 So we have a collection of wave functions that we've seen solve the Schrodinger equation and we understand why some of them need to have complex numbers in them. The one part we don't understand yet or at least don't have numerical value for yet is this constant N out in front of the wave function, the normalization constant. So we'll solve for that normalization constant the same way we did for particle in a box and that's by normalizing the wave function which means by guaranteeing that there's a hundred percent probability of finding the system in some state described by some values of our angles theta and phi for the orientation of this diatomic molecule that we're representing with this rigid rotor molecule. So if we integrate the probability over all thetas and all phi's we get one. So we need to ensure that that is true and remember that when you're doing an integration in spherical polar coordinates when you're integrating over theta and phi you don't just integrate over d theta d phi you need to integrate over sine theta d theta d phi that integration factor. The connection between this and the wave function is we remember that the probability is the wave function squared or more specifically for this quantum mechanical system the wave functions complex conjugate times the wave function is equal to the probability. So we need to guarantee that when we do this integral wave function times its complex conjugate sine theta d theta d phi integrated over all coordinates so let's write the limits of this integration I'm integrating over two variables theta has to run from 0 to pi and phi runs from 0 to 2 pi so that's a recipe for how to do the normalization. Take one of our wave functions plug into this integral integrate in this way and we need to make sure we get one out on the other end. So to see how that works let's take let's do one of the complex ones so let's take our wave function that looks like a normalization constant times sine theta e to the i phi and just so that we see how the complex numbers affect the integration if we want to normalize that function to find out what the value of this n is then we'll do exactly what's stated here I need to make sure that one is equal to the integral of phi from 0 to 2 pi theta from 0 to pi of complex conjugate so that's n sine theta e to the i phi with all the i's turned to negative i's so I turned that i into a negative i I should also say n I don't know the value then that could be i or it could be a complex number that needs to be complex conjugated as well so that's the complex conjugate of the wave function multiplied by the wave function itself so that's the n without a star sine theta e to the i phi without having turned it negative and then the integration factor sine theta d theta d phi so that's the integral we need to do we can clean it up a little bit the ends we can pull out of the integral those are just constants so n star times n that's n squared squared in this specific way we mean when we're talking about complex numbers I have a double integral 0 to pi 0 to 2 pi sine sine sine gives me three factors of sine the complex numbers turn out to be not a problem at all e to the minus something times e to the positive same thing those cancel each other and just make one so the e to the i phi e to the minus i for my terms just give me a 1 and I've already accounted for the sine theta so I just need d theta d phi the integral is looking much simpler already I've got an n squared the phi integral this outer integral from 0 to 2 pi d phi there's no fies in here I don't have to work hard to do that integration integral of d phi is just phi phi evaluated from 0 to 2 pi gives me 2 pi so the phi integral is done phi integral is equal to 2 pi and now I have I'll go ahead and rewrite the theta integral integral sine cubed theta d theta all right so what do we do here there's a few options probably you don't have this integral memorized you can use an integral table go to the internet look up that integral there's no shame in using integral table somebody's else has already done the work of doing this integral you can use the integral table to find out what it is but it's not actually that hard an integral to do so I'll show you the trick we need to actually perform this integral ourselves and that trick is use substitution so if we look back and think what I'd like to have is for one of these factors of sine theta d theta to be lumped into the du when I do my use substitution so if one of my factors is sine theta the reason I want to have one of these factors taken up by the du is if I have just sine squared left over I know a lot about how to do integrals of sine squared so what does you need to be in order for its differential to be sine theta it needs to be something like a cosine theta and now if I think about the signs if I make you equal to cosine theta its derivative is negative sine theta so if you use cosine theta du is minus sine theta d theta and that allows me to rewrite my integral I got n squared 2 pi when I transform the integral itself when I rewrite this using u's I need to remember also to transform the limits of the integral so my integral was theta running from 0 to pi but u is going to run from when theta is 0 u is equal to cosine of 0 which is 1 when theta is pi u is equal to cosine of pi which is negative 1 so my integral is running from 1 to negative 1 once I've converted into u and now my integral sine cubed I'd rather think of that as sine squared times sine sine squared let's do this one step at a time u squared is cosine squared so 1 minus u squared is sine squared so when I think of this sine cubed as sine squared times sine one of those sine squareds I can write as 1 minus u squared and the other sine I can write as a du with a negative sign so I'll stick a negative sign out in front of the entire thing so now I just have this integral to perform integral of 1 minus u squared and that's a much easier integral there's no trigonometric functions in there so I've got minus 2 pi n squared integral of 1 minus u squared is u minus integral of 1 is u integral of u squared is u cubed over 3 and I'm evaluating that between 1 and negative 1 inserting the top limits when u is equal to negative 1 u cubed is also equal to negative 1 so I have a minus negative 1 divided by 3 that's the two terms I get when I insert negative 1 for you and then I subtract what I get when I insert positive 1 so when u is equal to 1 u cubed is also equal to 1 so now I have a little bit of arithmetic left to do and I've run out of room so let's move up to the top so my normalization problem has become a 1 on the left side inside must be equal to negative 2 pi n squared times everything that's left in brackets here I've got a negative 1 minus 1 that looks like a negative 2 and a 1 third minus a negative 1 third so that looks like a plus 2 thirds negative 2 is negative 6 thirds if I add 2 thirds to the data all together I get negative 4 thirds so I've got 2 pi n squared times 4 thirds that negative sign cancels the negative sign on this negative 4 thirds and I need for 1 to be equal to 8 pi over 3 times n squared my goal is to solve for the value of n so if I leave n by itself move everything else over to the left hand side the three comes to the top the 2 times pi times 4 is on the bottom all this will be true my my wave function will be normalized its integral when I square it will be equal to 1 if the value of n squared is equal to 3 over 8 pi or if the value of n is equal to square root of 3 over 8 pi and now I don't have to worry about remember the reason we kept these vertical lines here on these magnitudes is because I didn't know if n was going to try out to be real or complex turns out I can get by with an n that's a real number square root of 3 over 8 pi and then when I square it I get 3 over 8 pi which is what I was looking for so we have solved for the value of n what that means is now we understand the wave function more completely this particular wave function the one we've been working with I can say that that wave function is not just equal to n some unknown constant but it's equal to square root of 3 over 8 pi times sine theta e to the i phi so that's what we've succeeded in doing is we've normalized that wave function if we someone gives us a wave function without the constant normalization constant specified we can go through this process to determine the normalization constant and be able to write down the fully correct wave function that process is not a lot of fun it's fairly tedious the main purpose of this video lecture has been to show you the in case you need to do it the procedure you need to go through how what a what exact integral you need to perform and set equal to one in order to solve for that value sometimes we don't end up needing the normalization constant so often we will just leave it unspecified as we did when we first wrote down these wave functions but if we do need to know the value then we can calculate it