I have this problem. I want to find the wave function in the momentum space for the particle in a 1D box.
We know that the wave function in the position space is:
$$Y_n(x) = A\sin{(n\pi x/L)}$$
Well, if I write the Schrödinger equation in the moment space I have:
$$\frac{p^2}{2m}Y_n(p) = E_nY_n(p)$$
So, this equation doesn't give me any information about the wave function $Y_n(p)$
I know that I can solve this problem just using the Fourier tranformation, but I'm asking myself if there is another posibility to solve this problem.
PD. If I use the Fourier transformation, do I have to integrate just from $O$ to $L$?