# Momentum space particle in a box

I am trying to formulate a solution to the particle in a box energy eigenvalue problem, without solving a differential equation, instead using eigenvectors of $p^2$. My idea is to do this. Within the box (let's say it is defined between $[-a,a]$ and within this region the hamiltonian is $H={p^2}/{2m}$ so the solution is $|\psi\rangle=c_+|p\rangle + c_- |-p\rangle$. This approach is really the free particle, but I cannot work out how to adjust the potential for the momentum representation. Any help would be appreciated.