Electron-spin operator components

In the limit of infinite atom separations, or if we switch off the Coulomb repui. sion between two electrons, all four wavefunctions have the same energy. But they correspond to different eigenvalues of the electron spin operator the first combination describes the singlet electronic ground state, and the other three combinations give an approximate description of the components of the first triplet excited state. 

Sharp and Lohr proposed recently a somewhat different point of view on the relation between the electron spin relaxation and the PRE (126). They pointed out that the electron spin relaxation phenomena taking a nonequilibrium ensemble of electron spins (or a perturbed electron spin density operator) back to equilibrium, described in Eqs. (53) and (59) in terms of relaxation superoperators of the Redfield theory, are not really relevant for the PRE. In an NMR experiment, the electron spin density operator remains at, or very close to, thermal equilibrium. The pertinent electron spin relaxation involves instead the thermal decay of time correlation functions such as those given in Eq. (56). The authors show that the decay of the Gr(T) (r denotes the electron spin vector components) is composed of a sum of contributions... 

The Hamiltonian (1) is spin free, commutative with the spin operator S2 and its z-component Sz for one-electron and many-electron systems. The total spin operator of the hydrogen molecule relates to the constituent one-electron spin operators as... 

The solid effect (SE) is a DNP mechanism which requires states mixing caused by the nonsecular component KpsE of the hyperfine coupling [16]. The pseudosecular term /i psE contains the form EzN and EzN (E Electron spin operator, N Nucleus spin operator), which leads to a mixing of the states of the system. In SE polarization, the new mix states are generated from the original states with a coefficient p, which can be calculated by the first order perturbation theory and is given by... 

The relationship between different components of orbital angular momentum such as Lz and Lx can be investigated by multiple SG experiments as discussed for electron spin and photon polarization before. The results are in fact no different. This is a consequence of the noncommutativity of the operators Lx and Lz. The two observables cannot be measured simultaneously. While total angular momentum is conserved, the components vary as the applied analyzing field changes. As in the case of spin or polarization, measurement of Lx, for instance, disturbs any previously known value of Lz. The structure of the wave function does not allow Lx to be made definite when Lz has an eigenvalue, and vice versa. 

The summation index n has the same meaning as in Eq. (31), i.e., it enumerates the components of the interaction between the nuclear spin I and the remainder of the system (which thus contains both the electron spin and the thermal bath), expressed as spherical tensors. are components of the hyperfine Hamiltonian, in angular frequency units, expressed in the interaction representation (18,19), with the electron Zeeman and the ZFS in the zeroth order Hamiltonian Hq. The operator H (t) is evaluated as ... 

Electrons (and many other particles) have associated with them an intrinsic angular momentum that has come to be called spin . One of the greatest successes of relativistic quantum mechanics is that spin is seen to arise naturally within the relativistic formalism, and does not need to be added post facto as it is in non-relativistic treatments. As with orbital angular momentum, spin angular momentum has x, y, and z components, and the operators 5, Sy, and S, together with orthonormal eigenfunctions a and fi of electron spin, are defined from ... 

Electron spin may interact directly 22,67,123 with the oscillating electromagnetic field of a photon. In this case the transition operator,, /(t) of Section IV, does not commute with the symmetric group S f, so that direct transitions between pure singlet and pure triplet states are allowed. Selection rules for this type of transition may be derived, as has been done by Chiu,22 by noting that f(t) has components which transform as different representations of the double group SF 0 (see Sect. VIII). 

A vectorial product will be defined below by (5.14), and V as a tensor of first rank is defined by (2.12). Operator L may be defined also in a more general way by the commutation relations of its components. Such a definition is applicable to electron spin s, as well. Therefore, we can write the following commutation relations between components of arbitrary angular momentum j ... 

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... 

The spin functions a and (3 (i.e., the components of a spin doublet) belong to the Ej/2 irrep of C2v- But what about singlet and triplet spin functions For this purpose, we look at the action of the symmetry operators on typical two-electron spin functions such as aa and ap. The results, displayed in Table 6, are easily verified. 

Sx, Sy, and Sz are the three components of the spin operator, D and E are the anisotropy constants which were determined via high-frequency electron paramagnetic resonance (D/kB 0.275 K and E/kB 0.046 K [10]), and the last term of the Hamiltonian describes the Zeeman energy associated with an applied field H. 

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