Yup. You don't have to think about it right now, but you might have heard in high school talking about p orbitals, the phase, sometimes you mark a p orbital as being a plus sign or negative sign. It doesn't even make sense now, they're not used in spectroscopy anymore, but this is where the names originally came from and they did stick. We'll call these here the 3 d x y, as the subscript, the 3 d y z, the 3 d z squared, the 3 d x z, and the 3 d x squared minus y squared. And when we talk about l it is a quantum number, so because it's a quantum number, we know that it can only have discreet values, it can't just be any value we want, it's very specific values. Privacy So when you solve certain types of problems, such as problems later on in the second half of your p-set, if you need to talk about the frequency of light emitted or absorbed for a one electron atom, such as lithium plus 2, for example, then you would need to plug in z, and remember the z value for lithium would just be 3. We can actually specify where those nodes are, which is written on your notes. Then we go negative and we go through zero again, which correlates to the second area of zero, that shows up also in our probability density plot, and then we're positive again and approach zero as we go to infinity for r. So, what this means is that when we're looking at an actual wave function, we're treating it as a wave, right, so waves can have both magnitude, but they can also have a direction, so they can be either positive or negative. It doesn't depend on theta, it doesn't depend on phi. The reason that we have no subscript to the s, is because the only possibility for m when you have an s orbital is that m has to be equal to 0. And the third one is called m, it's also m sub l. This is what we call the magnetic quantum number, and we won't deal with the fact of its being the magnetic quantum number here -- that kind of tells us the shape of the orbital or the way that the electron will behave in a magnetic field, but what's more relevant to thinking about the limits of this number is that it's also the z component of the angular momentum. The reason there are three quantum numbers is we're describing an orbital in three dimensions, so it makes sense that we would need to describe in terms of three different quantum numbers. 9 times 10 to the negative 18 joules. So I think we're safe to move on here. So if we say that l is just talking about our kinetic energy part, our rotational kinetic energy, and we know that electrons have potential energy, then it makes sense that l, in fact, can never go higher than n. And, in fact, it can't even reach n, because then we would have no potential energy at all in our electron, which is not correct. So we can switch back to our notes here. With more than 2,400 courses available, OCW is delivering on the promise of open sharing of knowledge. So, one way we could look at it is by looking at this density dot diagram, where the density of the dots correlates to the probability density. PROFESSOR: All right. So we came up with two formulas, which are similar to the two that I'm showing here. So, it's very easy to calculate, however, the number of radial nodes, and this works not just for s orbitals, but also for p orbitals, or d orbitals, or whatever kind of work of orbitals you want to discuss. 2 p what? So we can go on and do this for any orbital or any state function that we would like to. Hearing a little bit of a mix here. ... For a given atom, all wave functions that have the same values of both n and l form a subshell. When we first introduced the Schrodinger equation, what I told you was think of psi as being some representation of what an electron is. And then we also have the l being equal to 1 orbital, so those are going to be the 2 p x, the 2 p z, and the 2 p y orbital. Let's switch to a clicker question and just confirm that that is, in fact, true. We know that binding energy is always negative, we know that ionization energy is always positive. And when we talked about that, what we found was that we could actually validate our predicted binding energies by looking at the emission spectra of the hydrogen atom, which is what we did as the demo, or we could think about the absorption spectra as well. We're saying the probability of finding an electron at some distance from the nucleus in some very thin shell that we describe by d r. So if you think of a shell, you can actually just think of an egg shell, that's probably the easiest way to think of it, where the yolk, if you really maybe make it a lot smaller might be the nucleus. So this is our complete description of the ground state wave function. 1 a nought. We'll really think about what psi means. So we saw that our lowest, our ground state wave function is 1, 0, 0. For example, in the Bohr atom, the electron We can graph out what this is where we're graphing the radial probability density as a function of the radius. Hydrogen atom: h2 2 r2 d dr r2 dR(r) dr + " h2l(l+1) 2 r2 V(r) E # R(r) = 0 The solutions of the radial equation are the Hydrogen atom radial wave-functions, R(r). So we can think of a third case where we have the 3 s orbital, and in the 3 s orbital we see something similar, we start high, we go through zero, where there will now be zero probability density, as we can see in the in the density plot graph. OK. So, when we have, for example, l equal to 1, what kind of orbital is this? And what that is the probability of finding an electron in some shell where we define the thickness as d r, some distance, r, from the nucleus. And actually after the Schrodinger equation first was put forth, people had a lot of discussions about how is it that we can actually interpret what this wave function means, and a lot of ideas were put forth, and none of them worked out to match up with observations until Max Born here came up with the idea that we just square the wave function, and that's the probability density of finding an electron in a certain defined volume.