  The "particle in a box" problem is investigated for a relativistic particle
obeying the Klein-Gordon equation. To find the bound states, the standard
methods known from elementary non-relativistic quantum mechanics can only be
employed for "shallow" wells. For deeper wells, when the confining potentials
become supercritical, we show that a method based on a scattering expansion
accounts for Klein tunneling (undamped propagation outside the well) and the
Klein paradox (charge density increase inside the well). We will see that in
the infinite well limit, the wavefunction outside the well vanishes and Klein
tunneling is suppressed: quantization is thus recovered, similarly to the
non-relativistic particle in a box. In addition, we show how wavepackets can be
constructed semi-analytically from the scattering expansion, accounting for the
dynamics of Klein tunneling in a physically intuitive way
