 So let's take a look at our rigid rotor wave functions in even a little more detail. What we've seen so far is we've got a collection of wave functions and I've written more of them here than I've written before. Previously we've just seen these first four, now I've written a few more until I ran out of room at the bottom of the board here. And I've included the normalization constant. So every one of these different wave functions has its own individual normalization constant. If you normalize the wave function you can find these normalization constants given the wave function. So there's many of these wave functions this continues on forever. There's an infinite number of solutions that solve the rigid rotor Schrodinger equation. And if we look at enough of these we can see that there's several patterns that we can count on when describing the rigid rotor wave functions. So when we're talking about these wave functions they have some features in common with each other. First of all they all have this normalization constant, a constant that's different for every wave function. Some of them happen to be the same as each other but in general the way the normalization constants are different for every wave function. In addition to the constant out front there's a term that involves some sines or cosines of theta. And if we look closely that's we can describe that as a polynomial in sines and cosines of theta. Either a linear polynomial there's one term sine theta or cosine theta or sine theta or it might be a quadratic term with a sine squared or sine and cosine so degree two polynomial in sines and cosines. So there's a theta dependence of this function involves either one power or two powers or more powers of these sines and cosines so that's how the theta dependence looks. The phi dependence always shows up in the exponential I've either got e to the phi, e to the i phi with a plus sign or a negative sign, e to the 2i phi with a plus sign or a minus sign. So there's e to the i times phi times some small integer that integer might be one, integer might be negative one, it might be negative two, it might be positive two, it might even be zero, e to the zero phi gives us the phi dependence of this term and this term and this term. But in general it seems like I always have an e to the i times some integer times phi. So those are the characteristics of these wave functions and if we look a little closer particularly let's look at this polynomial dependence on theta in sines and cosines. If I divide the list up in this way then we can see that these are the terms that I've called linear. I've either got a sine or I've got a cosine or I've got a sine but there's one trigonometric function of theta so I'm going to call those the l equals one so in this, this set of three is the l equals one functions. The one simple constant wave function, wave function just equal to this normalization constant there is no dependence on theta so that's like a sine raised to the zero power or cosine raised to the zero power. So there's a zero-thorder polynomial, the theta dependence here is a zero-thorder polynomial or just constant. And now if I look below this line here I've got the functions I've called quadratic either a sine squared or maybe a cosine squared or a sine times a cosine but in general there's two powers of these sine and cosine functions so these I'll call the l equals two functions. So I'm using this variable l to describe the degree of the polynomial that describes the theta dependence of the function. So that number l that I've just described it's an integer. We call that number a quantum number it's an integer or a number that describes the quantum mechanical wave functions or the quantum properties of each of these wave functions. In particular we call that the angular momentum quantum number and it's not obvious right now why we would call it an angular momentum quantum number. Quantum number just means number describing quantum mechanical system. The reasons we call it an angular momentum quantum number will become clear soon enough when we move a little further and talk about more quantum mechanical properties. The preview of that is the larger the value of l the more oscillation this function has so as it oscillates up and down more often the angular dependence of this function has a higher angular momentum. So but right now that's just a name we call l the angular momentum quantum number and it's one of the numbers that helps us describe or distinguish these different wave functions from each other. The other way we can distinguish them from each other is via the phi dependence so I can also choose a quantum number that describes the phi dependence and that one I've already given a name to that's the integer m that describes the phi dependence in this exponential so I either have e to the i times negative one times phi or zero or positive one negative two negative one zero one positive two. So m is another quantum number that also has a name that's the magnetic quantum number again not obvious from these equations necessarily yet why it would be a magnetic quantum number the preview of that answer is each of these functions with a different value of m will behave differently from the others when you put it in a magnetic field so when you put these diatomic molecules with these wave functions in a magnetic field it's their magnetic quantum number that describes in part how they behave. So m is the value of the integer coefficient up in this exponent for the phi dependence of the function and notice also let's go ahead and label these functions by their m values so sometimes it jumps right out of this this function has m equals minus one this one has m equals plus one the functions with no phi dependence they have a quantum number of zero because e to the zero is just one so we've not written the the phi dependence there so likewise this term which has no phi dependence that's an m equals zero wave function for these l equals two functions I've got minus two minus one zero one and two so m is equal to minus two or minus one or zero or positive one or positive two and you might see a pattern now for the functions with l equals one m could be one or it could be negative one or it could be zero the value in between but we don't have any values of m this plus two or minus two when l is equal to two I have m equals minus two and m equals plus two and all the integers in between but m doesn't get any larger than two or any smaller than negative two when l is zero we could think of this as saying m is equal to zero or it's equal to negative zero or any integers in between there's only one of those integers so it happens to be the case that the valid values for m it could be negative l or one larger than that or and so on all the way up to l minus one or l so negative two up to two negative one up to one negative zero up to zero if I were to continue this list with some l equals three wave functions which exist then the values of m would range from negative three up to three and all the integer values in between so those are the possible values of the magnetic quantum number the angular momentum quantum number as we've seen here if we want to list the possible values of the magnet of the angular momentum quantum number they could be zero or one or two and that pattern continues it's any integer value of l beyond that so we have these quantum numbers that help us describe these wave functions and now notice that every single wave function that I've written down has a unique pair of quantum numbers if I pick one of these wave functions out of the list I can describe that either with the full quantum mechanical function or I can describe it as the l equals two n equals minus one wave function so what that means is when I write down this long list of wave functions I can describe those with indices if I give you the angular momentum quantum number and if I give you the magnetic quantum number I've uniquely identified one particular wave function and now we see that the general pattern for these functions I've got some normalization constant which depends on the value of l and m the normalization constant is different in general for every different l and m I've got some polynomial function that depends on l and depends on m the polynomial dependence on theta and I've got the phi dependence is just e to the i m phi so this part is pretty easy to predict if I say tell me what this looks like for an m equals five magnetic quantum number equal to five wave function you could just say e to the five i phi but if I say tell me what this function with the polynomial function would look like when l is equal to seven and m equals m is equal to four you'd have a hard time doing that right now so we do have some more to understand about the the behavior of this polynomial function that describes the theta dependence of these wave functions and there is a way to understand that more systematically and that's what we'll explore next.