  There exists a Klein-Gordon-like equation for a spin-1/2 particle in an
electromagnetic field with 2-spinors as wave functions that is a direct
consequence of the corresponding Dirac equation. Thus, it reproduces the same
binding energies for an electron in a hydrogen atom as the Dirac equation.
There is also a square-root equation for 2-spinors which can give the same
binding energies up to the forth order of the fine structure constant,
inclusively, what will be shown by means of a comparison of the
non-relativistic limit plus a first relativistic correction of both equations
for 2-spinors within the framework of a perturbation analysis. A parallel will
also be drawn to the spin-0 case.
