  In this paper, we analyze the propagation of a wave packet through a conical
intersection. This question has been addressed for Gaussian wave packets in the
90s by George Hagedorn and we consider here a more general setting. We focus on
the case of Schr{\"o}dinger equation but our methods are general enough to be
adapted to systems presenting codimension 2 crossings and to codimension 3 ones
with specific geometric conditions. Our main Theorem gives explicit transition
formulas for the profiles when passing through a conical crossing point,
including precise computation of the transformation of the phase. Its proof is
based on a normal form approach combined with the use of superadiabatic
projectors and the analysis of their degeneracy close to the crossing.
