  For the Landau problem with a rotating magnetic field and a potential in the
(changing) direction of the field, we derive a general factorization of the
time evolution operator that includes the adiabatic factorization as a special
case. We assume that the direction of the magnetic field changes with time in a
general way, so the Heisenberg equations of motion cannot be solved by
quadrature. Also, the potential is assumed to be of a general form. We use the
rotation operator associated with the solid angle Berry phase to transform the
problem to a rotating reference frame that follows the direction of the
magnetic field. In the rotating reference frame, we derive a natural
factorization of the time evolution operator by recognizing the crucial role
played by a gauge transformation. The major complexity of the problem arises
from the coupling between motion in the direction of the magnetic field and
motion perpendicular to the field. In the factorization, this complexity is
consolidated into a single operator that approaches the identity operator when
the potential confines the particle sufficiently close to a plane perpendicular
to the magnetic field. The structure of this operator is clarified by deriving
an expression for its generating Hamiltonian. The magnetic translation is the
most notable physical consequence in the adiabatic limit. This and the
non-adiabatic effects are studied as consequences of the general factorization.
