Recommend this question. The wave function for a particle in an in finite square? Well, right. And this is that it between X equals 20 to excuse to l and is your outside off this ridge, I'm gonna find the expectation value. So expectation value to find expectation very off a we function or you have to do we need to multiply its contra gate with the favorable that you want to find an expectation value in this case is x by X and not a body were found Should get any degree across Oh, space float negative fainting to positive in Haiti No, no case, because basically, we function only it's fairly from zero to l and it's, uh, empty out of ice I'll say so we can just leave me the range just so to hell. Let me carry of entities gets just you re function square Papa X, the X I mean, so it had to be a great dish. Um, what you want to do is to first called a vote o a saint square into ST, despite he double anger formula. So science way into just consigned to eggs and or you have to do is to drew the change through oh, into go integration by parts for this to this term can be very simply integrated. This no problem. But for distance, we have to do integration by parts. All right, so this by change rule, keep the first. Um Right. So over here you have two terms. Thanks. Every article saying so we keep you first toe in degree the second term, you might as a way, uh, we differentiate your first of and we need a greedy second toe. So this the integration by parts in, uh can I continue to simplify? So the first time? Just, uh, l square or two. And then for the second half Well, sorry for pain and face it. This is an injured. Your multiple pint. This would be cereal. The center will be zero. So you know that Have you got left? The two right, which is doing it to go and by integrating gets should get a negative sign again. Right? So this last home always to have to substitute in X equals two l and x coast with zero. So an execution bell to be course I for pie. Mary, Don't co sign pie. It's true. It's, uh, indigent. Multiple off course science to buy bank assigned to pi is just one. So this is just for by square this one minus away maybe substituting experts with zero because science hero and we know once again call signs You always one So this time goes to zero and we're left with over two. So this is the expectation value or for position X not to find it probably t that the particle lie between certain range you can do the same integration but result the x variable of this time. So you find probably it he that it exists a relative to a 0.5 I'm mutati range from 0.49 No. So you still fail 2.51? No. Every integrity, probability, density It's just site square. So a very similar integration. Yes. OK, step Yeah, necks So first ill minus away Hell for pie times slain for extra veil. And but this summer, integrating between 0.4 night no to 0.51 help. So first, don't you give us your point. T 02 right? And substituting in we should it signed to play for pie. Bring us a sign 1.6 by the circuit for a spot 5.26 instead. Poll minus five. Now we do the simple assess for the probability at around two point 25 l. It's just about around over four The great Between 2.24 and 0.26 l times again he integrate. You've already testy the and already same expression with you. Except we just started you in different values for our limits, right? We should get, actually, is your point night night instead of I s too Well, at first glance, it wouldn't make sense compared with Oh, but a answer upon a sea states he expected fair do expectation. Very youthful eggs is at over two. But we see that the probability for the particle to exist it's oh, over four. It's actually much greater, Dan probably t two axis around over to How I see is the case. That is because our probability density function in sexually symmetric seeing a trick a boat over to so it actually X is kind of like that tour picks it over four in trio over four, right? So the center off this symmetric probability density function even though it is zero right? He probably to you. But your sexual zero there is the expectation value, right? On average, because off the symmetry, this will give us our to. So there's no contradiction, right? Uh, and the expectation that is does not tell you d highest probability density, but rather it tells you what you would get on average, you forgot to take many, many experimental values from the experiments because you get equal pay for equal probability of getting helpful en trio full, on average, 90 weighted average would give us well.