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Hi guys, this assignment is driving me nuts! Thank you very much for the help!!

Consider the infinite square well described by V=0, -a/2<x<a/x, and V=infinity otherwise. At t=0, the system is given by the equation

[tex]\Psi(x,0) = C_{1} \Psi_{1}(x) + C_{2} \Psi_{2}(x)[/tex]

[tex]\Psi(x,0) = \frac{1}{\sqrt{2}} \sqrt{\frac{2}{a}}cos\left \frac{\pi x}{a} \right + \frac{1}{\sqrt{2}} \sqrt{\frac{2}{a}}sin\left \frac{2 \pi x}{a} \right[/tex]

(a) Obtain [tex]\Psi (x,t)[/tex]

(b) Use this [tex]\Psi (x,t)[/tex] to calculate <H>, delta H, <x> and <p>.

(c) What can you say about the result you obtained from part (b). Explain.

[tex]\psi(x,0)=\sum_{n=1}^{\infty}c_n\psi_n(x)[/tex]

[tex] \psi_{n}(x)= \sqrt{\frac{2}{a}}sin\left \frac{n \pi x}{a} \right[/tex]

[tex]E_{n}=\frac{n^2\pi^2\hbar^2}{2ma^2}[/tex]

[tex]c_{n}=\int_{0}^{a} \sqrt{\frac{2}{a}}sin\left \frac{n \pi x}{a} \right \psi(x,0)dx[/tex]

Um...this problem is kind of similar to the infinite well problem posted below earlier...I want to know if the formulae up there are the right one to use first...before I blindly apply it and do the integrals...

the second term in the wave function looks like an eigenfunction for the energy...but the first one is a cosine so I am not sure what to do there...do I need to...split them up?

The equations above are for the infinite well from o to a...but this question is from -a/2 to a/2...so I am not sure if the eigenfunctions [tex]\Psi(x)[/tex] change ...

I also know [tex]\Psi (x,t)[/tex]can be obtained from multiplying [tex]\Psi (x,0)[/tex]by the appropriate phase factor once the [tex]\Psi_{n} (x,0)[/tex] is written as a linear combination of the energy eigenfunctions...but then there's the cosine in the first term...

**1. The problem statement, all variables and given/known data**Consider the infinite square well described by V=0, -a/2<x<a/x, and V=infinity otherwise. At t=0, the system is given by the equation

[tex]\Psi(x,0) = C_{1} \Psi_{1}(x) + C_{2} \Psi_{2}(x)[/tex]

[tex]\Psi(x,0) = \frac{1}{\sqrt{2}} \sqrt{\frac{2}{a}}cos\left \frac{\pi x}{a} \right + \frac{1}{\sqrt{2}} \sqrt{\frac{2}{a}}sin\left \frac{2 \pi x}{a} \right[/tex]

(a) Obtain [tex]\Psi (x,t)[/tex]

(b) Use this [tex]\Psi (x,t)[/tex] to calculate <H>, delta H, <x> and <p>.

(c) What can you say about the result you obtained from part (b). Explain.

**2. Relevant equations**[tex]\psi(x,0)=\sum_{n=1}^{\infty}c_n\psi_n(x)[/tex]

[tex] \psi_{n}(x)= \sqrt{\frac{2}{a}}sin\left \frac{n \pi x}{a} \right[/tex]

[tex]E_{n}=\frac{n^2\pi^2\hbar^2}{2ma^2}[/tex]

[tex]c_{n}=\int_{0}^{a} \sqrt{\frac{2}{a}}sin\left \frac{n \pi x}{a} \right \psi(x,0)dx[/tex]

**3. The attempt at a solution**Um...this problem is kind of similar to the infinite well problem posted below earlier...I want to know if the formulae up there are the right one to use first...before I blindly apply it and do the integrals...

the second term in the wave function looks like an eigenfunction for the energy...but the first one is a cosine so I am not sure what to do there...do I need to...split them up?

The equations above are for the infinite well from o to a...but this question is from -a/2 to a/2...so I am not sure if the eigenfunctions [tex]\Psi(x)[/tex] change ...

I also know [tex]\Psi (x,t)[/tex]can be obtained from multiplying [tex]\Psi (x,0)[/tex]by the appropriate phase factor once the [tex]\Psi_{n} (x,0)[/tex] is written as a linear combination of the energy eigenfunctions...but then there's the cosine in the first term...

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