In this question. We have a pocket in the infinite square brown. Uh, with this initial wave function, Sigh X zero is good to a times Taiwan X plus side to act. Okay, so, um, for Infinite Square. Wow. Yeah, And that's where our problem. Okay, so the diagram looks like this. He from zero to a, uh, and the potential is infinity. Okay. And then so the re function Sai wan x is 12 over a sorry pi X or a and then side two X. Yes, Uh, ritual a sorry two pi X or a Okay, we have the one because two hi square HPR square over two and a square. And then you have me to he goes to, uh, for high square H Bar square by to m a square. Okay, so, um, first, there are five parts in the question. Okay? So in part, they want to find a normalization constant. Okay, so we'll use the normalization condition. Yeah, which states that, uh, the integral from negative infinity to infinity model. Big Sigh, x zero square, E X. A secret one. Okay, so So what you are going to have is, uh, a square well, Taiwan side too. Star times. I want side too. Okay. And the X And then this is a good one. And then you're going to have what's so one square plus one say to a square. Hi. Um, Taiwan star, like two plus trying to start Taiwan. Okay. And the X goes to one. Since all the functions are real. So, um, and then we have our uh huh energy. I can function. Normalize. So this is because to one, and we have, uh, one side too Square yanks. It's also it was one and then someone inside to, ah, diagonal to each other. So we had Sai Wan side, too. Yeah. Thanks. Is it good to zero? Okay. And similarly inside to Taiwan. Okay, so the above, uh, the line above, we'll get two times more a square. Is he going to one? Which means that on a is equal to 1/2. Okay, so this is our normalization Constant. Probably came to be, um, we want to find, uh uh huh. The re function in time. And also the probability density at Time T. Okay. So, um uh huh. So we are given this. It's the would you show a function? Okay. And we want to find, uh, we function at time T. Okay. What we need to do is just to attach the face sector into the negative i e one t or h five and then to each, uh, come Okay, foresight to will just be negative. I e to t over h by. Okay, then we define omega s, uh, e one over h bar. So which is hi square each bar over to M E Square so we can simplify the txt to become Taiwan ex into the minus. I Omega Key has. And then e two is four times of e one, so the Omega would be four times minus I for Omega T. Yeah. So this is, uh, re function at time key. Okay, then if you want to find a probability density, okay, which is, uh, Mark's Square. Mm. So, uh huh. Yeah. So just record. Just remember, that's I want X is, uh, to over a, um sorry. Hi. X over a inside two. X is to a Sorry, um, two packs of a Okay. Yeah. So I'm not putting the exact conscience here yet, but if you want to substitute. Yeah, you can just substitute. Okay, So you do the more square, then you have half. What? Uh, so yeah, so just okay. Complete. Okay, so the more square will be Sorry, star. Sorry. Right? Yes. Uh, 1/2 will become half mhm. And then you have a Sidewinder X. Yeah, because space the spatial wave function is real. So I'm not going to put a star. Yeah, but, uh, time based factor from minus I becomes I he and then decide to x into the eye for Megawati. Okay, so this is the side star, and then the sigh Uh uh, we got the star will be you can just copy. Okay. And then, um then you just multiply So you have Sai wan square. Okay, so the this term multiplied this times the exponential term cancer out. And then, um, you also have the site to square as well. If these two times world, there's no time dependence. And then you have, um, Taiwan Side two. Me to the minus. I trio Megawati and then you also have the side one side, two e to the I tree omega t mhm. He didn't continue to algebra. If so, here you can pull out the Sai Wan inside, too. You have each of the I three omega T plus into the minus. I tree omega T. Okay, this can be, uh, we return there. So to Hussein, using the oil a formula Jose tree. Omega T. Yeah. And you want to substitute ah, functions inside here. Um, so you have house, um, times. So there's a two over a mhm I'm saying. And then, uh, science Square. Hi. Excellent. A plus science square to my ex or a class, um, to sign. Hi. Absolutely sorry. Two packs over a Jose tree. Omega T. Okay. And then you can simplify further. Okay, so this is, uh, the function. The probability density? Yeah. After we function at time T. Okay. Okay. So that's your baby. Then we go to Passy. Passy, you want to find the expectation value of position? Okay, So expectation value opposition is according to the definition. Spicer X Sorry. Um, the X And since X doesn't operate, X can be moved everywhere. So you actually get this x Times months? I sexy star square. What square? So, uh, then you just need to copy what you have in the previous question. Okay to here. Okay, Go from zero to a, um X Times one over a a square high X over a plus. Thanks. Where two packs of a thus to sign. Hi. X over a sorry two pi x over a cause. I try Omega T and then the X. Okay, So here there are three integral that we need to do. Okay. I'm just going to do them one by one. It makes it easier. Okay. So, uh, zero to a x Zion square. Hi X over a, uh, the X Mhm. This gives us a square over to Okay. This is also the same call. Um, x Times Square to buy X over a the x Mhm. Yeah. And then the integral Jeez. Zero to a X. Yeah, to hear, um, sign Hi. X over a sorry two packs over a Hussein trio Omega t the X, Then if you use so that the cosine omega T doesn't participate in the integration is a constant in this case. And then So if you use a wolfram Okay. All right. And you would get, um we are going to get negative It's a square or like high square. There's a two in friends. And that's because I try Omega T. Okay, So here use. Uh huh. Yeah. Okay. You sure? The answer. Okay. See you sometime. Mhm. And so, uh, then we just put everything together. Okay, So the expectation value of X, we have won over a in front, and then we have a square over to us. Uh, the square over to uh huh. Mm. You know, I think this one is a square before. Yeah, okay. And then plus, uh, minus. Okay. Minus 16. A square over nine Pi square. Of course. I trio McGaughey. Okay, then you can cancel the A. So you have me over to minus 16. 