We derive optimal order a posteriori error bounds for a fully discrete Crank-Nicolson finite element scheme for linear Schroedinger equations. The derivation of the estimators is based on the reconstruction technique; in particular, we introduce a novel elliptic reconstruction that leads to estimates which reflect the physical properties of the equation. Our analysis also includes rough potentials. Using the obtained a posteriori error estimators, we further develop and analyze an existing time-space adaptive algorithm, and we apply it to the one-dimensional Schrodinger equation in the semiclassical regime. The adaptive algorithm reduces the computational cost drastically and provides efficient error control for the solution and the observables of the problem, especially for small values of the Planck constant.