Dirac Bispinor

For the Dirac bispinor, the irreducible representation matrix Dab for each helicity component is a Pauli spin matrix a multiplied by ti/2. Then... 

We start this chapter with a discussion of the non-relativistic limit (nrl) of relativistic quantum theory (section 2). The Levy-Leblond equation will play a central role. We also discuss the nrl of electrodynamics and study how properties differ at their nrl from the respective results of standard non-relativistic quantum theory. We then present (section 3) the Foldy-Wouthuysen (FW) transformation, which still deserves some interest, although it is obsolete as a starting point for a perturbation theory of relativistic corrections. In this context we discuss the operator X, which relates the lower to the upper component of a Dirac bispinor, and give its perturbation expansion. The presentation of direct perturbation theory (DPT) is the central part of this chapter (section 4). We discuss the... 

The basic reason for the divergences observed at the end of the previous subsection is that there is a fundamental difference between the upper large) component g> of the Dirac bispinor, and the 2-component FW spinor... 

It is important that the exact relation between the upper and lower components of the Dirac bispinor is, in the presence of a magnetic field with vector potential A modified to... 

Figure 1. The radial parts of the large component of the 6pi/2 bispinor and the corresponding pseudospinor obtained in equivalent Dirac-Fock and 21-electron GRECP/SCF calculations for the state averaged over the relativistic 65 /26 1/2 configuration of thallium. Their difference is multiplied by 1000. The GRECP is generated for the state averaged over the nonrelativistic 6s 6p 6d configuration.

In this section I will outline the different methods that have been used and are currently used for the computation of parity violating effects in molecular systems. First one-component methods will be presented, then four-component schemes and finally two-component approaches. The term one-component shall imply herein that the orbitals employed for the zeroth-order description of the electronic wavefunction are either pure spin-up spin-orbitals or pure spin-down spin-orbitals and that the zeroth-order Hamiltonian does not cause couplings between the two different sets ( spin-free Hamiltonian). The two-component approaches use Pauli bispinors as basic objects for the description of the electronic wavefunction, while the four-component schemes employ Dirac four-spinors which contain an upper (or large) component and a lower (or small) component with each component being a Pauli bispinor. 

Using matrix multiplication rules, the Dirac equation [Eq. (3.54)] with bispinors can be rewritten in the form of two equations with spinors and 0 ... 

How should we interpret a bispinor wave funetion Does the Dirae equation describe a single fermion, an electron, a positron, an electron and a Dirac sea of other electrons (infinite number of particles), or an effective electron or effective positron (interacting with the Dirac sea) After eighty years, these questions do not have a clear answer. 

We will use the Dirac equation [Eq. (3.59)]. First, the basis set composed of two bispinors will be created q 2 = wave function will be sought as a linear... 

An analysis of the transformation properties of the Fock-Klein-Gordon equation and of the Dirac equation leads to the conclusion that satisfaction of the first of these equations requires the usual (i.e., scalar) wave function, whereas the second equation requires a bispinor character of the wave function. Scalar functions describe spinless particles (because they cannot be associated with the Pauli matrices), while bispinors in the Dirac equation are associated with the Pauli matrices, and describe a particle of 1/2 spin. 

In practice, this looks as follows. The many-electron wave function (let us focus our aUmtiim on a two-electron system only) is constructed from those trispinors, which correspond to positive energy solutions of the Dirac equation. For example, among two-electron functions built of such bispinors, no function corresponds to E, and, most importantly, to E". This means that carrying out computations with such abasis set, we do not use the full DC Hamiltonian, but insleacL its projection on the space of states with positive energies. 

The Dirac equation represents an approximation- and refers to a single particle. What happens with larger systems Nobody knows, but the first idea is to construct the total Hamiltonian as a sum of the Dirac Hamiltonians for individual particles plus their Coulombic interaction (the Dirac-Coulomb apjmmmation). This is practised routinely nowadays for atoms and molecules. Most often we use the mean-field approximation (see Chapter 8) with the modification that each of the one-electron functions represents a four-component bispinor. Another approach is extreme pragmatic, maybe too pragmatic we perform the non-relativistic calculations with a pseudopotential that mimics what is supposed to happen in a relativistic case. 

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