Dirac-Pauli representation

In the standard (Dirac-Pauli) representation, the Dirac equation for an electron in the field of a stationary potential V reads... 

Abstract. An application of the Rayleigh-Ritz variational method to solving the Dirac-Coulomb equation, although resulted in many successful implementations, is far from being trivial and there are still many unresolved questions. Usually, the variational principle is applied to this equation in the standard, Dirac-Pauli, representation. All observables derived from the Dirac equation are invariant with respect to the choice of the representation (i.e. to a similarity transformation in the four-dimensional spinor space). However, in order to control the behavior of the variational energy, the trial functions are subjected to several conditions, as for example the kinetic balance condition. These conditions are usually representation-dependent. The aim of this work is an analysis of some consequences of this dependence. 

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. 

The Dirac-Pauli representation is most commonly used in all applications of the Dirac theory to studies on electronic structure of atoms and molecules. Apart of historical reasons, there are several features of this representation which make its choice quite natural. Probably the most important is a well defined symmetry of and in the case of spherically-symmetric potentials V. The Dirac Hamiltonian... 

Hence, = I + 1 if k > 0 and = I — 1 if k < 0. Consequently, in the Dirac-Pauli representation and have definite parity, (—1) and (—1) respectively. It is customary in atomic physics to assign the orbital angular momentum label I to the state fnkm.j- Then, we have states lsi/2, 2si/2) 2ri/2, 2p3/2, , if the large component orbital angular momentum quantum numbers are, respectively, 0,0,1, ,... while the corresponding small components are eigenfunctions of to the eigenvalues 1,1,0,2,. 

According to Eqs. (2) and (13) the Hamiltonian eigenfunction in the Dirac-Pauli representation may be written as... 

Another important feature of the Dirac-Pauli representation is its natural adaptation to the non-relativistic limit. If V —E l << x( then Eq. (2) transforms directly to its non-relativistic counterpart known as the Levy-Leblond equation ... 

If p 7 0 Eqs. (21) and (22) are coupled, but the relations between components of the wavefunction are much simpler than in the standard Dirac-Pauli representation. By the elimination of and respectively from Eq. (21) and from Eq. (22), we get two decoupled second-order equations for and ... 

Eq. (36) may also be expressed as a system of two first-order equations, i.e. as the radial Dirac equation in the representation of Biedenharn. Let us rewrite the radial Dirac-Pauli equation (18) with V = —Zjr in the form... 

We demonstrated that by the selection of a representation of the Dirac Hamiltonian in the spinor space one may strongly influence the performance of the variational principle. In a vast majority of implementations the standard Pauli representation has been used. Consequently, computational algorithms developed in relativistic theory of many-electron systems have been constructed so that they are applicable in this representation only. The conditions, under which the results of these implementations are reliable, are very well understood and efficient numerical codes are available for both atomic and molecular calculations (see e.g. [16]). However, the representation of Weyl, if the external potential is non-spherical, or the representation of Biedenharn, in spherically-symmetric cases, seem to be attractive and, so far, hardly explored options. 

In performing the similarity transformation above, we have changed our representation of the particle from a Dirac representation to the so-called Foldy-Wouthuysen representation. This new representation provides a very simple link with the non-relativistic Schrodinger-Pauli representation. The latter, which is a two-component representation, just corresponds to the two upper components of the Foldy-Wouthuysen representation (3.125). It must be noted that under a similarity transformation such as (3.105), the operators which represent physical observables are also transformed. For example, the position observable whose operator is R in the Dirac representation is transformed to R " in the Foldy-Wouthuysen representation where... 

The Pauli form factor also generates a small contribution to the Lamb shift. This form factor does not produce any contribution if one neglects the lower components of the unperturbed wave functions, since the respective matrix element is identically zero between the upper components in the standard representation for the Dirac matrices which we use everywhere. Taking into account lower components in the nonrelativistic approximation we easily obtain an explicit expression for the respective perturbation... 

The paper of Louis de Broglie, associating a wavelength to any particle, had appeared in 1925.27 The five notes of Schrodinger on wave mechanics appeared in 1926.28 The various papers on the representation of physical quantities by matrices by Born, Heisenberg, Jordon, Dirac, and Pauli were published between 1925 and 1927.29 The results of diffraction experiments by Davisson and Germer30 of electrons scattered by a single... 

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

The transformation of the Dirac Hamiltonian to two-component form is accompanied by a corresponding reduction of the wavefunction. As discussed in detail in section 2, the four-component Dirac spinor will have only two nonvanishing components, as soon as the complete decoupling of the electronic and positronic degrees of freedom is achieved, and can thus be used as a two-component spinor. This feature can be exploited to calculate expectation values of operators in an efficient manner. However, this procedure requires that some precautions need to be taken care of with respect to the representation of the operators, i.e., their transition from the original (4 x 4)-matrix representation (often referred to as the Dirac picture) to a suitable two-component Pauli repre-... 

Pi its components form the 4x4 Dirac matrices acting on the ith electron, a may be formulated in the standard representation in terms of the three 2x2 Pauli spin matrices = ([Pg.631]

This fact is the basis of Pauli s principle, which is therefore also termed Pauli s exclusion principle. In addition, we have the Fermi-Dirac statistics for other states (in occupation number representation)... 

We now turn to the Gaunt interaction, and use the terms from the modified Dirac representation in (15.54) to derive the Breit-Pauli operators. These terms need no renormalization, because they are all of order 1/c. The three classes of operators defined in (15.54) are considered in turn. 

The above discussion leads to the Pauli approximation to the Dirac equation, which was considered in depth in chapter 3 of Ref, 51, The Pauli approximation amounts to ignoring the small components(S), Since the Pauli method leads to a natural two-component representation of spin, the method is quite popular in several chemical applications. 

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