  The theory of Bloch electron dynamics for carriers in homogeneous electric
and magnetic fields of arbitrary time dependence is developed in the framework
of the Liouville equation. The Wigner distribution function (WDF) is determined
from the single particle density matrix in the ballistic regime, i.e.,
collision effects are excluded. The single particle transport equation is
established with the electric field described in the vector potential gauge,
and the magnetic field is treated in the symmetric gauge.
  The general approach is to employ the accelerated Bloch state representation
(ABR) as a basis so that the dependence upon the electric field, including
multiband Zener tunneling, is treated exactly. In the formulation of the WDF,
we transform to a new set of variables so that the final WDF is gauge invariant
and is expressed explicitly in terms of the position, kinetic momentum, and
time.
  The methodology for developing the WDF is illustrated by deriving the exact
WDF equation for free electrons in homogeneous electric and magnetic fields.
The methodology is then extended to the case of electrons described by an
effective Hamiltonian corresponding to an arbitrary energy band function. In
treating the problem of Bloch electrons in a periodic potential, the
methodology for deriving the WDF reveals a multiband character due to the
inherent nature of the Bloch states. In examining the single-band WDF, it is
found that the collisionless WDF equation matches the equivalent Boltzmann
transport equation to first order in the magnetic field. These results are
necessarily extended to second order in the magnetic field by employing a
unitary transformation that diagonalizes the Hamiltonian using the ABR to
second order. The work includes a discussion of the multiband WDF transport
analysis and the identification of the combined Zener-magnetic field induced
tunneling.
