In this note the rough Dirac approximation is replaced by the accurate equation (24) which is subjected to Schroedinger and Dirac quantization defined in Eqs. (25) and (32) respectively. The relativistic momentum p in the denominator is computed using the original and important new procedures of UFT324 – UFT328, and in two dimensions, expectation values can be computed using the two dimensional rotational wavefunctions (33). There should be significant differences in energy levels between the rough approximations used by Dirac and the accurate atom. In the next note the procedure will be extended to three dimensions and the H atom.