I am trying to figure out what will happen to my particle, if it is initially in the ground state of an infinite square well, and suddenly becomes free.
$$V(x) = 0, -\frac{a}{2} \leq x \leq \frac{a}{2}$$ and infinity elsewhere. Suddenly the potential is removed, and I want to see how my probability distributions in x and p will change.
I have worked out that $$\psi(x,0) = \sqrt{\frac{2}{a}} \sin(\frac{\pi x}{a} + \frac{\pi}{2})$$
Which is my initial wavefunction that will change over time once the potential is removed. I thought I could use Plancherel's theorem to find $\phi(p)$ and then use $$\psi(x,t) = \frac{1}{\sqrt{2 \pi \hbar}}\int^{\infty}_{-\infty} dp \phi(p)e^{ipx/\hbar}e^{-ip^2t/2\hbar m} $$
but neither $\psi(x,0)$, nor $\phi(p)$ are normalisable over free space, so how do I proceed?
I have calculated $\phi(p)$ inside the potential well, but I don't think that helps me.