  The relativistic Foldy-Wouthuysen transformation is used for an advanced
description of planar graphene electrons in external fields and free
(2+1)-space. It is shown that the initial Dirac equation should by based on the
usual $(4\times4)$ Dirac matrices but not on the reduction of matrix dimensions
and the use of $(2\times2)$ Pauli matrices. The latter approach does not agree
with the experiment. The spin of graphene electrons is not the one-value spin
and takes the values $\pm1/2$. The exact Foldy-Wouthuysen Hamiltonian of a
graphene electron in uniform and nonuniform magnetic fields is derived. The
exact energy spectrum agreeing with the experiment and exact Foldy-Wouthuysen
wave eigenfunctions are obtained. These eigenfunctions describe multiwave
(structured) states in (2+1)-space. It is proven that the Hermite-Gauss beams
exist even in the free space. In the multiwave Hermite-Gauss states, graphene
electrons acquire nonzero effective masses dependent on a quantum number and
move with group velocities which are less than the Fermi velocity. Graphene
electrons in a static electric field also can exist in the multiwave
Hermite-Gauss states defining non-spreading coherent beams. These beams can be
accelerated and decelerated.
