Authors’ summary: The time-dependent linear Schrödinger equation for nuclei on the whole space is semidiscretized using Hermite and Gauss-Hermite basis functions. These are well suited, on the one hand, for the conservation properties of the numerical solution and, on the other hand, for their remarkable approximation properties. We investigate theoretically and numerically the convergence of the spectral and pseudo-spectral Gauss-Hermite semidiscretization schemes.