# Momentum space particle in a box

• tomothy
In summary, the conversation discusses the problem of finding a solution to the particle in a box energy eigenvalue problem without solving a differential equation. The proposed approach involves using eigenvectors of p^2 within the box. However, there is difficulty in adjusting the potential for momentum representation. The suggestion is to use linear combinations of e^{ipx} to match the boundary conditions. It is noted that this approach does not work for the infinite square well. The possibility of using Fourier transform of the potential is mentioned, but it is complicated due to the uncertain boundary conditions in momentum space. It is also mentioned that the momentum operator is not hermitean and does not have real eigenvalues for the bound problem.

#### 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.

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.

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).