  We consider a one-dimensional optical lattice of three-dimensional Harmonic
Oscillators which are loaded with neutral fermionic atoms trapped into two
hyperfine states. By means of a standard variational coherent-state procedure,
we derive an effective Hamiltonian for this quantum model and the hamiltonian
equations describing its evolution. To this end, we identify the algebra
$\mathcal L$ of two-fermion operators --describing the relevant microscopic
quantum processes of our model-- whereby the natural choice for the trial state
appears to be a so(2r) coherent state. The coherent-state parameters, playing
the role of dynamical variables for the effective Hamiltonian, are shown to
identify with the $\mathcal L$-operator expectation values thus providing a
clear physical interpretation of this algebraic mean-field picture.
