  We discuss the one-dimensional, general quadratic Hamiltonian and the
bi-dimensional charged particle in time-dependent electromagnetic fields
through the Lie algebraic approach. Such method consists in finding a set of
generators that form a closed Lie algebra in terms of which it is possible to
express the Hamiltonian and the therefore the evolution operator. The evolution
operator is then the starting point to obtain the propagator as well as the
explicit form of the Heisenberg picture position and momentum operators. First,
the set of generators forming a closed Lie algebra is identified for the
general quadratic Hamiltonian. This algebra is later extended to study the the
Hamiltonian of a charged particle in electromagnetic fields, given the
similarities between the terms of these two Hamiltonians.
