# Hamiltonian for coupling between current and potential

Let's look at a general multi-particle Hamiltonian $H_0$ (e.g. many non-interacting particles). Now we add a background electric field. The new Hamiltonian is then $H=H_0+ \Delta H$ with $\Delta H=-J \cdot A$, where $J$ is the quantum operator associated to the electric current and $A$ is the magnetic vector potential (gauge: $A_t=0)$.

My question is: Where does this expression come from?

• Do your particles have charge? I assume, that yes. You need to look for the derivation of the QED hamiltonian, or the integration hamiltonian of the charged particle with EM field. – MsTais Apr 15 '17 at 20:14
• @MsTais: Yes, the particles are charged. Does this Hamiltonian in QED have a special name or is it related to some other topics that could help me find some good resources? – Quasar Apr 17 '17 at 12:31
• From what I know, "QED Hamiltonian" is the name. For instance, in Nuclear Theory the interaction term in Hamiltonian has exactly the structure $H_{int}=-\int d^3 r \langle \vec{j} \cdot \vec{A} \rangle$, where $\vec{j}$ is the nuclear current. I have never seen the derivation, but hopefully searching by these key-words could be helpful to you. – MsTais Apr 19 '17 at 14:44