 All right, so we spent a little bit of time understanding what the wave functions mean when they're spherically symmetric. Turns out the S orbitals we're familiar with from general chemistry result from the wave functions with L equals 0 and M equals 0 with no angular dependence. So now let's see if we can understand what the wave functions look like when they have some angular dependence. So for example, when L equals 1, then we have some theta terms in the wave function. When M is non-zero, when it's plus or minus 1 in this case, we have some e to the minus i phi or e to the plus i phi terms that show up in the exponential in the wave function. So we'll take these first two cases and see if we can understand what the wave function would look like. So we're going to mostly ignore the radial part. We've talked about what the radial parts look like already. These exponentially dying terms look turn into oscillating and then decaying wave functions with radial nodes. So for now we're just going to focus on the angular part of the wave function. So if the angular behavior of the wave function just looks like cosine theta, if I draw wave function as a function of theta, cosine theta, I only need to worry about the value of theta from 0 to pi, and cosine theta looks like this function. So remember, we're not really interested in the wave function so much as we're on the probability and the probability looks like wave function squared with an r squared in front of it. The r squared isn't particularly necessary this time because we're only plotting the theta dependence. But if I square this wave function, then what I discover is regardless of whether the value is negative or positive or negative, the squared value is positive, but it does have a node. There's a place where the wave function goes through 0, so that means there's a place where there's 0 probability of finding the electron. And that's that the node occurs at a value of pi over 2. So if we now try to sketch what that wave function looks like, if I take the nucleus and I ask where am I likely to find the electron relative to that nucleus, I'm quite likely to find it at an angle of theta equals 0, meaning pointing straight up along the z-axis. I'm likely to find it in the direction of the z-axis. I'm somewhat less likely to find it if I look at a slight angle from the z-axis because the probability is decaying. When the angle is pi over 2, in other words, when I'm in this xy-plane, which I'll draw like this, so in the xy-plane, I'm not at all likely to find the wave function. The value of the probability is 0, so this whole plane represents a node, which I'll write so that you can see the entire word. And in this case, it's an angular node. Rather than a radial node, instead of saying there's an R at which the electron can't exist, at this time there's an angle, theta equals pi over 2, for which the wave function is 0 and the probability is also 0. If we continue, I'm also very likely to find the electron at an angle of pi, meaning bent down a full angle of pi away from the z-axis, so pointing going to the negative z-axis. So what that means is I'm likely to find the electron here, I'm likely to find the electron here, and if I put in a lot more of these points, what I find is this two-lobed wave function, where I'm quite likely to find the electron above the straight up. As I move the angle sideways, I get less and less likely to find the electron, and as I move closer to the horizontal axis, I'm significantly less likely to find the electron. And then the plane itself represents an angular node. So it won't surprise you much at this point to hear that this wave function with that dumbbell shape that you're familiar with, that's the wave function that we call a P orbital or a P wave function. It's a Pz wave function because it's oriented along the z-axis, and it's a 2p orbital because it has a quantum number, a principal quantum number of n equals 2. Things do get a little bit more complicated when I consider the wave functions with m equals 1 and negative 1 that introduce these complex components into the wave function. Complexity affects the wave function because when I square the wave function, I need to remember to get the complex conjugate times itself, but notice that the i equals 1 and the i equals negative 1 wave functions, I'm sorry, not i, the m equals 1, the m equals negative 1 wave functions are already complex conjugates of each other. So the square of these two wave functions are the same as each other. So what we typically do in order to visualize these wave functions is we can take the 2, 1, 1 wave function. If I add it to the 2, 1, negative 1 wave function, so I've taken a linear combination of these two wave functions, I'll go ahead and write a 1 over square root of 2 out front, which is important if I want to keep these wave functions normalized, but that's, for understanding the shape of these wave functions, we don't have to worry about the 1 over square root of 2, but if I take this particular sum of the two wave functions, then without going through all the details, so if that's the z-axis pointing upwards, x-axis pointing this way, there's a y-axis receding into the