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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.

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https://www.physicsforums.com/showthread.php?t=694158.

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The momentum operator is not hermitean either (->no real eigenvalues).

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