Eigenvalue spectrum Dirac

In the one-electron theory of Dirac, one finds that the eigenvalue spectrum consists of two parts. The positive-energy spectrum comprises states corresponding to those found in non-relativistic theory, but with energies shifted by mt a.u., corresponding to the rest-mass energy. The energies of these states contain... 

Two critical questions about the removal of the spin dependence from the Dirac equation are, What effect does this have on the eigenvalue spectrum and How is the wave function modified These questions may be answered by a similar process to that used for the atomic Dirac equation in chapter 7. The second question has as an auxiliary question, How does the wave function behave in the nonrelativistic limit ... 

The eigenvalue spectrum of (15.14) may be deduced from the behavior of the equations in the region far from the nuclei, where V 0 and consequently all terms involving V are negligible. The spin-free modified Dirac equation in this region may be written (in Hartree atomic units)... 

If En represents the lowest positive energy eigenvalue of the bound Dirac spectrum, the further evaluation of (23) is straightforward employing standard methods. In the present work this is the case as we consider only the lS 1/2 state. The numerical evaluation scheme is similar to that described already. After including the results for wf irred and AE wf irred, we obtain a total... 

In perturbation theory, one often considers the essential spectrum. It consists of the continuous spectrum together with the accumulation points of the eigenvalues. Whenever the potential vanishes at infinity, the essential spectrum of the Dirac equation is the set... 

Here E, are the Dirac eigenvalues, index s runs over the complete spectrum of the Dirac Hamiltonian. This spectrum includes the solutions with positive and negative energy values E,. For the bound-state solutions the... 

Such a transformation converts the 4 x 4 Dirac Eq.4.12 into two separate eigenvalue problems. The one which corresponds to what is called the positive (electronic) part of the spectrum ... 

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