Given the way functions, including the mixed eight wave function, we're going to start with the ground state, and we're gonna take the expectation value for the position operator and proceeding. We get the following integral, and we see that this is going to be an integral over and even or symmetric interval, meaning that it's symmetric about the vertical axis, the Y axis. In other words, it's an even interval or symmetric interval about the origin from minus infinity to infinity. But this is an integral of an odd function. So from that observation, we can already conclude that this integral should be zero. You could also look this up in an integral table or real quick. You can. You can actually do the integral Um, by making a quick change of variables a Z I'll show you. So there's a real quick way to do that. You can write. This is to in a girls so an integral from minus infinity to zero and an integral from zero to infinity, and you'll perform a change of integral, a change of variables on just the first integral. So we basically changed X two minus X. That's the change of variables. And when we do that, we get an integral from infinity to zero. We leave the second integral alone, and this is the former that we now have. So in doing that we come to this next form. We flipped the integral on the first integral itself and we have a minus sign. Now notice that the minus Sign on Negative X right here has cancelled with the minus sign in in de negative X and the minus Sign in negative X up here has gone away because it's negative X squared. So that leaves us with X dx in the next line. And we just have e to the negative a X squared. So we just have a minus sign right here simply because we flipped this integral right here. So now this integral will cancel with the integral that we left alone on the rights and we therefore have zero. So the expectation value for X in the ground state of zero. Next we continue and we calculate the expectation value for X and the first excited state. But again, we have the integral of an odd function over a symmetric interval, and we get a similar result, we should get zero, but we go ahead and just write out the same trick, the same procedure right here, and it works the same way, except that it's X cubed. Had this been an even function then, because you're squaring or cute or taking to the fourth power, etcetera. This method would not work, and you would not have any girls that cancel out. But if you have 135 etcetera for odd functions, they do cancel. And so you see once again that these entered rules will vanish because they add together, the minus sign causes them to cancel. So now we finally come to the expectation value for X in the mix state, and we get three terms when you when you substitute in the mix state in the expectation value formula, you get, ah, an expansion and their three terms that result in that expansion and one of them is across term, which gives us this to in the cross term, and the other two are square terms. The square terms will look familiar. We just calculated them and notice that we have the two there in the cross term because you had really you had four terms since it's ah wasn't expansion. So they should add together to give you the two. We just calculated the square terms. They vanished. So this leaves us with just what's left over two times the cross terms. But the two with 1/2 are going to cancel. And so now we just have that our expectation value equals the integral of X times, the to wave functions. And the only reason I have absolute value signs is in case we were dealing with complex valued functions. And I like to be general and just include what goes into a formula in its most general form. But we don't really need them because we're dealing with real valued functions. So when you substitute everything in for these particular wave functions, this is what you obtain and it looks like this. Now notice we finally haven't even function over ah symmetric interval, and you could evaluate this using particular methods. You could research galaxy and in a girls, for example, and see how they're evaluated. You can just look in a table of any girls, which is what a lot of people dio and because we don't have a lot of time. We're just going to state what this integral is equal to. So when you look it up in a table, for example, or if you do like to just see how it's done, you could look it up. You can use polar coordinates to evaluate this type of integral Um, this is what you get. And so there's some cancellation that will go on when you when you have all these new miracle or basically all these factors multiplying each other and you finally get one divided by the square root.