Time Evolution of Schrodinger's Equation

[doublemint]

Hello!

Here is my question:
Consider a particle of mass m, whose initial state has wavefunction \psi(x), in an infinite potential box of width a. Show that the evolution under the Schrodinger equation will restore the initial state (possibly with a phase factor) after time T=\frac{4ma^{2}}{\pi\hbar}.

I am not quite sure what to do.
So far i wrote down this:
\Psi(x,t) = \psi(x,t) e^{\frac{-iEt}{\hbar}} .. (1)
\frac{d\Psi}{dt} = \frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}\Psi(x,t) .. (2)
Now subbing (1) into (2),
\frac{-iE}{\hbar}\psi(x,t)e^{\frac{-iEt}{\hbar}}=-E\Psi(x,t) using Schrodinger's Time-independent Equation. Where V(x)=0 for a infinite potential well.
Finally: \Psi(x,t)=\frac{i}{\hbar}\psi(x,t)e^{\frac{-4iEma^{2}}{\pi\hbar}}

I do not think I did it correctly. However its something..
Any help would be appreciated!
Thanks
Shaun

 

[ideasrule]

Write out the stationary states explicitly, in terms of a and m. Do they come back to the initial state after <br /> \frac{4ma^{2}}{\pi\hbar}<br />? They're sine functions, so it should be easy to tell.

 

[doublemint]

I got it! Your ideas rule!