Pauli spin operators

The Pauli spin operators, S, encountered in Margenau and Murphy, p. 402, are another example. 

The Dirac operators a, . ..04, work on the basis vectors ei,. ..04 just as the Pauli spin operators work on the spin eigenvectors a, / . The only other property to be noted is that the Dirac spin-orbitals have two large components and two small components , their ratio being of the order (2moc)" ... 

PROBLEM 3.6.5. Define the 4x4 Dirac spin operator a from the three 2x2 Pauli spin operators [Pg.153]

It is often convenient to use the Pauli spin operator a = 2S, whence ... 

The Pauli spin operator for the component at an angle 9 with respect to the z-axis is given by... 

There are several ways of reducing the one-electron relativistic equation (11.2.7) to a form involving the Pauli spin operators. The method we use here is due to Lowdin (1964) and utilizes the matrix partitioning technique of Section 2.5. 

This is an example of Eq. (8-197). The ensemble average of positive spin represented by the operator az, or the Pauli spin matrix Q is quite trivially... 

The spin operators may be taken to be the Pauli spin matrices.7... 

Spin operators, taken as Pauli spin matrices, 730... 

The wave funetion obtained eorresponds to the Unrestricted Hartree-Fock scheme and beeomes equivalent to the RHF ease if the orbitals (t>a and (()p are the same. In this UHF form, the UHF wave funetion obeys the Pauli prineiple but is not an eigenfunction of the total spin operator and is thus a mixture of different spin multiplicities. In the present two-eleetron case, an alternative form of the wave funetion which has the same total energy, which is a pure singlet state, but whieh is no longer antisymmetric as required by thePauli principle, is ... 

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... 

One of the pedagogically unfortunate aspects of quantum mechanics is the complexity that arises in the interaction of electron spin with the Pauli exclusion principle as soon as there are more than two electrons. In general, since the ESE does not even contain any spin operators, the total spin operator must commute with it, and, thus, the total spin of a system of any size is conserved at this level of approximation. The corresponding solution to the ESE must reflect this. In addition, the total electronic wave function must also be antisymmetric in the interchange of any pair of space-spin coordinates, and the interaction of these two requirements has a subtle influence on the energies that has no counterpart in classical systems. 

Though the true electron spin operators were employed here as well as in the Breit-Pauli Hamiltonian, the phenomenological Spin Hamiltonian, in which the spin coupling is an exchange effect, is in sharp contrast to the Breit-Pauli Hamiltonian, that is including the (magnetic) spin-spin interactions. Since the exchange effect is an effect introduced by the Pauli principle imposed on the wave function, we may write the electron-electron interaction as an expectation value,... 

Of course, the Spin Hamiltonian as given in Eq.(73) could also be directly derived from Eq. (77) for Dirac (81) pointed out that any permutation operator can be written in terms of vectors of Pauli spin matrices Oj and q,- as... 

The spin operators Sx, and Sg which occur in the Breit-Pauli Hamiltonian form a basis for the Lie algebra of SU(2). The concept of the electron as a spinning particle has arisen through the isomorphism between SU(2) and the angular momentum operators. This analogy is unnecessary and often undesirable. 

SX Sy Sz spin operators whose matrix representatives are the Pauli matrices... 

In quantum mechanics, spin is described by an operator which acts on a spin wavefunction of the electron. In the present case this operator describes an angular momentum with two possible eigenvalues along a reference axis. The first requirement fixes commutation rules for the spin components, and the second one leads to a representation of the spin operator by 2 x 2 matrices (Pauli matrices [Pau27]). One has... 

Scalar relativistic DKH2 approach with Pauli spin-orbit operator, reported in [ 114] with respect to WF6... 

Pauli introduced slightly different spin operators known as the Pauli spin matrices. They are defined by... 

A very useful equation employing the Pauli spin matrices is the so-called Dirac relation. For any pair of vector operators u and v... 

The Breit-Pauli spin-orbit operator has one major drawback. It implicitly contains terms coupling electronic states (with positive energy) and posi-tronic states (in the negative energy continuum) and is thus unbounded from below. It can be employed safely only in first-order perturbation theory. 

As may be seen by comparing Eqs. [103] and [105], the no-pair spin-orbit Hamiltonian has exactly the same structure as the Breit-Pauli spin-orbit Hamiltonian. It differs from the Breit-Pauli operator only by kinematical factors that damp the 1/rfj and l/r singularities. 

---