  The Casimir force between two parallel uncharged closely spaced metallic
plates is evaluated in ways alternatives to those usually considered in the
literature. In a first approximation we take in account the suppressed quantum
numbers of a cubic box, representing a cavity which was cut in a metallic
block. We combine these ideas with those of the MIT bag model of hadrons, but
adapted to non-relativistic particles. In a second approximation we consider
the particles occupying the energy levels of the Bohr atom, so that the Casimir
force depends explicitly on the fine structure constant alpha. In both
treatments, the mean energies which have explicit dependence on the particle
mass and on the maximum occupied quantum number (related to the Fermi level of
the system) at the beginning of the calculations, have these dependences
mutually canceled at the end of them. Finally by comparing the averaged
energies computed in both approximations, we are able to make an estimate of
the value of the fine structure constant alpha.
