  In this paper we work with the approximation of unitary groups of operators
of the form $e^{-itH}$ where $H\in\mathscr{L}(\mathcal{H})$ is the Hamiltonian
of a given quantum dynamical system modeled in the discretizable Hilbert space
$\mathcal{H}=\mathcal{H}(G)$, to perform such approximations we implement some
techniques from operator theory that we name particular projection methods by
compatibility with quantum theory conventions. Once particular representations
are defined we study the interelation between some of them properties with the
original operators that they mimic. In the end some estimates for numerical
implementation are presented to verify the theoretical discussion.
