 All right, so now we're going to talk a little bit about quantum mechanics in this lecture. So quantum mechanics we've seen so far has been with the model systems summarized here, either particle in a box or rigid rotor or harmonic oscillator. And quantum mechanics has been very useful so far. We've been able to take the results of quantum mechanical calculations, take those energy levels and use them to do statistical mechanics, make predictions about what systems are found in what states, extend that a little further to talk about thermodynamics. And we've gotten a lot of knowledge out of being able to talk about the thermodynamic properties of lots of different systems. But I'll point out now that the only cases where you've actually solved quantum mechanical problems, solved Schrodinger's equation, has been for fairly approximate systems. It's not actually true that molecules behave like particles in a box or perfect rigid rotors or perfect harmonic oscillators. These are only approximations, they're only model systems. It turns out there is one system, the next system that we'll study, that we can study not just the model system but the way this system actually behaves in the real world. So the next quantum mechanical problem we're going to study is the hydrogen atom. So just like for particle in a box, rigid rotor and harmonic oscillator, what we need to do first is to figure out what the potential energy is that the system obeys. So the hydrogen atom, of course, consists of one proton in the nucleus, which has a charge of plus one electron, same charge as an electron, and also in a hydrogen atom we have an electron, which has a charge of minus one in units of electrons. So of course, quantum mechanically, the way we think about that is the nucleus, which is a point at the center of the atom, is surrounded by the electron in some sense. So if we solve for the wave function, that's going to end up telling us the probability of finding the electron at various different locations around the nucleus of the atom. But at any given position, we can consider the electron to be located at a position space, and the wave function ends up telling us something about the probability of finding that electron and generates this probability cloud. So what we need to understand is how to calculate the potential energy of the electron, which is at some point in three-dimensional space around the center of the atom, so at some point in r, theta, phi coordinates. But of course, it doesn't matter if we think of this atom at the moment as just located at some point in space around the nucleus. The interaction between these two, between this plus charge and this minus charge, is Coulombic. So it doesn't matter whether that electron is north of the nucleus or south or east or west or in what direction it is. So the angular coordinates don't matter, don't affect the potential energy. So we can end up writing this potential energy expression using only the distance between the proton and the electron. And what Coulomb tells us is that interaction, potential energy of interaction between two point charges is charge of one of them times the charge of the other one divided by r. So in this case, the charge is negative one times positive one. So negative one electron times positive one electron charge units, so it's minus e squared over r. And to get the units right, that expression actually looks like minus e squared over 4 pi epsilon naught times r, where epsilon naught is the permittivity of vacuum. So you can think of that 4 pi epsilon naught as essentially just a unit conversion that allows you to calculate a charge squared divided by a distance and end up with units of potential energy. This is the expression for the potential energy of an electron at some distance away from a proton. Turns out, so we'll write down Schrodinger's equation next, but it turns out we can get a little more mileage by thinking not just of a hydrogen atom, this expression would be almost exactly the same for, let's say, a helium plus ion, right? Helium plus ion has two protons in the nucleus and because it's a positively charged ion, it only has one electron in the electron crowd surrounding the nucleus. So the only difference between this diagram I've written for a hydrogen atom and a comparable diagram for a helium plus ion would be that the nuclear charge would be plus two instead of plus one for the helium plus. And likewise, if I make ions of lithium or beryllium and so on, as long as I make the ions strongly charged enough to only have one electron surrounding the nucleus, the only thing that's different is the nuclear charge is plus z, the atomic number of the ion that we're talking about, the electron charge is still minus one. So in this case, the product of the two charges is going to be minus z times e times e. So I can use the same expression, just include the atomic number of the element that we're talking about and we can use this expression not only for a hydrogen atom, but also for some particular ions that we call hydrogen-like ions and they're hydrogen-like in the sense that they only have one electron. So it's a lot like a hydrogen atom with only one electron in the electron cloud with a differently charged nucleus. All right. So this is our expression for the potential energy. We can use that in the Schrodinger equation to find the equation that we're going to have to solve for the wave function of the electrons in the hydrogen atom. So Schrodinger's equation, of course, is h psi times e psi. That begins to get more meaningful if we write down what the Hamiltonian operator is and that's this question of constants minus h squared over 8 pi squared mass times del squared plus the potential energy times the wave function. So we're going to end up solving for the wave function, which is going to have dependence on r, theta, and phi, the three-dimensional spherical polar coordinates. Because we're working in spherical polar coordinates because it's easy to talk about distance and angle away from the center of some coordinate system, that means we also need to write down del squared in spherical polar coordinates. So if I take this term, the minus h squared over 8 pi squared mass, del squared in polar coordinates is a little bit complicated as we recall from the days when we were thinking about the rigid rotor. But what that looks like is 1 over r squared d dr r squared d dr of the wave function. So that's the r derivatives. We also need to include 1 over r squared sine theta d d theta sine theta derivative of the wave function with respect to theta. And there's a term that looks like 1 over r squared sine squared theta and then second derivative of the wave function with respect to phi. So that thing I've just written in brackets, that's the expression for del squared in spherical polar coordinates. We still need to include potential energy times the wave function. So that's going to look like not a plus. So plus the potential energy, but the potential energy is negative. So I'll write minus z e squared over 4 pi epsilon not r multiplied by the wave function. So this whole long expression on these two lines is the left-hand side of Schrodinger's equation on the right side of Schrodinger's equation. That's all equal to the energy times psi. And if I clean that up a little bit, the mass here, think for a second about what the mass is that I'll use in this expression. Ask yourself, is that going to be the mass of a proton, mass of an electron? And the answer is that it's the mass of the electron because that's the particle that we're solving for the wave function of. If we want the wave function of the electron, we'll use the mass of the electron here. And again, just to point out a few features of this equation, the r theta and phi, when we're done, we'll end up solving for a wave function as a function of r theta and phi. Those are the coordinates of the electron relative to the nucleus. So at some distance and some angle, theta and phi relative to the nucleus. And in these derivatives, d dr, d d theta, d d phi, those are derivatives with respect to those coordinates of the electron. And this r, again, is the same r, distance of the electron away from the proton. So what we have here is the Schrodinger equation, the version of the Schrodinger equation for the hydrogen atom or for a hydrogen like ion that we'll need to solve if we want to understand the wave function of a hydrogen atom.