  It is shown the central field Dirac equation can be simplified through the
use of real conjugate spinors to substitute for the upper and lower components
of the bi-spinor eigensolutions. This substitution reduces the Dirac equation
for the hydrogen atom to the problem of solving a single second order
differential equation similar to a Klein-Gordon equation but containing
additional terms to take account of the spin on the electron. The bi-spinor
wave functions are readily constructed once the solution is known in terms of
the conjugate spinors.
