Hello everyone. Welcome back to Exploring Quantum Physics. I'm Charles Clark. We're going to conclude this lecture by another application of pertubation theory to a problem that can be solved exactly, important problem. But this will show you how you can use pertubation theory to test against a result to see if you're getting the right answer in the leading order. We now return to a problem that we studied previously, the spectra of the isotopes of hydrogen. But we're going to approach it from the standpoint of perturbation theory rather than a direct solution of the Schröder equation. So this is a fairly important problem, the history of science, and it's a useful testing ground for perturbation theory because we do have available to us an exact solution. So we can compare perturbation theory in the limit. And then, perturbation theory also gives us some guidance as to how to approach the problem of multi-electron atoms, where the isotope shifts are also important. But where there typically is not an easily tractable solution available. Certainly not an exact solution. And the basic idea, as you recall on perturbation theory, there's a small parameter, and we take that to be the ratio of the electron mass to the nuclear mass. So let's start. We have the electronic coordinates, the mass, the position, the momentum, the designated the the same for the nucleus. Mass capital M, and its own position. It's a two particle system. The Hamiltonian consists of the electron kinetic energy, the nuclear kinetic energy, and then the interaction potential. Now we split this up into two pieces shown here. The one in the red box contains the electron kinetic energy and the interaction energy. And then the term in the blue box is the nuclear kinetic energy. The energy of motion of the nucleus. And we have chosen to extract the small parameter by this method so that we get an equation that looks like the beginning of the perturbation theory. We have a 0 total Hamiltonian, and then a small quantity, which we'll conventionally call lambda times another potential. And here are the values of that small parameter for the several stable or nearly stable isotopes of hydrogen. A protium is the name for electron bound to a proton, mass one, mass proton mass, deuterium, electron bound to a deuteron, and tritium, the electron bound to a triton. Nuclear mass is roughly in the ratio one, two, three. To tell you the truth, I can never remember the equations of perturbation theory. I always return to first principles, which are defined by this equation. And the basic idea is you have some model Hamiltonian and then a correction to it, which is proportional to the parameter lambda. And then both the wave function and the energy are to be expanded in powers of lambda. And so you just put these things in and expand away. And it's very easy to generate the various terms. So here's the zeroth order. Here's the term that's first order in lambda and so on. Now in previous examples on perturbation theory, we were focused mostly on the second order of energy shifts. But in this case, we have a concrete application of the first order energy, which happens to not to vanish in this system and gives the leading correction to the energy levels. So back to our defining equations, now let's do an explicit construction of the zeroth order solution. So the value of our coupling concept lambda goes to zero in the limit of nuclear mass goes to infinity. So we just take that limit, and we choose the coordinate origin to be the position of the nucleus. Then we get our h nought solely in terms of the variables of the electron. And we're going to solve this equation. So now a brief in-video quiz to give you an opportunity to understand what this equation really means. So, I hope that in thinking about that, you came to the conclusion that E0, well, first of all, that it is an energy of the hydrogen atom with an infinite nuclear mass. And that, in fact, it can be any such energy. In other words, although perturbation theory is often applied to the ground states of systems, it's not restricted to those. And, in this case, we can apply to any Eigenfunction of the original Hamiltonian. That is, to a state with any principal quantum number greater than or equal to one. This is very important because it allows us to calculate the isotope shift of a transition wavelength or a transition frequency, which is the difference in the isotope shifts of two different energy levels. So it's really critical that we be able to apply perturbation theory to any level in order to get a result that's useful. So, we have available to us the E0, and the sine 0 can be obtained by solving the infinite mass equation. So now we are to evaluate the first order energy, and so, there it is, it's the nuclear kinetic energy evaluated over an electronic wave function. How are we going to handle that? Well, it's always convenient to choose a good coordinate reference frame. And if we use the center of mass frame then we have this very useful identity that the momentum of the nucleus is equal and opposite to the momentum of the electron. So now we've converted that expectation value into one that involves electronic coordinates only. And what's more, by use of the virial theorem, we don't even need to do an explicit calculation of that matrix element because we showed previously that for the 1 over r potential, the expectation value of the average kinetic energy, which is what this is, is just equal and opposite to the total energy. So, now we can complete the calculation, the first order in little m over big M. So, the energy is just E0 + lambda E1, and E1 is minus E0, so we get this result for the energy level shift. So at this point maybe you should pause and verify that these two results are in agreement to first order in the ratio of electrons to nuclear mass, which is the goal of the exercise. So this use of the virial theorem is something that is also takes place in modern calculations of the of many electron atoms. Because typically, one doesn't have any exact answers and perturbation theory is the only way to make progress. Okay, that's it for this lecture, hope to see you again soon.
