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When we normalize the free particle by putting it in a box with periodic boundary conditions, we avoid the "pathological" nature of the momentum representation that take place in the normal problem of a particle in a box with the usual boundary conditions of Ψ=0 at the two borders. Thus, the momentum operator is self-adjoint; something that does not hold for the normal problem of the infinite well.

My question is, what happens to the position representation in the case of the box normalization with periodic boundary conditions? Is the position operator self-adjoint and/or hermitian? How do we prove these things?

Thanks in advance!

Note: For information on some things I mentioned, you can go to https://www.physicsforums.com/threads/particle-in-a-box-in-momentum-basis.694158/page-2

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# I Box normalization for the free particle-Position operator

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