Application of lie systems to quantum mechanics: Superposition rules

Abstract

We prove that t-dependent Schrödinger equations on finite-dimensional Hilbert spaces determined by t-dependent Hermitian Hamiltonian operators can be described through Lie systems admitting a Vessiot-Guldberg Lie algebra of Kähler vector fields. This result is extended to other related Schrödinger equations, e.g. projective ones, and their properties are studied through Poisson, presymplectic, and Kähler structures. This leads to deriving nonlinear superposition rules for them depending on a lower (or equal) number of solutions than standard linear ones. As an application, we study n-qubit systems and special attention is paid to the one-qubit case.