  We present a study of the elastic exciton--electron ($X-e^-$) and
exciton--hole ($X-h$) scattering processes in semiconductor quantum wells,
including fermion exchange effects. The balance between the exciton and the
free carrier populations within the electron-hole plasma is discussed in terms
of ionization degree in the nondegenerate regime. Assuming a two-dimensional
Coulomb potential statically screened by the free carrier gas, we apply the
variable phase method to obtain the excitonic wavefunctions, which we use to
calculate the 1$s$ exciton--free carrier matrix elements that describe the
scattering of excitons into the light cone where they can radiatively
recombine. The photon emission rates due to the carrier-assisted exciton
recombination in semiconductor quantum-wells (QWs) at room temperature and in a
low density regime are obtained from Fermi's golden rule, and studied for
mid-gap and wide-gap materials. The quantitative comparison of the direct and
exchange terms of the scattering matrix elements shows that fermion exchange is
the dominant mechanism of the exciton--carrier scattering process. This is
confirmed by our analysis of the rates of photon emission induced by
electron-assisted and hole-assisted exciton recombinations.
