Momentum space particle in a box

[tomothy]

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.

 

[Khashishi]

You know the solutions to the free particle are e^{ipx}, so look for linear combinations of those such that they match the boundary conditions. If you are talking about a perfect box, then look for linear combinations such that the edges go to 0.

 

[Ravi Mohan]

I tried to do the same thing for infinite square well, but it turns out that it cannot be done. Check out this thread
https://www.physicsforums.com/showthread.php?t=694158.

 

[persundqvist]

Fourier transform of the potential? It would be an oscillating and nasty thing. Also the boundary condition in the momentum space would be difficult to concider. Also remember that the momentum is not an eigenvalue for the bound problem (i.e., momentum and posistion are both uncertain).
The momentum operator is not hermitean either (->no real eigenvalues).