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Quantization of the mesoscopic circuit with an external power source has been studied in this paper. The exact quantum invariant and wave function of the time-dependent mesoscopic LC circuit with an external power source, has been obtained via the field invariant method. The auxiliary functions in the quantum invariant operator and wave function corresponds to the external power source in the Hamiltonian of the mesoscopic circuit. Introduction As the miniaturization of integrated circuit size, quantum mechanical effects of circuit must be considered[1-2]. Therefore, it is urgently needed to establish a quantum theory of the mesoscopic circuit[2]. Up to now, many works have been done to solve the mesoscopic circuit whose Hamiltonians are explicitly time-dependent[3-7]. There are mainly two kinds of methods: path integral method and invariant operator method. In 1969, via the so-called invariant operator method, Lewis and Riesenfeld presented the exact wave function of the time-dependent harmonic oscillator by finding out the quantum invariant operator of the time-dependent Hamiltonian[3]. The quantization process of a L-C design mesoscopic circuit is similar to harmonic oscillator’s[4]. In 1992, Pedrosa et al. calculated the exact wave function and quantum fluctuations of the harmonic oscillator with time-dependent mass and frequency by quantum invariant operator method[5]. Later, they used the quantum invariant operator method to calculate the mesoscopic RLC circuit[6-7]. However, when an external power source applies to the mesoscopic circuit, the new quantum effect of the mesoscopic circuit will be produced[8-10]. Hereby, with the aid of quantum invariant operator method, we will study the time-dependent mesoscopic LC circuit with an external power source and derive its exact quantum invariant and wave function. Calculation of exact quantum invariant and wave function The Hamiltonian of the time-dependent LC mesoscopic circuit with an external power source can be written as[11] 2 2 2 1 ( ) ( ) ( ) ( ) 2 ( ) 2 p H t L t t q t q L t ω ξ = + − (1) Where ( ) t ξ is the electromotive force of external power source, p denotes the generalized current, q denotes the charge, L stands for the inductance and ω stands for frequency. According to the canonical transformation, H q p H p q ∂  =  ∂   ∂  = −  ∂

Exact Wave functions of Time-dependent Mesoscopic LC Circuit with an External Power Source