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 ... 

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... 

In addition to being normalized and anticommutative, these matrices, of course, must be Hermitian. These conditions are similar to those for the three components Ojc, Oyy spin operator a and of their Pauli representations as 2D matrices ... 

In the traditional approaches to nuclear physics, nuclear structure and nuclear reactions at low energies are studied using nonrelativistic many-body theory. The state of the nucleons is described by a nonrelativistic Hamiltonian, and the interactions are taken to be static potentials which are assumed to arise from meson exchange. The average kinetic energy of a bound nucleon and the nuclear binding potential (in the two-component Pauli representation) are both much less than the nucleon mass, so that, in the traditional approach, neither relativistic kinematics nor dynamics should be needed in descriptions of low energy nuclear phenomena. 

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