Separation of Variables in the Dirac Equation

As in the nonrelativistic case, most of the salient features of the atomic systems are exposed in the treatment of the simplest of these, the hydrogen-like one-electron atoms. In Hartree atomic units the time-independent Dirac equation yields the coupled equations 

For the Dirac equation, we know already that the large and small components have different radial functions, so we must seek a separation of the form 

P and Q are the radial large and small components of the wave function. The factor of i has been introduced to make the radial components real. The angular functions are two-component spinors, that is, a product of angular and spin functions the spin variable r has been explicitly shown. 

To achieve this separation of variables in the Dirac equation, we must be able to factorize the operator a p. The nuclear potential V is already a function of r only. We follow the procedure in Schiff (1968) and introduce the radial operators for momentum and velocity 

The scalar product r I vanishes because = r x p. The vector product r x f may be evaluated using standard vector relations  

---