  We consider the Schr\"odinger-Poisson system in the two-dimensional whole
space. A new formula of solutions to the Poisson equation is used. Although the
potential term solving the Poisson equation may grow at the spatial infinity,
we show the unique existence of a time-local solution for data in the Sobolev
spaces by an analysis of a quantum hydrodynamical system via a modified
Madelung transform. This method has been used to justify the WKB approximation
of solutions to several classes of nonlinear Schr\"odinger equation in the
semiclassical limit.
