  We study conceptional problems of a photo-electron emission (PEE) process
from a free exciton in insulating crystals. In this PEE process, only the
electron constituting the exciton is suddenly emitted out of the crystal, while
the hole constituting the exciton is still left inside and forced to be
recoiled back to its original valence band. This recoil on the hole is surely
reflected in the spectrum of the PEE with a statistical distribution along the
momentum-energy curve of the valence band. This distribution is nothing but the
square of the exciton wave function amplitude, since it shows how the electron
and the hole are originally bound together. Thus, the momentum-resolved PEE can
directly determine the exciton wave function. These problems are clarified,
taking the $\Gamma$ and the saddle point excitons in GaAs, as typical examples.
New PEE experiments are also suggested.
