power of h bar to the 6th and the 6th power of the electric charge and the third power of the reduced mass. I think that's probably more analytically interesting than many of the results that you've seen so far. This does have a dimension of volume. Welcome to check that, keeping mind that e is in the CGS system. So in other words e squared over r has the dimensions of energy. Now there is something I want to draw your attention to. It's the behavior of the polarizability of hydrogen as the plot constant goes to 0. I mean, as if we could set the plot at constant 0, but this is the limit of classical mechanics. And so what you see is that, in this limit, the polarizability goes to 0. So in other words in a classical world, the hydrogen atom would not respond to an electric field at all, it'd just be too tightly bound. And this is consistent, I think, with the idea that you'd have in the Bohr model that if the quantum of action plots constant were smaller, the radius of the atom would shrink and shrink and shrink and the electron would become so tight, would become ever more tightly bound, the polarizability would go to 0. So this shows, really, the opposite effect in the transition of the classical limit that we saw for the polar molecule. And one of the home work problems this week, which was actually suggested by one of the students, or at least I interpreted their remarks as a suggestion, shows, well I don't want to reveal the answer, but just say, the behavior of that problem, which is relevant to trapping of atoms in the Earth's gravitational field is, I'll just say, not exactly the same as either of these two cases. Okay, so that's a good introduction to perturbation theory. I realize it's brief and of an overview nature, but we've used it to attack some real problems of atomic and molecular physics. I hope that gives you the confidence that you can go and use it on the homework problem and in other things that you're interested in.
