Relativistic single-particle spectrum

Although Dirac s equation does not directly admit of a completely self-consistent single-particle interpretation, such an interpretation is physically acceptable and of practical use, provided the potential varies little over distances of the order of the Compton wavelength (h/mc) of the particle in question. It allows, for instance, first-order relativistic corrections to the spectrum of the hydrogen atom and to the core-level densities of many-electron atoms. The latter aspect is of special chemical importance. The required calculations are invariably numerical in nature and this eliminates the need to investigate central-field solutions in the same detail as for Schrodinger s equation. A brief outline suffices. 

The surprising implication is that Dirac s equation does not allow of a self-consistent single-particle interpretation, although it has been used to calculate approximate relativistic corrections to the Schrodinger energy spectrum of hydrogen. The obvious reason is that a 4D point particle is without duration and hence undefined. An alternative description of elementary units of matter becomes unavoidable. Prompted by such observation, Dirac [3] re-examined the classical point model of the electron only to find that it has three-dimensional size, with an interior that allows superluminal signals. It all points at a wave structure with phase velocity > c. 

The spectrum of the single-electron Dirac operator Hd and its eigenspinors (/> for Coulombic potentials are known in analytical form since the early days of relativistic quantum mechanics. However, this is no longer true for a many-electron system like an atom or a molecule being described by a many-particle Hamiltonian H, which is the sum of one-electron Dirac Hamiltonians of the above kind and suitably chosen interaction terms. One of the simplest choices for the electron interaction yields the Dirac-Coulomb-Breit (DCB) Hamiltonian, where only the frequency-independent first-order correction to the instantaneous Coulomb interaction is included. 

The Dirac equation is rigorously invariant with respect to the Lorentz transformation, which is certainly the most important requirement for a relativistic theory. Therefore, it would seem to be a logically sound approximation for a relativistic description of a single quantum particle. Unfortunately, this is not true. Recall that the Dirac Hamiltonian spectrum contains a... 

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