Energy and momentum balance equations: An approach to quantum transport in closed circuits

Using the Heisenberg equations of motion, we have derived a set of gauge-invariant quantum-mechanical energy and momentum balance equations governing the transport of charged particles in a closed electric circuit. Because the driving electric field needs not to be uniform along the circuit, the balance equations provide a suitable tool for the investigation of quantum transport in mesoscopic systems. As an example, we investigated the global and local properties of energy dissipation in a quantum wire circuit under the influence of a localized elastic scatterer, modeled by a Dirac-delta potential. In the low-temperature linear-response regime we obtain the Landauer formula for the resistance of the elastic-scattering barrier. The localized electric field originating from this barrier is obtained using a self-consistent solution of the Poisson equation.