  Approximate solutions of the Dirac equation are found for ultrarelativistic
particles moving in a periodic potential, which depends only on one coordinate,
transverse to the largest component of the momentum of the incoming particle.
As an example we employ these solutions to calculate the radiation emission of
positrons and electrons trapped in the planar potential found between the (110)
planes in Silicon. This allows us to compare with the semi-classical method of
Baier, Katkov and Strakhovenko, which includes the effect of spin and photon
recoil, but neglects the quantization of the transverse motion. For high-energy
electrons, the high-energy part of the angularly integrated photon energy
spectrum calculated with the found wave functions differs from the
corresponding one calculated with the semi-classical method. However, for lower
particle energies it is found that the angularly integrated emission energy
spectra obtained via the semi-classical method is in fairly good agreement with
the full quantum calculation except that the positions of the harmonic peaks in
photon energy and the photon emission angles are shifted.
