  Available laser technology is opening the possibility of testing QED
experimentally in the so-called strong-field regime. This calls for developing
theoretical tools to investigate strong-field QED processes in electromagnetic
fields of complex spacetime structure. Here, we propose a scheme to compute
electron wave functions in tightly focused laser beams by taking into account
exactly the complex spacetime structure of the fields. The scheme is solely
based on the validity of the Wentzel-Kramers-Brillouin (WKB) approximation and
the resulting wave functions, unlike previously proposed ones [Phys. Rev. Lett.
\textbf{113}, 040402 (2014)], do not rely on approximations on the classical
electron trajectory. Moreover, a consistent procedure is indicated to take into
account higher-order quantum effects within the WKB approach depending on
higher-and-higher powers of the Planck constant. In the case of a plane-wave
background field the found wave functions exactly reduce to the Volkov states,
which are then written in a new and fully quasiclassical form. Finally, by
using the leading-order WKB wave functions to compute the probabilities of
nonlinear Compton scattering and nonlinear Breit-Wheeler pair production, it is
explicitly shown that, if additionally the energies of the charges are
sufficiently large that the latter are not significantly deflected by the
field, the corresponding Baier's formulas are exactly reproduced for an
otherwise arbitrary classical electron/positron trajectory.
