Separation of orbit and spin

We have considered only a spin component. In a general case there is also an orbital contribution to the observed structure factor so that the moment density will actually be determined. The orbital contribution is important for non-orbitally singlet transition metal ions and for rare earth metals and ions. The separation of orbital and spin components has been discussed by Moon (74) and Marshall and Lovesey (1). 

Separation of orbit and spin in holmium 34 5.2.1. Magnetism and superconductivity ... 

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]... 

The second term s may be called the operator for spin angular momentum of the photon. However, the separation of the angular momentum of the photon into an orbital and a spin part has restricted physical meaning. Firstly, the usual definition of spin as the angular momentum of a particle at rest is inapplicable to the photon since its rest mass is zero. More importantly, it will be seen that states with definite values of orbital and spin angular momenta do not satisfy the condition of transversality. 

In this section we will give a qualitative summary of the role of spin-orbit coupling in molecules. If the separation of spatial and spin parts of the wave function for a state was rigorously justifiable on the grounds that the spin and orbital motions of the electron were completely independent, then electronic transitions between states of different multiplicity would be rigorously forbidden. The ground state of such a system, < o, may be... 

Several factors determine the magnitude of the transition-moment integral in Eq. (1) and these are reflected in the observed oscillator strengths. The complete electronic wave function for the state of a molecule can be expressed as a product of orbital and spin wave fimctions for the molecule, and integration in Eq. (1) can be carried out separately over the orbital (space) part and the spin part of the integral. Transitions, therefore, can be weak if either part of the integrand in Eq. (1) is small. 

It would appear that identical particle pemuitation groups are not of help in providing distinguishing syimnetry labels on molecular energy levels as are the other groups we have considered. However, they do provide very usefiil restrictions on the way we can build up the complete molecular wavefiinction from basis fiinctions. Molecular wavefiinctions are usually built up from basis fiinctions that are products of electronic and nuclear parts. Each of these parts is fiirther built up from products of separate uncoupled coordinate (or orbital) and spin basis fiinctions. Wlien we combine these separate fiinctions, the final overall product states must confonn to the pemuitation syimnetry mles that we stated above. This leads to restrictions in the way that we can combine the uncoupled basis fiinctions. 

Equation (9-392) together with (9-394) and (9-395) are the proofs of the assertions that x is the position operator in the Foldy-Wouthuysen representation.16 (Note also that x commutes with /J the sign of the energy.) We further note that in the FTP-representation the operators x x p and Z commute with SFW separately and, hence, are constants of the motion. In the F W-representation the orbital and spin angular momentum operators are thus separately constants of the motion. The fact that... 

For systems of two or more particles orbital and spin angular momenta are added separately by the rules of vector addition. Integral projections of the shorter vector on the direction of the longer vector, are added to the long vector, to give the possible eigenvalues of total angular momentum. 

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. 

In the oxide MnO, Mn ions occupy octahedral holes in an oxide iattice. The degeneracy of the 3c/ levels of manganese are split into two, as for Ti ". The five d electrons of the Mn ions occupy separate d orbitals and have parallel spins. Explain why the absorption lines due to transitions between the two 3 c/levels are very weak for... 

Because electrons have a natural repulsion for one another, they do not begin to pair up in the same orbital until all the other orbitals at the same energy level are singly occupied. Electrons in separate orbitals tend to spin in the same direction, and so the arrows should be shown all pointing in the same direction until pairing is necessary. For these reasons, the two 2p electrons of a carbon atom in its lowest energy state are in separate 2p orbitals and are drawn pointing in the same direction ... 

Usually the Born-Oppenheimer separation of nuclear and electronic coordinates is assumed and small terms in the hamiltonian, such as spin-orbit coupling, are neglected in the first approximation. Perturbation... 

One wishes to calculate exactly the energy of A B — C (Eq. 3.50) relative to a situation where A and B — C are separated, in the effective VB Hamiltonian framework, as in the preceding exercise. Rewrite Equation 3.50 so that the two determinants exhibit maximum orbital and spin correspondence. Calculate the energies of the unnormalized determinants abc and acb, and the Hamiltonian matrix element ( afoc // acfo ). The following simplifications will be used... 

The spin selection rule, AS = 0, might be expected to be of universal applicability, since it does not require the molecule under consideration to have any geometrical symmetry. However, spin-forbidden transitions are also frequently observed. The spin rule is based again on the idea of separability of wavefunctions, this time of the spin and spatial components of the electronic wavefunction. However, the electron experiences a magnetic field as a result of the relative motion of the positive nucleus with respect to it, and this field causes some mixing of spatial and spin components, giving rise to spin-orbit... 

Let us now study exactly the same approach when we like to separate the space and spin functions, and let us assume that = (i. O2. 3. Is a linearly Independent orbital basis of order... 

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