Geometrical approach to two-level Hamiltonians

Two-level systems were shown to be fully described by a single function, known sometimes as the Stueckelberg parameter. Using concepts from differential geometry, we give geometrical meaning to the Stueckelberg parameter and to other related quantities. As a result, a generalization of the Stueckelberg parameter is introduced, and a relation obtained between two-level systems and spatial one-dimensional curves in three-dimensional space. Previous authors used this Stueckelberg parameter to solve analytically several two-level models. We further develop this idea, and solve analytically three fundamental models, from which many other known models emerge as special cases. We present the detailed analysis of these models.