Negative energy branch

The Furry bound state interaction picture of quantum electrodynamics76 relies on an expansion of the second-quantized electron field operator in terms of single-particle solutions of the Dirac equation for a static external field. This external field may be thought of as some mean atomic or molecular potential, whose single-particle spectrum can be divided into positive- and negative-energy branches. This can always be done for the usual elements of the Periodic Table, although problems arise for super-heavy atomic nuclei. 

Unless carefully implemented the representation of the Dirac spectrum obtained within the algebraic approximation may exhibit undesirable properties which are not encountered in non-relativistic studies. In particular, an inappropriate choice of basis set may obliterate the separation of the spectrum into positive and negative energy branches. So-called intruder states may arise, which are impossible to classify as being of either positive or negative energy character. The Furry bound state interaction picture of quantum electrodynamics is thereby undermined. 

Some work has been reported on relativistic coupled cluster methods most notably by Kaldor, Ishikawa and their collaborators.232 These calculations are carried out within the no virtual pair approximation and are therefore analogous to the non-relativistic formulation. Perturbative analysis of the relativistic electronic structure problem demonstrated the importance of the negative energy branch of the spectrum in the calculation of energies and other expectation values. 

A consequence of this conjecture, which survives in more modern formulations of relativistic quantum mechanics, is that even the simple hydrogenic atom is a many-body problem The electrons filling the negative energy branch of the Dirac spectrum are... 

Upon taking the square root, both signs must be considered, leading to a positive- and a negative-energy branch separated by a gap of 2mc. In classical mechanics free particles... 

Although Eq. (11.103) seems to be unnecessarily complicated, it can be solved by purely numerical iterative techniques, and the matrix representation of the operator Q is obtained [616]. This result appears to be the best representation of the operator Q that can be achieved within a given basis and is only limited by machine accuracy. Note that all expressions occurring in Eq. (11.103) and hence the matrix representation of the operator Q depend only on the squared momentum rather than on the momentum variable itself, which is the key feature of the computational feasibility of this approach. This is an essential trick for actual calculations of transformed two-component operators first noticed by Hess [623] (compare also section 12.5.1). Eq. (11.103) is still nonlinear and therefore bears the possibility of negative-energy solutions for the operator Q. The choice toward the positive-energy branch has to be implemented via the boundary conditions imposed on the numerical iterative technique. Essentially, Q and hence R have to be small operators with operator norms much smaller than unity. 

This indicates that the R branch lines occur at energies which grow closer and closer together as J increases (since the 15.88 - Ji term will cancel). The P branch lines occur at energies which lie more and more negative (i.e. to the left of the origin). So, you can... 

There is another important feature to note in the curve of Figure 10.1, the second derivative of the energy with respect to Ns is discontinuous at the ground state multiplicity and must be negative in both directions, due to the fact that both branches in the plot have negative curvatures. This second derivative, as in the... 

---