How to get the energy eigenvalue of the Hamiltonian: H0+λp/m ?

[Jiangwei Du]

TL;DR Summary

We have already know the energy eigenvalue E0 of initial Hamiltonian H0. So when we add the extra item-λp/m, how the energy eigenvalue will vary?

Someone says we can choose the new eigenstate: exp(-iλx/hbar)*ψ,and let the momentum operator p acts upon this new state. At the same time, so does p^2. Something miraculous will happen afterwards. My question is: how to image this point? Thank you very much.