On interacting fermions and bosons with definite total momentum

Any {\it exact} eigenstate with a definite momentum of a many-body Hamiltonian can be written as an integral over a {\it symmetry-broken} function $\Phi$. For two particles, we solve the problem {\it exactly} for all energy levels and any inter-particle interaction. Especially for the ground-state, $\Phi$ is given by the simple Hartree-Fock/Hartree ansatz for fermions/bosons. Implications for several and many particles as well as a numerical example are provided.