  We present an exact computation of effective Hamiltonians for an elementary
model obtained from the Yukawa theory by going to the limit of bare fermions
being infinitely heavy and bare bosons being at rest with respect to the
fermions that emit or absorb them. The coupling constant can be arbitrarily
large. The Hamiltonians are computed by solving the differential equation of
the renormalization group procedure for effective particles (RGPEP). Physical
fermions, defined in the model as eigenstates of the effective Hamiltonians,
are obtained in the form of an effective fermion dressed with a coherent state
of effective bosons. The model computation illustrates the method that can be
used in perturbative computations of effective Hamiltonians for realistic
theories. It shows the mechanism by which the perturbative expansion and
Tamm-Dancoff approximation increase in accuracy along the RGPEP evolution.
