Dirac equation, spin

It took some time until it was realized that the Dirac theory describes the spin correctly because it is a spinor-field theory, and not because it is relativistic [16]. In fact, if one takes the nonrelativistic limit of the Dirac equation, spin survives, and this is consistent with the observation that the Galilei group has spinor representations as well. So, without any doubt, spin is not a relativistic effect. 

The most common description of relativistic quantum mechanics for Fermion systems, such as molecules, is the Dirac equation. The Dirac equation is a one-electron equation. In formulating this equation, the terms that arise are intrinsic electron spin, mass defect, spin couplings, and the Darwin term. The Darwin term can be viewed as the effect of an electron making a high-frequency oscillation around its mean position. 

The Dirac equation automatically includes effects due to electron spin, while this must be introduced in a more or less ad hoc fashion in the Schrodinger equation (the Pauli principle). Furthermore, once the spin-orbit interaction is included, the total electron spin is no longer a good quantum number, an orbital no longer contains an integer number of a and /) spin functions. The proper quantum number is now the total angular momentum obtained by vector addition of the orbital and spin moments. 

Neglect of relativistic effects, by using the Schrodinger instead of the Dirac equation. This is reasonably justified in tlie upper part of the periodic table, but not in the lower half. For some phenomena, like spin-orbit coupling, there is no classical counterpart, and only a relativistic treatment can provide an understanding. 

In summary, the OP-term introduced by Brooks and coworkers has been transferred to a corresponding potential term in the Dirac equation. As it is demonstrated this approach allows to account for the enhancement of the spin-orbit induced orbital magnetic moments and related phenomena for ordered alloys as well as disordered. systems by a corresponding extension of the SPR-KKR-CPA method. 

Particles spin Vz, 517 Dirac equation, 517 spin 1, mass 0,547 spin zero, 498 Partition function, 471 grand, 476 Parzen, E., 119,168 Pauli spin matrices, 730 PavM, W., 520,539,562,664 Payoff, 308 function, 309 discontinuous, 310 matrix, 309... 

Despite the complication due to the interdependence of orbital and spin angular momenta, the Dirac equation for a central field can be separated in spherical polar coordinates [63]. The energy eigenvalues for the hydrogen atom (V(r) = e2/r, in electrostatic units), are equivalent to the relativistic terms of the old quantum theory [64]... 

Abstract. The Dirac equation is discussed in a semiclassical context, with an emphasis on the separation of particles and anti-particles. Classical spin-orbit dynamics are obtained as the leading contribution to a semiclassical approximation of the quantum dynamics. In a second part the propagation of coherent states in general spin-orbit coupling problems is studied in two different semiclassical scenarios. 

Keywords Dirac equation, semiclassics, spin-orbit coupling, coherent states... 

Spin-orbit coupling problems are of a genuine quantum nature since a priori spin is a quantity that only occurs in quantum mechanics. However, already Thomas (Thomas, 1927) had introduced a classical model for spin precession. Later, Rubinow and Keller (Rubinow and Keller, 1963) derived the Thomas precession from a WKB-like approach to the Dirac equation. They found that although the spin motion only occurs in the first semiclassical correction to the relativistic classical electron motion, it can be expressed in merely classical terms. 

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. 

Semiclassical studies of the propagation of coherent states have proven useful in many circumstances see, e.g., (Klauder and Skagerstam, 1982 Perelomov, 1986). Here we consider spin-orbit coupling problems that result from the Dirac equation either in a semiclassical or in a non-relativistic approximation (see, e.g., the Hamiltonians (30) and (31)). The Hamiltonians H that arise in such a context can be viewed as Weyl quantisations of symbols... 

A particularly interesting feature of the theory [9] is the incorporation of deviations from Coulomb scattering due to the nonvanishing size of the projectile nucleus. The very fact that the theory is based on the Dirac equation and that spin dependences enter nontrivially indicates that quantum mechanics is essential here. Moreover, at the highest energies considered, pair production becomes important, i.e., an effect that does not have a classical equivalent [57]. 

Four-component relativistic molecular calculations are based directly on the Dirac equation. They include both scalar relativistic effects and spin-orbit... 

We now consider how to eliminate either all relativistic effects or exclusively the spin-orbit interaction from the relativistic Hamiltonian. We start from the Dirac equation in the molecular field... 

We now consider how to eliminate the spin-orbit interaction, but not scalar relativistic effects, from the Dirac equation (25). The straightforward elimination of spin-dependent terms, taken to be terms involving the Pauli spin matrices, certainly does not work as it eliminates all kinetic energy as well. A minimum requirement for a correct procedure for the elimination of spin-orbit interaction is that the remaining operator should go to the correct non-relativistic limit. However, this check does not guarantee that some scalar relativistic effects are eliminated as well, as pointed out by Visscher and van Lenthe [44]. Dyall [12] suggested the elimination of the spin-orbit interaction by the non-unitary transformation... 

The second term on the right-hand side of the equation gives for point nuclei directly the one-electron spin-orhit operator (2) of the Breit-Pauli Hamiltonian and can he eliminated to give a spin-free equation that becomes equivalent to the Schrddinger equation in the non-relativistic limit. In a quaternion formulation of the Dirac equation the elimination becomes particularly simple. The algebra of the quaternion units is that of the Pauli spin matrices... 

In the quaternion modified Dirac equation the spin-free equation is thereby obtained simply by deleting the quaternion imaginary parts. For further details, the reader is referred to Ref. [13]. 

We then turn to the question of how to eliminate the spin-orbit interaction in four-component relativistic calculations. This allows the assessment of spin-orbit effects on molecular properties within the framework of a single theory. In a previous publication [13], we have shown how the spin-orbit interaction can be eliminated in four-component relativistic calculations of spectroscopic properties by deleting the quaternion imaginary parts of matrix representations of the quaternion modified Dirac equation. We show in this chapter how the application of the same procedure to second-order electric properties takes out spin-forbidden transitions in the spectrum of the mercury atom. Second-order magnetic properties require more care since the straightforward application of the above procedure will extinguish all spin interactions. After careful analysis on how to proceed we... 

It is also common in the literature to write the time-independent Dirac equation in terms of Pauli-spin matrices... 

A detailed study of the Dirac equation and its solutions will not be required it will simply be assumed, as already indicated, that the S3rstem of N electrons above the negative-energy sea may be described using a wavefunction constructed from antisymraetrized products of (positive energy) spin-orbitals of type (29). It is, however, necessary to know the basic properties of the operators Q/i, which appear in the Dirac equation... 

Here we have used the natural expansion (33), with spin-orbitals written in the form (29). The second term in (41), absent in a Pauli-type approximation, contains the correction arising from the use of a 4roomponent formulation it is of order (2tmoc) and is usually negligible except at singularities in the potential. As expected, for AT = 1, (41) reproduces the density obtained from a standard treatment of the Dirac equation but now there is no restriction on the particle number. 

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