  The inclusive probability to record an electron in elastic electromagnetic
scattering of an electron by a spin one-half hadron is obtained, the initial
quantum states of the electron and the hadron being described by the density
matrices of a general form. Contrary to the Rosenbluth formula for the
differential cross-section for this process, the first nontrivial contribution
to the inclusive probability turns out to be of order $\alpha$ and not
$\alpha^2$. This contribution describes the interference between the trivial
contribution to the $S$-matrix and the leading contribution to its connected
part. The explicit expression for this interference terms is derived. For
example, for Gaussian wave packets the interference contribution to the
inclusive probability is of relative order $2\times 10^{-2}$ in comparison with
the contribution of the free passed particle for the nonrelativistic proton and
the $10$ MeV electron with the normal deviation of momenta in the electron and
hadron wave packets of order $10$ keV. It is shown that the same interference
term arises when the electron is scattered by the classical electromagnetic
field produced by the hadron electromagnetic current averaged with respect to
the free evolving density matrix of the hadron, even in the case of a single
hadron. The interference term describes coherent scattering of the electron by
the hadron wave packet and is immune to the quantum recoil experienced by a
hadron due to scattering. The effective electron mass operator is found on the
mass-shell.
