 Okay, so let's continue with this question of what makes a transition allowed or forbidden using this transition dipole moment. One feature of this expression that I didn't point out in the last video lecture is where this comes from. We haven't derived this expression. This comes from time-dependent quantum mechanics, if you remember the fifth postulate of quantum mechanics that explains the time dependence of the wave function. Later on in the course we'll be able to derive this expression using time-dependent quantum mechanics. But we'll put that off for now and we'll just say that this expression is given to us. So remember that this expression tells us the intensity of an absorption, a transition that evolves light, and going from some rigid rotor level lm to some different level l' m' And if this integral is equal to zero, then that transition is forbidden, it won't happen, light won't be absorbed to make that transition. On the other hand if that integral is non-zero, then the transition is allowed. So we can compute the integral, this integral has a value, it may be zero, it may be some non-zero value, the value, whether it's a large non-zero value, a small non-zero value can tell us something about whether the transition is strongly allowed or only weakly allowed, whether a lot of light is absorbed or less light is absorbed. But for the most part what we're interested in is whether the transition never happens at all or whether it can happen at all either weakly or strongly. So we're often interested in just knowing whether this integral will come out to be zero or something non-zero. And doing so will tell us something about the selection rules for those processes. We've already seen the gross selection rule, one way to make this integral zero, a forbidden transition would be any one for a molecule with a dipole moment of zero, that's certainly one way to make the integral zero, that's the gross selection rule. But now we're more interested in more specific selection rules about what changes in L and what changes in M are allowed. And we'll be able to determine some rules that determine those changes. So the easiest way to understand that at first is maybe with an example. So let's consider a transition from the zero zero level, the ground state of the rigid rotor, L equals zero, M equals zero, up to a state, let's say the one zero level, excitation of one quantum number in L, and not changing the value of M. If we recall what the rigid rotor wave functions look like, the psi zero zero wave function is just a constant, some normalization constant. The psi one zero wave function is a normalization constant, a different constant times cosine theta. So those are the rigid rotor wave functions. So in order to determine whether this transition is allowed or forbidden, we just need to take these wave functions and insert them into the transition dipole moment expression and determine whether that integral comes out to be zero or non-zero. So if we do that, this transition dipole moment integral looks like a double integral. The initial wave function, psi zero zero, is just this constant. The dipole moment, mu times cosine theta, the dipole moment in the z direction. The destination wave function, the one zero state, in our case is n one zero times cosine theta. And then for our integration variables, we have d theta d phi, not forgetting that in polar coordinates, we need sine theta d theta d phi. All right, so that's the integral we have to do. We can simplify that a little bit. The constants we can pull out of the integral. So there's an n, there's another n, mu. If the dipole moment of the molecule is not changing, we can pull that out of the integral. The integral is over both theta and phi. So if I look at any of the pieces that depend on phi, there are none. So the phi integral is particularly easy. That's just the integral of d phi. The theta integral is a little more complicated. I have a cosine theta, another cosine theta, a sine theta. So that's the integral we have to perform. And that's not a terribly difficult integral. The integral over d phi from zero to pi, that one's just going to be 2 pi. But again, keep in mind that we don't care about the number. We don't care whether that phi integral came out to be 2 pi or pi or 3 pi. As long as it comes out to be a number that's non-zero, then everything's OK. If any one of the contributions to this transition dipole moment is zero, if this integral is zero or if this integral is zero, that will kill the whole thing and make the transition forbidden. So really, all we need to know is whether the integral is going to come out to be equal to zero or not. So the theta integral, certainly, we could do this integral. It's not too difficult to do u substitution with u equal to cosine theta would resolve it pretty quickly. But we can take advantage of this shortcut that we don't care what the numerical value is. We don't care what specific value that integral comes out to be. We just want to know, does this theta integral come out to be zero or doesn't it? So to do that, let's take a look at the symmetry of that function. So if I just plot a few of these components of this integral from zero up to, and that's a typo, we should be integrating theta from zero up to pi, not from zero up to itself. If I integrate theta from, so this is the theta variable, I need to know what this function cosine squared sine looks like. So sine theta looks like this, so that's sine theta. I'll draw on a different graph what cosine theta looks like. Cosine starts at 1 and becomes negative by the time we get to pi. So that's what the graph of cosine theta looks like. We're not interested in cosine. We're interested in cosine squared. So if I square cosine, it's still zero where the function is equal to zero. But both the negative and the positive portions of this function become positive. So that cosine squared looks something like this. And then what we're really interested in is this product cosine squared times sine. So if I multiply sine times this pink curve here, the cosine squared term, it's equal to zero in a few places. Sine is equal to zero at zero and at pi. Cosine squared is equal to zero at halfway through, at pi over 2. And everywhere else, I'm multiplying a positive number times a positive number until it hits zero. Or a positive number times a positive number until the sine function hits zero. So this function, generally speaking, is going to have a shape like that. It's largest here at pi over 4, 3 pi over 4. It hits zero at pi over 2. Even those specific details are too much information. I don't need to know where the zeros are. All I need to know is that function is either positive everywhere or it's sometimes zero, but it never dips negative. That's enough for me to say that the area under this curve is non-zero and is positive. That's, after all, what we're interested in here. This integral of cosine squared sine theta, that's asking for the area underneath this curve, cosine squared sine. And what that area comes out to be is not relevant at the moment. All we need to know is that that integral is, in fact, greater than zero, non-zero. Likewise, this term is greater than zero. So without knowing the specific value, that's enough for me to say this total integral is non-zero. So this 0, 0 to 1, 0 transition for the rigid rotor is, in fact, an allowed transition. If I have a rigid rotor or a diatomic molecule that we're treating as a rigid rotor in the L equals 0, M equals 0 state, and I shine the appropriate frequency of light at it, it can absorb a photon of that light to excite up to the L equals 1, M equals 0 state. So that's an example of how we would determine whether a transition is forbidden or allowed. Transition dipole moment came out to be non-zero, so it's an allowed transition. We could work another example where the integral comes out to be zero, and that would be a forbidden transition. But obviously, it's going to get a little bit tedious to work one of these integrals for every possible transition that we can imagine. It's much more useful to come up with a general set of rules for which changes in L, which changes in M are allowed or forbidden. So that's the next thing we'll do.