Thermal State for the Capacitance Coupled Mesoscopic Circuit with a Power Source

The Schrödinger equation of the mesoscopic capacitance coupled circuit with an arbitrary power source is solved by means of two step unitary transformation. The original Hamiltonian transformed to a very simple form by unitary operators so that it can be easily treated. We derived the exact full wave functions in Fock state. By making use of these wave functions and introducing the Lewis--Riesenfeld invariant operator, the thermal state have been constructed. The fluctuations of charges and currents are evaluated in thermal state. For T→ 0, the uncertainty products between charges and currents in thermal state recovers exactly to that of Fock state with n, m=0.