Dirac calculations eigenfunctions

Apart of historical reasons, there are several features of the Dirac-Pauli representation which make its choice rather natural. In particular, it is the only representation in which, in a spherically-symmetric case, large and small components of the wavefunction are eigenfunctions of the orbital angular momentum operator. However, this advantage of the Dirac-Pauli representation is irrelevant if we study non-spherical systems. It appears that the representation of Weyl has several very interesting properties which make attractive its use in variational calculations. Also several other representations seem to be worth of attention. Usefulness of these ideas is illustrated by an example. 

EDE in the external Coulomb field in Fig. 1.6. The eigenfunctions of this equation may be found exactly in the form of the Dirac-Coulomb wave functions (see, e.g, [10]). For practical purposes it is often sufficient to approximate these exact wave functions by the product of the Schrodinger-Coulomb wave functions with the reduced mass and the free electron spinors which depend on the electron mass and not on the reduced mass. These functions are very convenient for calculation of the high order corrections, and while below we will often skip some steps in the derivation of one or another high order contribution from the EDE, we advise the reader to keep in mind that almost all calculations below are done with these unperturbed wave functions. 

Let us briefly discuss the idea of and the results obtained from coupled-channel calculations with momentum eigenfunctions as it has been employed recently (Tenzer et al. 2000b). The collision is considered in a coordinate system, where the scattering ions have equal but opposite velocities v = voez. With respect to this system the total time-dependent Hamiltonian H is decomposed into the unperturbed (free) Dirac Hamiltonian... 

The average value of the dipole moment will be calculated by means of Dirac s perturbation theory for nonstationary. states, up to third order the zero order refers to the free molecules in the absence of the field. Let the wave function of the system of the two interacting molecules in- the external field be specified by y, an eigenfunction of the total Hamiltonian H. This wave function y> may be expanded in a complete set of the energy eigenfunctions unperturbed system the index n labels the various unperturbed eigenstates characterized by the energy En. We may then write... 

The possible severity of the problem has been shown by M. Stanke and J. Karwowski, Variationalprinciples in the Dirac theory Spurious solutions, unexpected extrema and other traps in A eir Trends in Quantum Systems in Chemistry and Physics, vol. I, pp. 175 190, eds. J. Maniani et al., Kluwer Academic Publishers, Dordrecht (2001). Sometimes an eigenfunction corresponds to a quite different eigenvalue. Nothing of that sort appears in non-relativistic calculations. 

---