We discuss the adaptive numerical solution of the Schrödinger-Poisson equation on a truncated finite domain with an underlying space discretization by conforming piecewise polynomial finite elements, where we truncate to a sufficiently large finite domain and impose homogeneous Dirichlet boundary conditions. The motivation for this approach is the possibility to treat the Poisson equation separately by dedicated solvers for the arising linear equations. The classical convergence orders in both the time and space discretization are established theoretically under natural assumptions on the regularity of the exact solution and illustrated by numerical experiments. Adaptive time-stepping relying on a defect-based error estimator is shown to correctly reflect the solution behaviour.