Momentum Eigenstate with a non-zero potential

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1. Feb 20, 2017

QFT25

1. The problem statement, all variables and given/known data
Prove that if a particle starts in a momentum eigenstate it will remain forever in a eigenstate given the potential c*y where c is a constant and y is a spatial variable.

2. Relevant equations

(h/i)d/dx is the momentum operator and a momentum eigenstate when put in the momentum operator gives an eigenvalue times the momentum eigenstate.

3. The attempt at a solution

If p commutes with H then a eigenstate of H is an eigenstate of p always. My problem is that p does not commute with H and I always thought that you can only have momentum eigenstates for systems with with zero potential. So I'm at a loss where to begin because if I workout the Heisenberg equations I get dp/dt where p is the operator in the Heisenberg picture to be none zero. Any help will be much appreciated.

2. Feb 20, 2017

QFT25

Can an momentum eigenstate have a eigenvalue which is time dependent.