Dirac equation development

Following the hypothesis of electron spin by Uhlenbeck and Goudsmit, P. A. M. Dirac (1928) developed a quantum mechanics based on the theory of relativity rather than on Newtonian mechanics and applied it to the electron. He found that the spin angular momentum and the spin magnetic moment of the electron are obtained automatically from the solution of his relativistic wave equation without any further postulates. Thus, spin angular momentum is an intrinsic property of an electron (and of other elementary particles as well) just as are the charge and rest mass. 

Our approach is based on a systematic semiclassical study of the Dirac equation. After separating particles and anti-particles to arbitrary powers in h, a semiclassical expansion of the quantum dynamics in the Heisenberg picture is developed. To leading order this method produces classical spin-orbit dynamics for particles and anti-particles, respectively, that coincide with the findings of Rubinow and Keller Hamiltonian relativistic (anti-) particles drive a spin precession along their trajectories. A modification of that method leads to a semiclassical equivalent of the Foldy-Wouthuysen transformation resulting in relativistic quantum Hamiltonians with spin-orbit coupling. 

This experimental development was matched by rapid theoretical progress, and the comparison and interplay between theory and experiment has been important in the field of metrology, leading to higher precision in the determination of the fundamental constants. We feel that now is a good time to review modern bound state theory. The theory of hydrogenic bound states is widely described in the literature. The basics of nonrelativistic theory are contained in any textbook on quantum mechanics, and the relativistic Dirac equation and the Lamb shift are discussed in any textbook on quantum electrodynamics and quantum field theory. An excellent source for the early results is the classic book by Bethe and Salpeter [6]. A number of excellent reviews contain more recent theoretical results, and a representative, but far from exhaustive, list of these reviews includes [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. 

The shaded area in this sketch is not arbitrary, as it is determined by the right-hand side of Eq. (587). The line integrals OA and AO change sign, and this accounts for reflection of matter waves and for the Sagnac and Young effects in matter waves, such as electron waves. Therefore, the electron is an 0(3) invariant entity, as shown by the Sagnac effect for electron waves [44]. It follows that the Dirac equation should be developed as an 0(3) invariant equation. 

The prediction, and subsequent discovery, of the existence of the positron, e+, constitutes one of the great successes of the theory of relativistic quantum mechanics and of twentieth century physics. When Dirac (1930) developed his theory of the electron, he realized that the negative energy solutions of the relativistically invariant wave equation, in which the total energy E of a particle with rest mass m is related to its linear momentum V by... 

Dirac s development of TDHF theory invokes the Heisenberg equation of motion for the density operator as a basic postulate,... 

For numerical evaluation (to sum over the entire spectrum of Dirac equation) B-splines are used [28], in particular the version developed by I.A. Goidenko [29]. Earlier the full QED calculations were carried out only for the ground (lsi/2)2 state He-like ions for the various nuclear charges Z. At that ones used either B-splines or the technique of discretization of radial Dirac equations [27]. As well as in [27] we used the Coulomb gauge. For control we reproduced the results of the calculation of (lsi/2)2 state and compared them with ones of [27]. Coulomb-Coulomb interaction is reproduced for every Z with the accuracy, on average, 0.01 %, Coulomb-Breit is with the accuracy 0.05 % and Breit-Breit (with disregarding retardation) is with the accuracy 0.1%. The small discrepancy is explained by the difference in the numerical procedures applied in [27] and in this work. 

Where the Schrodinger or Dirac equations apply, quantization will appear from sundry mathematical conditions on the existence of physically meaningful solutions to these differential equations. However, in nonrela-tivistic terms, spin does not come from a differential equation It comes from the assumptions of spin matrices, or from "necessity" (the Dirac equation does yield spin = 1/2 solutions, but not for higher spin). So we must posit quantum numbers (see Section 2.12) even when there are no differential equations in the back to "comfort us." This is especially true for the weak and strong forces, where no distance-dependent potential energy functions have been developed. 

The most unsatisfactory features of our derivation of the molecular Hamiltonian from the Dirac equation stem from the fact that the Dirac equation is, of course, a single particle equation. Hence all of the inter-electron terms have been introduced by including the effects of other electrons in the magnetic vector and electric scalar potentials. A particularly objectionable aspect is the inclusion of electron spin terms in the magnetic vector potential A, with the use of classical field theory to derive the results. It is therefore of interest to examine an alternative development and in this section we introduce the Breit Hamiltonian [16] as the starting point. We eventually arrive at the same molecular Hamiltonian as before, but the derivation is more satisfactory, although fundamental difficulties are still present. 

Relativistic effects may be also considered by other methods than pseudopotentials. It is possible to carry out relativistic all-electron quantum chemical calculations of molecules. This is achieved by various approximations to the Dirac equation, which is the relativistic analogue to the nonrelativistic Schrodinger equation. We do not want to discuss the mathematical details of this rather complicated topic, which is an area where much progress has been made in recent years and where the development of new methods is a field of active research. Interested readers may consult published reviews . A method which has gained some popularity in recent years is the so-called Zero-Order Regular Approximation (ZORA) which gives rather accurate results ". It is probably fair to say that... 

An overview of the development of the finite difference Hartree-Fock method is presented. Some examples of it axe given construction of sequences of highly accurate basis sets, generation of exact solutions of diatomic states, Cl with numerical molecular orbitals, Dirac-Hartree-Fock method based on a second-order Dirac equation. 

During the last few decades, general computer programs, based on the Dirac equation, for extensive four-component electronic structure calculations for atoms have been developed by several groups. These programs are being improved and extended continually, incorporating the explicit treatment of the Breit interaction as well as an ever more sophisticated consideration of interelectron correlation and even of QED effects. 

After the development of the Dirac equation one might have guessed that, within a framework in which the state of a many-electron system is described by a multi-Dirac spinor, the velocity factors Vj in (1.2) should simply be replaced by their formal counterparts in Dirac theory, viz, coj. This yields the operator... 

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