  It is known that Schr\"odinger equation fails in describing the dynamics of
highly energetic particles. We propose to quantify this lack of Lorentz
covariance by evaluating the probability for a particle to be measured outside
the set of light cones which are compatible to its initial wave function. We
consider a simple case of a particle released from a box, which, in turn, is
inside a larger container. It is shown that besides the increasing error at
relativistic energies, there may be a complete breakdown, with Schr\"odinger
dynamics implying in deterministic, superluminal signaling for Lorentz factors
above 129. In addition, we give an exact asymptotic expression for the
violation in local causality by employing the stationary exponent method, from
which the Compton wave length of the particle naturally arises as the relevant
scale for the stationary points.
