I have the following very basic question, and i'd really like your help! If we have a system, that is described by a Hamiltonian H, then we can expand the state of the system to the basis of H. And we say that, if we measure the observable H the state will collapse to one of H's eigenstates. Is it correct to expand the state ket to the basis of a different, random Hamiltonian H'? Remember that the system is being described by H and not H'! If this is possible, then when measuring the energy of the system, what are we going to measure? We are going to measure the eigenvalues of H, or the eigenvalues of H'? Thanks in advance!