We study the Schrödinger equation in systems with several degrees of freedom using symmetry, separation of variables, and other exact methods. We solve some of them with different techniques. We discuss quantum particles in central potentials, with emphasis on the potential \(\alpha /r\) ; we analyze it in arbitrary dimension n, not just in \(n=3\) (the hydrogen atom): only looking at the problem from this broader perspective one can grasp the deep web of its algebraic and geometric structures. We solve the system both with Representation Theory methods (the n-dimensional generalization of the Runge-Lenz vector) and analytically by separation of variables in either spherical and parabolic coordinates. We also study the quantum rotator and its generalizations using Peter-Weyl theory. The last three sections are dedicated to the motion of a charged particle in a magnetic field. We describe the Landau levels for an electron moving in an uniform magnetic field, as well as for the electron moving in the Poincaré plane in presence of a homogeneous magnetic field. The motion of a charged particle in the field of a magnetic monopole is also studied in detail.