  We formulate a wave atom optics theory of the Collective Atomic Recoil Laser,
where the atomic center-of-mass motion is treated quantum mechanically. By
comparing the predictions of this theory with those of the ray atom optics
theory, which treats the center-of-mass motion classically, we show that for
the case of a far off-resonant pump laser the ray optics model fails to predict
the linear response of the CARL when the temperature is of the order of the
recoil temperature or less. This is due to the fact that in theis temperature
regime one can no longer ignore the effects of matter-wave diffraction on the
atomic center-of-mass motion.
