Relativistic spin-0 particle in a box: Bound states, wave packets, and the disappearance of the Klein paradox

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