distance, then this wave function looks an awful lot like the 2pz wave function, except by doing this I've rotated it and it points along the x-axis now, and that's the wave function that we call the 2px wave function is a sum of the m equals 1 and m equals negative 1 wave functions, so these are not the 2px and 2py wave functions, if I take the sum of them I get the 2px wave function, if I take the difference of them, so 1 over square root of 2 psi 2, 1, 1 minus psi 2, 1, negative 1, that will be the 2py wave function, which this one I won't be able to draw quite as well because it involves drawing into this third dimension, but a wave function that is behind and in front, but with these p-lobes, so the 2px and 2py wave functions are linear combinations of the m equals 1 and m equals negative 1 wave functions, so we've seen that the l equals 1 wave functions look like p orbitals, it's also worth understanding what those p orbitals look like when n is not 2 but when n equals 3, so we've added, by moving up to a larger value of n, we've introduced some extra complexity, not complex number complexity, but some extra difficulty in the radial part of the wave function, so the polynomial which in the simpler wave functions was just an r times 1 is now an r times a polynomial that has a 0 in it, so that's going to introduce a radial node, so again cosine theta looks exactly like cosine theta for the 2px wave function, but if I draw, so this would be, if I'm going to draw the 3, what are we doing, 310 wave function, a sketch of that wave function, cosine theta means the wave function is p-shaped aligned along the z-axis, however I've also introduced a radial node, so now I have 2 nodes, I have a node, an angular node, so let me go ahead and erase the x-y-axis, use this plane to represent them, so I've got an angular node at pi over 2 because of the cosine theta term, I also, as it turns out, have a radial node, so there's a sphere at which I'm not allowed to observe the electron because the probability is equal to 0, that sphere comes because the wave function is equal to 0 at some particular value of r, so if I attempt to sketch the 3D picture of this wave function where these scribbles indicate places where I'm allowed to find the electron, turns out I'm allowed to find it in some region close to the nucleus or far away from the nucleus, those regions still occupy a sort of dumbbell p-sort of shape, and in the negative z direction, the same sort of behavior is taking place, I've got a region close to the nucleus and a region further away from the nucleus where I can observe the electron, but at this particular value of r I have a node, so I've got a radial node, and this plane, the theta equals pi over 2 plane, also represents a node, and now we can distinguish between an angular node and a radial node, and this 3p wave function, the 3pz wave function, has both an angular node and a radial node, and for one last example, we can ask what happens when the value of l is not equal to 1 but equal to 2, so what we saw is that for the principal quantum number, when I increase n from 2 to 3 I introduced a radial node, what happens with the angular nodes is when I increase l from 1 to 2 that's going to introduce an angular node, and we can see why that's happening because the angular behavior of the function, this 3 cosine squared theta minus 1 term is going to have, because it's quadratic, it's going to have 2 places where it can equal 0 instead of just a single place like it did for the 2p wave function, so again if I just attempt to draw the sketch, so here's the z axis, I'll find out now that instead of just pi over 2 being the single value of theta that is a node and is disallowed, I've got two different values of theta, so when theta equals pi over 3 and when theta equals 2 pi over 3, those are both nodes, meaning I cannot find the electron with those coordinates, so what I get is a more elongated lobe above and below the plane where I can find the electron and then in between because theta equals pi over 3 is not just in this direction but all phi directions, so I actually have a cone, so there's a cone where I'm not allowed to find the electron and the wave function sort of occupies that cone above and below the origin and then in between the origin I can also find the electron over here and over here and I end up with sort of a donut shaped, a donut goes behind as well, so I end up with a lobe above, a lobe below and then a donut ring surrounding the axis, so that might be familiar to you as a wave function we call the 3D z squared wave function and that's just the name we give to it but what's happened by increasing the angular quantum number from 1 to 2 is that we've increased from a P orbital to a D type orbital and then we could continue examining the shape and drawing what the wave function looks like for the other L equals 2 orbitals and it would not be a surprise to you to find that we'd get the other D shaped orbitals, the clover leaf shaped orbitals that you're probably familiar with from general chemistry, so in the next lecture we'll show you some pictures that are more representative of what these look like than my poor sketches on the board.