  The time evolution is studied for the Landau problem with a general time
dependent electric field ${\bf E}(t)$ in a plane perpendicular to the magnetic
field. A general and explicit factorization of the time evolution operator is
derived with each factor having a clear physical interpretation. The
factorization consists of a geometric operator (path-ordered magnetic
translation), a dynamical operator generated by the usual time-independent
Landau Hamiltonian, and a nonadiabatic operator that determines the transition
probabilities among the Landau levels. Since the path-ordered magnetic
translation and the nonadiabatic operators are, up to completely determined
numerical phase factors, just ordinary exponentials whose exponents are
explicitly expressible in terms of the canonical variables, all of the factors
in the factorization are explicitly constructed. The numerical phase factors
are quantum mechanical in nature and could be of significance in interference
experiments. The factorization is unique from the point of view of the quantum
adiabatic theorem and provides a demonstration of how the quantum adiabatic
theorem (incorporating the Berry phase phenomenon) is realized when infinitely
degenerate energy levels are involved. Since the factorization separates the
effect caused by the electric field into a geometric factor and a nonadiabatic
factor, it makes possible to calculate the nonadiabatic transition
probabilities near the adiabatic limit. A formula for matrix elements that
determines the mixing of the Landau levels for a general, non-adiabatic
evolution is also provided by the factorization.
