wigglywinks
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Homework Statement
A particle of mass m is confined to a space 0<x<a in one dimension by infinitely high walls at x=0 and x=a. At t=0, the particle is initially in the left half of the well with a wavefunction given by,
$$\Psi(x,0)=\sqrt{\dfrac{2}{a}}$$
for 0<x<a/2
and,
$$\Psi(x,0)=0$$
for a/2 < x < a
Find the particle's time dependent wavefunction $$\Psi(x,t)$$
Homework Equations
I think the following equations are relevant (let me know if I don't have them written down correctly),
$$\Psi(x,t)=\sum_n^\infty c_n\psi_n(x) e^{-i E_n t/\hbar}$$
where $$\psi_n(x)=\sqrt{\dfrac{2}{a}}\sin\left(\dfrac{n\pi x}{a}\right)$$
and
$$E_n=\dfrac{n^2\hbar^2 \pi^2}{2ma}$$
and
$$c_n=\int_0^{a/2} \psi_n(x)\Psi(x,0)~dx$$
The Attempt at a Solution
I tried to find the time dependent wavefunction for the left half of the well. I'm not exactly sure what to do about the right half, but since the wavefunction at x=0 is equal to zero on the right half, the probabilities c_n would also be zero, so I'm thinking that the wavefunction is probably zero for the right half (so the full time dependent wavefunction for both halves would be piecewise continuous).
So for the left half of the well, I first tried to find c_n,
$$c_n=\int_0^{a/2}\sqrt{\dfrac{2}{a}}\sin\left(\dfrac{n\pi x}{a}\right)\sqrt{\dfrac{2}{a}}~dx$$
$$c_n=\frac{2}{a}\int_0^{a/2}\sin\left(\dfrac{n\pi x}{a}\right)~dx$$
so I get,
$$c_n=\dfrac{4\sin^2\left(\frac{n\pi}{4}\right)}{n\pi}$$
Then I plug this into
$$\Psi(x,t)=\sum_n^\infty c_n\psi_n(x) e^{-i E_n t/\hbar}$$
and also plugging in the equation for ψ_n(x), to get,
$$\Psi(x,t)=\sum_n^\infty \dfrac{4\sin^2\left(\frac{n\pi}{4}\right)}{n\pi}\sqrt{\dfrac{2}{a}}\sin\left(\dfrac{n\pi x}{a}\right) e^{-i E_n t/\hbar}$$
I tried to compute this on mathematica, but it didn't work (it looks like it probably diverges, but I don't know).
Does anyone know what I did wrong, or if this approach is even correct to begin with?