  For cosmic neutrinos we study the conditions and the effects of the coherence
loss as well as coherent broadening of the spectrum. We evaluate the width of
the neutrino wavepacket produced by charged particles under various
circumstances: in an interaction-free environment, in a radiation-dominated
medium (typical of the sources of the gamma ray bursts) and in the presence of
a magnetic field. The effect of the magnetic field on the wavepacket size
appears to be more important than the scattering. If the magnetic field at the
source is larger than $\sim$10 Gauss, the coherence of neutrinos will be lost
while traveling over cosmological distances. Various applications of these
results have been considered. We find that for large magnetic fields ($B> 10^9$
Gauss) and high energies ($E_\nu>{\rm PeV}$), "coherent broadening" can modify
the energy spectrum of neutrinos. In the coherent case, averaging out the
oscillatory terms of the probabilities does not induce any statistical
uncertainty beyond what expected in the absence of these terms. A deviation
from the standard quantum mechanics that preserves average energy and unitarity
cannot alter the picture.
