  In this paper we investigate the problem of fermion creation inside a three
dimensional box. We present an appropriate wave function which satisfies the
Dirac equation in this geometry with MIT bag model boundary condition. We
consider walls of the box to have dynamic and introduce the time evolution of
the quantized field by expanding it over the 'instantaneous basis'. We explain
how we can obtain the average number of particles created. In this regard we
find the Bogliubove coefficients. We consider an oscillation and determine the
coupling conditions between different modes that can be satisfied depending on
the cavity's spectrum. Assuming the parametric resonance case we obtain an
expression for the mean number of created fermions in each mode of an
oscillation and their dynamical Casimir energy.
