  We investigate wavepacket dynamics across supercritical barriers for the
Klein-Gordon and Dirac equations. Our treatment is based on a multiple
scattering expansion (MSE). For spin-0 particles, the MSE diverges, rendering
invalid the use of the usual connection formulas for the scattering basis
functions. In a time-dependent formulation, the divergent character of the MSE
naturally accounts for charge creation at the barrier boundaries. In the Dirac
case, the MSE converges and no charge is created. We show that this
time-dependent charge behavior dynamics can adequately explain the Klein
paradox in a first quantized setting. We further compare our semi-analytical
wavepacket approach to exact finite-difference solutions of the relativistic
wave equations.
