  We study the coupled translational, electronic, and field dynamics of the
combined system "a two-level atom + a single-mode quantized field + a
standing-wave ideal cavity". We derive Hamilton -- Schr\"odinger equations for
probability amplitudes and averaged position and momentum of a point-like atom
interacting with the quantized field in a standing-wave cavity. They
constitute, in general, an infinite-dimensional set of equations with an
infinite number of integrals of motion which may be reduced to a dynamical
system with four degrees of freedom if the quantized field is supposed to be
initially prepared in a Fock state. This system is found to produce semiquantum
chaos with positive values of the maximal Lyapunov exponent. At large values of
detuning $|\delta|\gg 1$, the Rabi atomic oscillations are usually shallow, and
the dynamics is found to be almost regular. The Doppler -- Rabi resonance, deep
Rabi oscillations that may occur at any large value of $|\delta|$ to be equal
to $|\alpha p_0|$, is found numerically and described analytically (with
$\alpha$ to be the normalized recoil frequency and $p_0$ the initial atomic
momentum). Two gedanken experiments are proposed to detect manifestations of
semiquantum chaos in real experiments. In the chaotic regime values of the
population inversion $z_{out}$, measured with atoms after transversing a
cavity, are so sensitive to small changes in the initial inversion $z_{in}$
that the probability of detecting any value of $z_{out}$ in the admissible
interval becomes almost unity in a short time. Chaotic wandering of a two-level
atom in a quantized Fock field is shown to be fractal. Fractal-like structures,
typical for chaotic scattering, are numerically found in the dependence of the
time of exit of atoms from the cavity on their initial momenta.
