  We develop a Hamiltonian that describes the time-dependent formation of a
molecular Bose-Einstein condensate (BEC) from a Bardeen-Cooper-Schrieffer (BCS)
state of fermionic atoms as a result of slowly sweeping through a Feshbach
resonance. In contrast to many other calculations in the field (see e.g.
[1-4]), our Hamiltonian includes the leading post-adiabatic effects that arise
because the crossover proceeds at a non-zero sweep rate. We apply a path
integral approach and a stationary phase approximation for the molecular zero
momentum background, which is a good approximation for narrow resonances (see
e.g. [5, 6]). We use two-body adiabatic approximations to solve the atomic
evolution within this background. The dynamics of the non-zero momentum
molecular modes is solved within a dilute gas approximation and by mapping it
onto a purely bosonic Hamiltonian. Our main result is a post-adiabatic
effective Hamiltonian in terms of the instantaneous bosonic
(Anderson-)Bogoliubov modes, which holds throughout the whole resonance, as
long as the Feshbach sweep is slow enough to avoid breaking Cooper pairs.
