  We consider the ground state of a mixture of two pseudospin-$\1/2$ Bose gases
with interspecies spin exchange in a trapping potential. In the mean field
approach, the ground state can be described in terms of four wave functions
governed by a set of coupled Gross-Pitaevskii-like (GP-like) equations, which
differ from the usual GP equations in the existence of an interference term due
to spin-exchange coupling between the two species. Using these GP-like
equations, we calculate such ground state properties as chemical potentials,
density profiles and spin density profiles, which are directly observable in
experiments. We compare the cases with and without spin exchange. It is
demonstrated that the spin exchange between the two species lowers the chemical
potentials, tends to equalize the wave functions of the two pseudospin
components of each species, and thus homogenizes the spin density. The novel
features of the density and spin density profiles can serve as experimental
probes of this novel Bose system.
