Abstract
In the second lesson, we tackled the problem of writing the Hamiltonian in an arbitrary quantum circuit. For atoms, the quantum Hamiltonian is simply obtained by treating the conjugate variables of the classical problem - electron position and momentum - as non-switching operators. What about circuits ? The role of position and impulse is played by flux and charge. These canonically conjugate operators are defined from the time integrals of voltage and current in one branch of the circuit. The validity of the switching relations between charge and flux can be seen from the switching relations between electric and magnetic fields in quantum electrodynamics. Among the surprising notions of quantum circuits is that of generalized flux across any element, not necessarily an inductor. This allows us to quantize the electromagnetic modes of a transmission line, treating it as a chain of coupled LC oscillators.
