  We calculate the differential cross sections for Compton scattering of
photons described by Hermite Gaussian (HG) wave function in the framework of
relativistic quantum mechanics. The HG wave gamma-rays propagating along the
z-direction have quantum numbers of nodes of nx and ny in the x- and
y-directions, respectively. The calculated differential cross section is
symmetric with respect to both the zx- and zy-planes. The nodes whose number is
identical with nx and ny appear in the energy spectrum measured in zx- and
zy-planes, respectively. These results indicate that it is possible to identify
the HG wave photon and its quantum numbers nx and ny by measuring Compton
scattering. The present proposed method can be also applied to gamma-ray
astronomy.
