Separation of the center-of-mass in independent particle systems

It is shown that if an independent-particle Hamiltonian can be separated into a center-of-mass and internal part, it must be a second degree polynomial in the momentum and position operators. In particular the spin can enter only through an additive term which is independent of the orbital quantities. If the masses of the particles are the same then the single particle Hamiltonians must be the same; this rules out the possibility of treating protons and neutrons differently in a separable, independent particle Hamiltonian. Invariance requirements further restrict the form of the Hamiltonian.