  We study the crossover of a quasi-two-dimensional Fermi gas trapped in the
radial plane from the Bardeen-Cooper-Schrieffer (BCS) regime to the
Bose-Einstein condensation (BEC) regime by crossing a wide Feshbach resonance.
We consider two effective two-dimensional Hamiltonians within the mean-field
level, and calculate the zero-temperature cloud size and number density
distribution. For a model 1 Hamiltonian with renormalized atom-atom
interaction, we observe a constant cloud size for arbitrary detunings. For a
model 2 Hamiltonian with renormalized interactions between atoms and dressed
molecules, the cloud size deceases from the BCS to BEC side, which is
consistent with the picture of BCS-BEC crossover. This qualitative discrepancy
between the two models indicates that the inclusion of dressed molecules is
essential for a mean-field description of quasi-two-dimensional Fermi systems,
especially on the BEC side of the Feshbach resonance.
