Solving for Wave Function in an Infinite Square Well

[silverthorne]

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

Homework Statement

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

$$\Psi(x,0) = C_{1} \Psi_{1}(x) + C_{2} \Psi_{2}(x)$$
$$\Psi(x,0) = \frac{1}{\sqrt{2}} \sqrt{\frac{2}{a}}cos\left \frac{\pi x}{a} \right + \frac{1}{\sqrt{2}} \sqrt{\frac{2}{a}}sin\left \frac{2 \pi x}{a} \right$$

(a) Obtain $$\Psi (x,t)$$
(b) Use this $$\Psi (x,t)$$ to calculate <H>, delta H, <x> and <p>.
(c) What can you say about the result you obtained from part (b). Explain.

Homework Equations

$$\psi(x,0)=\sum_{n=1}^{\infty}c_n\psi_n(x)$$

$$\psi_{n}(x)= \sqrt{\frac{2}{a}}sin\left \frac{n \pi x}{a} \right$$

$$E_{n}=\frac{n^2\pi^2\hbar^2}{2ma^2}$$

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

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 $$\Psi(x)$$ change ...

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

 

[v0id]

In this case, both cosine and sine forms are eigenfunctions of the energy operator. This is because we no longer have the boundary conditions $$\Psi(0,0) = \Psi(a,0) = 0$$.

I would think the complete form $$\Psi(x,t)$$ is easily obtained from the initial condition by simply appending the time-dependence factor to $$\Psi(x,0)$$ to yield $$\Psi(x,t) = \Psi(x,0) \exp{\left( \frac{iEt}{\hbar} \right)}$$. But then again, I'm one of your fellow students who also needs to get this done by Monday 12:00h.

Moreover, once you have $$\Psi(x,t)$$, the rest of the question is very easy. You know what the Hamiltonian operator simplifies to in this case (I'm assuming "delta H" refers to the variance $$\sigma$$ and not the standard deviation).

 

[silverthorne]

Ah...thanks! Yes, I forgot the fact that the reason why only the sines are the eigenfunctions is because of the initial conditions! So yeah, the cosine here works just fine. So I guess it's just $$\Psi(x,t) = \frac{1}{\sqrt{2}} \sqrt{\frac{2}{a}}cos\left \frac{\pi x}{a} \right \exp{\left( \frac{iE_{0} t}{\hbar} \right)}+ \frac{1}{\sqrt{2}} \sqrt{\frac{2}{a}}sin\left \frac{2 \pi x}{a} \right \exp{\left( \frac{iE_{1} t}{\hbar} \right)}$$, with $$E_{0}=\frac{\pi^2\hbar^2}{2ma^2}$$ and $$E_{1}=\frac{2^2\pi^2\hbar^2}{2ma^2}$$?

 

[v0id]

This is very strange. If we run with the assumption that the time-dependence factor can be tacked on so primitively, then the expected value of the Hamiltonian is a function of time.

I see no other way to derive the time-dependent form of $$\Psi(x,t)$$ from only $$\Psi(x,0)$$.

 

[silverthorne]

Um...well, I am stuck then if we can't just tack on the exponential factor. How should I go about in doing this problem?

 

[v0id]

Sorry, maybe it does work. I just tried solving for $$\langle H \rangle$$ in both Maple and Maxima, and the two CASs give different results. Maple gives a single value, but I think Maxima is unable to simplify the final expression to that value.

EDIT: My mistake. Maxima is able to simplify the expression perfectly. The input was wrong.

 

[silverthorne]

It's ok! So I guess the answer to part a is as above...

Anyway, I very much appreciate your help!