Quantum Effect in the Mesoscopic RLC Circuits with aSource**The project supported by National Natural Science Foundation ofChina under Grant No. 10405009 and the Natural Science Foundation ofHebei Province of China under Grant No. 103143

Abstract

The research work on the quantumeffects in mesoscopic circuits has undergone a rapid developmentrecently, however the whole quantum theory of the mesoscopiccircuits should consider the discreteness of the electric charge.In this paper, based on the fundamental fact that the electriccharge takes discrete values, the finite-differenceSchrödinger equation of the mesoscopic RLC circuit with asource is achieved. With a unitary transformation, theSchrödinger equation becomes the standard Mathieuequation, then the energy spectrum and the wave functions of thesystem are obtained. Using the WKBJ method, the average of currentsand square of the current are calculated. The results show theexistence of the current fluctuation, which causes noise in thecircuits. This paper is an application of the whole quantummesoscopic circuits theory to the fundamental circuits, and theresults will shed light on the design of the miniation circuits,especially on the purpose of reducing quantum noise coherentcontrolling of the mesoscopic quantum states.