  We solve the Schr\"{o}dinger equation for two electrons plus one hole by
writing it in the electron-exciton basis. The main advantage of this basis is
to eliminate the exciton contribution from the trion energy in a natural way.
The interacting electron-exciton system is treated using the recently developed
composite boson many-body formalism which allows an exact handling of electron
exchange. We numerically solve the resulting electron-exciton Schr\"{o}dinger
equation, with the exciton levels restricted to the lowest $1s, 2s$ and $3s$
states, and we derive the trion ground state energy as a function of the
electron-to-hole mass ratio. While our results are in reasonable agreement with
those obtained through the best variational methods using free carrier basis,
this electron-exciton basis is mostly suitable to easily reach the bound and
unbound trion excited states. Through their wave functions, we here calculate
the optical absorption spectrum in the presence of hot carriers for 2D quantum
wells. We find large peaks located at the exciton levels, which are attributed
to electron-exciton (unbound) scattering states, and small peaks identified
with trion bound states.