8/9. Pi square for sign. She Omega T, and this is equal to able to one minus 32 a over nine point square. Course. I tree Omega T. Mm. Right. So this is the expectation value of the position. Then we need to determine the empty too. Okay. The M P two is, uh, 32 a or nine pi square A. That should be It is not here. Sorry. Okay. After you find out a So there is not there. 8 30 to over 95 square times a day or two. You can use your calculator. You get 0.360 times are over to Yeah, and then the frequency. Hey, the frequency. Is he good to tree omega? Okay, which is three pi square, each bar the right by two m a square. Okay. Next, we want to find, uh, expectation, value of the momentum, the expectation, value or momentum. By definition, it is, um, size scar, and then the momentum The coordinate representation of momentum is, uh, Bob, uh, I I show up at your ex. Uh huh. And this is a star sign. XT Yes. Okay, So this thing we need to do it, uh, carefully and step by step. Okay. So, uh, yeah, we need to do the differentiation first. So So let's just write down what our sexy is. Society is one or a sorry. Hi x or a me to the minus. I only got t Uh huh. Sorry. Two Pi x or a you to the minus. I call my body Mhm. Passion. Passion X Sorry. XT. You go to one over a um hi. Over a cause I Hi X over a. You do the minus. I only got 80 plus to buy over a co sign two pi x over a e to the minus I for Omega T. Yeah. And then, um you can simplify by pulling out. Uh, pyro a. Okay, I'm going to ignore the H Power I for the moment. It just put it back three. No. Okay. We are just going to focus on multiplying functions. Okay, so do this, Um, before that, Yeah, it's just right down the complex contributors. Wow. Well, okay, so this is the complex congregate, and then we are going to multiply. That's I start with the G d x of the sign. So, uh, we have pie over a square. What? Sign packs over a call. Sign packs over a. Okay, so the we first write down the terms that exponential terms cancer out. Oh, okay. And then the next times Yeah, I'm not. Wow. Yeah. Okay. Okay. And then we are going to integrate this thing. Okay, Then you can use, uh, from again. You realize that, uh, the science Jose with the same arguments. They just cancer out. They just give you zero, okay? They don't cancel, but they give you zero okay? Yeah. Yeah, because they are Darwin. Okay. Yeah. So basically, this is equal to zero after integration. Okay? After integration. Okay, not in this, uh, when you multiply the two functions, but when you integrate and then using them again, Okay, so if you use your friend or the other two terms too. Sorry. Hi, Axl. A sorry. Two packs of a the X, and this gives you minus for a trip. I and then you do the other 10 to a I'm sorry. Two pi x over a was saying I actually a Yes. This gives you, um for a or tree pie T. So, um, so when it comes to the integral Okay, the integral of Mm hmm. So I x c the x Uh huh. I I star 60 d d x I x T. Yes. Can you have mhm um, hi. Over a square. And then you have, um, negative for a trip. I into the minus. I tree omega t plus for a or tree pine. Eat the eye tree. Omega T. Okay. Okay. So the pie cancels the A also cancer. Okay, so you have 43 A e tree omega T minus E minus. I treat omega t mhm so far over three a, uh, times, too. I sign tree omega heat. Mhm. And so the expectation value of P Yes. You go to thus far over. I did you grow off. Sorry, star XT the V X I XT the x. Okay, so this is equal to H bar Over I for three. A two. I sign tree omega T. Okay, so the eye cancels and you get meet each power P a Sorry. Three omega t no. Okay, so this is the expectation value of the momentum. Okay, so just want to highlight you can do from how far Here. Okay. Yeah. Okay. In part five, you're in part E. Um, You want to find, uh, probability of getting each energy? Okay, So probability of e equals two e one. His house? Yeah, and the probability of getting equals e to It's also hot. Okay. It's basically because we wrote our sy x zero s in the form of, um see, I, uh yes, I I right. And so the probability of getting any egos to ei, it's just mon see, I square. Okay. And each c I, in this case is one of the two. So when you square it, you get half okay. Mhm. Then you need to find the expectation value of the Hamiltonian. Hey, expectation value of the Hamiltonian is is equal to the probability time stuff, uh, individual energy and sum them up. So it's half e one plus half too. Yeah, so half b one C two. And you know that E two is for everyone. So we have five over to E one. Yes, Yes, 5/2. Hi. Square H bar square over to M E Square. Okay. Yeah. So, um, the expectation value of the Hamiltonian is, uh, in between you on and you too. Okay, So this is the expectation value of Hamiltonian, and then this is how it compares to, uh, you want to need to, and that's all for this question.