Dirac energy bounds

Such comparisons promise interesting tests of QED. Unfortunately, however, the theory of hydrogen is no longer simple, once we try to predict its energy levels with adequate precision [36]. The quantum electrodynamic corrections to the Dirac energy of the IS state, for instance, have an uncertainty of about 35 kHz, caused by numerical approximations in the calculation of the one-photon self-energy of a bound electron, and 50 kHz due to uncalculated higher order QED corrections. 

The radial functions Pmi r) and Qn ir) may be obtained by numerical integration [16,17] or by expansion in a basis (for recent reviews see [18,19]). Since the Dirac Hamiltonian is not bound from below, failure to observe correct boundary conditions leads to variational collapse [20,21], where admixture of negative-energy solutions may yield energies much below experimental. To avoid this failure, the basis sets used for expanding the large and small components must maintain kinetic balance [22,23]. 

It is known (Chap. A) that Koopmans theorem is not vahd for the wavefunctions and eigenvalues of strongly bound states in an atom or in the cores of a solid, i.e. for those states which are a solution of the Schrodinger (or Dirac) equation in a central potential. In them the ejection (or the emission) of one-electron in the electron system means a strong change in Coulomb and exchange interactions, with the consequent modification of the energy scheme as well as of the electronic wavefunction, in contradiction to Koopmans theorem. 

The application of G-spinor basis sets can be illustrated most conveniently by constructing the matrix operators needed for DCB calculations. The DCB equations can be derived from a variational principle along familiar nonrelativistic lines [7], [8, Chapter 3]. It has usually been assumed that the absence of a global lower bound to the Dirac spectrum invalidates this procedure it has now been established [16] that the upper spectrum has a lower bound when the trial functions lie in an appropriate domain. This theorem covers the variational derivation of G-spinor matrix DCB equations. Sucher s repeated assertions [17] that the DCB Hamiltonian is fatally diseased and that the operators must be surrounded with energy projection operators can be safely forgotten. 

In the representation advanced above, our 3D world is bounded by a hypersurface, whose normal points into our world. This is interpreted as the surface of Dirac s sea of energy momentum. Sources and sinks correspond to punctures on the hypersurface driven by. 1%, identified with particles and antiparticles, respectively. In this way, particles and antiparticles become solitons in the 4D ether. 

Figure 5.8 Energy eigenvalues (dots) of the discretized and complex rotated one-particle Dirac Hamiltonian with s-symmetry. Bound states that are well represented in the box and on the grid are lying on the x-axis. The low-energy pseudo-continuum states are rotated with an angle of approximately 20 (where 0 is the complex rotation angle) down from the real axis, as indicated with the straight solid line.

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... 

Here pj7m) denotes the spherical-wave free-electron function with the usual notations for Dirac angular quantum numbers. The numbers jlm are fixed by the overlap with the bound-electron wave function a) = njlm) where n is the principal quantum number. Integration over p is interpreted as integration over energies Ep = /p2 + m2. 

Therefore the negative-energy solutions for the Dirac equation are not a mathematical fiction In principle, each fundamental particle does have its corresponding antiparticle (which has the opposite electrical charge, but the same spin and the same nonnegative mass). Equation (3.6.15) also shows the formation of a transient Coulomb-bound electron-positron pair ("positronium"), whose decay into two photons is more rapid if the total spin is S = 0 than if it is S = 1, and is dependent on the medium. 

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