Figure 1

Hamiltonian. Functions \(f(x)\) and \(g(x)\) in Eq. (10), ranging from \(a x=1\), which corresponds to \(v=0\) and \(a x=20\), which corresponds to \(v=0.998749\) (in \(c=1\) units). There is a singularity at the point where the denominators are 0, which corresponds to a point where the dynamics cannot be described by a Dirac equation. We see as well the linear region in the non-relativistic regime for very small ax \((v\simeq 0)\) and the transition to the region \(v=1\) \((ax\rightarrow \infty )\), where the dynamics is equivalent to \(v=0\), due to Lorentz-invariance