Time evolution of two-dimensional quadratic Hamiltonians: A Lie algebraic approach

We develop a Lie algebraic approach to systematically calculate the evolution operator of the generalized two-dimensional quadratic Hamiltonian with time-dependent coefficients. Although the development of the Lie algebraic approach presented here is mainly motivated by the two-dimensional quadratic Hamiltonian, it may be applied to investigate the evolution operators of any Hamiltonian having a dynamical algebra with a large number of elements. We illustrate the method by finding the propagator and the Heisenberg picture position and momentum operators for a two-dimensional charge subject to uniform and constant electro-magnetic fields.