Most importantly: can that potential energy in the QM Hamiltonian able to describe the motion of a single particle in an external electromagnetic field?
In non-relativistic QM using the Schrodinger Equation, fields (like from the electromagnetic field or other forces of nature) was not part of the "system" whose energy is described by the Hamiltonian operator--i.e., the wave function. But the field is certainly part of the Hamiltonian, since it contributes to the potential energy that appears in that Hamiltonian. But that potential energy is the potential energy of the particle whose wave function the equation gives the dynamics for; it's not the potential energy of the field."
Is this valid for all forces of nature. That is, they can all contribute to the potential energy that appears in the Hamiltonian. Can you mention a field that has no effect on it?