Extended relativistic representation of spin-1/2 particles

The variational approach to the Dirac-Coulomb Hamiltonian is extended to include simultaneously real electron and positron states. The spurious root previously obtained using finite-basis sets for states with kappa >0 is now clearly identified as the variational ground state of a negative-energy positron, resulting in a variational spectrum that is symmetric around E=0 and free of spurious roots. Moreover, a criterion for the accuracy of the variational results is obtained in terms of the (unphysical) positron component of the electron wavefunction. This approach is then further extended non-trivially to mix the non-optimized variational electron and positron states in the presence of a Coulomb potential. Exact solutions are presented. It is found that shifts in the non-optimized variational Dirac energy levels are obtained, expressed in terms of an arbitrary angle. This property is used to avoid variational collapse with any basis set and, as a consequence, the projector over positive-energy states is derived in a simple analytic form.

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