In this talk, we consider computational aspects of different quantum dmechanical problems in form of a linear or nonlinear eigenvalue problem. After introducing the density-matrix formalism, whose elements can be identified with elements of the Grassmann manifold and that allows to elegantly deal with clusters of eigenvalues and arbitrary multiplicities, we will proceed with three applications of the formalism. First, we will discuss a posteriori error estimators for the density-matrix error for linear Schrödinger eigenvalue problems. In the second part, we consider ab-initio Born-Oppenheimer Molecular Dynamics (BOMD) at the level of Kohn-Sham Density Functional Theory (DFT) and exploit the geometric structure of the Grassmann manifold computationally by making use of the Grassmann exponential and logarithm to provide accurate initial guesses of the resolution of the nonlinear DFT-equations at each time-step. Finally, we show how the reduced basis method allows breaking the curse of dimensionality for the ground-state computation of an array of excited Rydberg atoms where the dimension of the Hilbert space grows exponentially with the number of particles.