  We have developed a numerical model for the temporal evolution of particle
and photon spectra resulting from nonthermal processes at the shock fronts
formed in merging clusters of galaxies. Fermi acceleration is approximated by
injecting power-law distributions of particles during a merger event, subject
to constraints on maximum particle energies. We consider synchrotron,
bremsstrahlung, Compton, and Coulomb processes for the electrons, nuclear,
photomeson, and Coulomb processes for the protons, and knock-on electron
production during the merging process. The broadband radio through gamma-ray
emission radiated by nonthermal protons and primary and secondary electrons is
calculated both during and after the merger event. Using ROSAT observations to
establish typical parameters for the matter density profile of clusters of
galaxies, we find that typical merger shocks are weak and accelerate particles
with relatively soft spectra. We consider the prospects for detecting
nonthermal radio and gamma-ray emission from clusters of galaxies and
implications for the origin of ultra-high energy cosmic rays and the diffuse
gamma-ray background. Our results suggest that only a few of the
isotropically-distributed unidentified EGRET sources are due to shocks formed
in cluster mergers, and that only a minor contribution to the diffuse
extragalactic gamma-ray background can originate from cluster merger shocks.
Cluster merger shocks can accelerate protons to $\lesssim 10^{19}$ eV for the
standard parameters considered here. We predict that {\it GLAST} will detect
several cluster mergers and, depending on the mean magnetic fields in the
intracluster medium, the Low Frequency Array could detect anywhere from several
to several hundred.
