The goal of this paper is to obtain a high order full discretization of the initial value problem for the linear Schrödinger equation in a finite computational domain. For this we use a high order finite element discretization in space together with an adaptive implementation of local absorbing boundary conditions specifically obtained for linear finite elements, and a high order symplectic time integrator. The numerical results show that it is possible to obtain simultaneously a very good absorption at the boundary and a very small error in the interior of the computational domain.