Mesoscopic dual transmission line with discrete charge

In this work, we consider a mesoscopic dual transmission line. First we analyze the classic case and we found the Lagrangian and Hamiltonian for this system. Next, we analyze the quantum case and consider the charge discreteness. We found a Hamiltonian that in the limit when q_e → 0 reproduces the correct motion equations. This Hamiltonian differs from the other one in a previous work [1]. Using Caldirola-Kanai method we incorporate electrical resistance in the system and obtain the correct motion equations.