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The unitary operators U(t), describing the quantum time evolution of systemswith a time-dependent Hamiltonian, can be constructed in an explicit mannerusing the method of time-dependent invariants. We clarify the role ofLie-algebraic techniques in this context and elaborate the theory for SU(2) andSU(1,1). We show that the constructions known as Magnus expansion andWei-Norman expansion correspond with different representations of the rotationgroup. A simpler construction is obtained when representing rotations in termsof Euler angles. The many applications are postponed to Part II of the paper.

Solving the von Neumann equation with time-dependent Hamiltonian. Part I: Method