Dirac Electron in the Coulomb Field

Even in the framework of nonrelativistic quantum mechanics one can achieve a much better description of the hydrogen spectrum by taking into account the finite mass of the Coulomb center. Due to the nonrelativistic nature of the bound system under consideration, finiteness of the nucleus mass leads to substitution of the reduced mass instead of the electron mass in the formulae above. The finiteness of the nucleus mass introduces the largest energy scale in the bound system problem - the heavy particle mass. 

The relativistic dependence of the energy of a free classical particle on its momentum is described by the relativistic square root 

The kinetic energy operator in the Schrodinger equation corresponds to the quadratic term in this nonrelativistic expansion, and thus the Schrodinger equation describes only the leading nonrelativistic approximation to the hydrogen energy levels. 

The classical nonrelativistic expansion goes over jp- jm . In the case of the loosely bound electron, the expansion in jp IrrP corresponds to expansion in (Za) hence, relativistic corrections are given by the expansion over even powers of Za. As we have seen above, from the explicit expressions for the energy levels in the Coulomb field the same parameter Za also characterizes the binding energy. For this reason, parameter Za is also often called the binding parameter, and the relativistic corrections carry the second name of binding corrections. 

For the bound electron, calculation of the relativistic corrections should also take into account the contributions due to its spin one half. Account for the spin one half does not change the fundamental fact that all relativistic 

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