In this paper, we try to present a way in terms of which one can analytically obtain the Hartree selfconsistent potential instead of computing it by the numerical iterative procedure as usual, which is convenient for us to describe the current flow through a mesoscopic conductor. In our treatment, we expand the action function S(x)by Planck constant h, then the self-consistent potential and the wavefunction can be solved analytically order by order starting from the Poisson equation and quantum Hamilton-Jacobian equation, the differential conductance and quantum capacitance can thus be obtained naturally. In our paper, we show the quantum corrections up to the second order, and the electron-electron interaction is considered only at the Hartree approximation level.
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