Spin operator general

The first theoretical handling of the weak R-T combined with the spin-orbit coupling was carried out by Pople [71]. It represents a generalization of the perturbative approaches by Renner and PL-H. The basis functions are assumed as products of (42) with the eigenfunctions of the spin operator conesponding to values E = 1/2. The spin-orbit contribution to the model Hamiltonian was taken in the phenomenological form (16). It was assumed that both interactions are small compared to the bending vibrational frequency and that both the... 

Thus, the quantum-mechanical treatment of generalized angular momentum presented in Section 5.2 may be applied to spin angular momentum. The spin operator S is identified with the operator J and its components Sx, Sy, Sz with Jx, Jy, Jz- Equations (5.26) when applied to spin angular momentum are... 

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. 

We will first give a discussion of some results of general spin-operator algebra not much is needed. This is followed by a derivation of the requirements spatial functions must satisfy. These are required even of the exact solution of the ESE. We then discuss how the orbital approximation influences the wave functions. A short qualitative discussion of the effects of dynamics upon the functions is also given. 

Here the components of excited state J are expressed in a representation that diagonalizes the spin-orbit operator. In general, this will be a complex representation. The principle of spectroscopic stability can again be used to express the components of Jin a representation that we denote jM. This representation is made up of space and spin parts where the spin part diagonalizes the spin operator. 

In general, a spin Hamiltonian involving only electron-spin and nuclear-spin operators can be found that will satisfactorily account for the experimental results. The spin Hamiltonian has become the crossroad for the path followed by the experimentalist and the theorist. Experimentally, the spin Hamiltonian and its constants are determined from the ESR spectra, whereas, theoretically, the spin Hamiltonian and its constants are computed from the wave function of the ion. 

In magnetic resonance we are often confronted with the problem of obtaining a solution to a Hamiltonian which has only spin operators. To find the allowed energies and eigenfunctions, we generally start out with a convenient set of spin functions < , which represent the spin system but are not eigenfunctions of the Hamiltonian,. The eigenfunction ifi can, however, be constructed from a linear sum of the s ... 

S denotes a vector operator comprising three components Sx, Sy, and Sz. Note that generally a spin Hamiltonian replaces all the orbital coordinates required to define the system by spin coordinates. With the total spin operator S = SA + SB and S2 = SA + Sg + 2SaSb, the Hamiltonian can be rewritten as... 

We recall from section 5.2.4 that, from the general theory of angular momentum, j can take half-integral (more strictly half-odd) values as well as integral ones. The particular case of j = 1 /2 deserves special mention because of its importance in the discussion of electron or proton spin. For j = 1/2, there are two possible states l/2, 1/2) and 11/2, -1/2) which are often denoted a) and /3) respectively. The spin operators which define these states are particularly simple. For example,... 

Higher terms, biquadratic in the spin operators opS, also can occur.18 Spin-spin anisotropic interactions of the type D [S S2] are known for electrons,19 where D is a parameter vector this expression is the antisymmetric part of the most general bilinear spin-spin interaction. Observation of a field-induced magnetization in SrCu2(B03)2 via 11B NMR has revealed the presence of such an interaction therein.20... 

It appears that the three-space basis associated with each spin vector operator is arbitrary. Thus operator opS (and/or opI) need not necessarily be taken as quantized along some obvious physical direction, such as that of an applied magnetic field B. In other words, spin operators opS and opI need not be expressed in the same space, that is to be quantized along the same spatial directions (i.e. the spin projection quantum numbers may be measured along different selected directions in our three-space). The most general case, which occurs when the two quantization axes are not aligned, prevents the parameter matrices from being tensors. 

We apply to this equation the same remarks as those adopted for the comparison among Eqs. (316)—(318). We note, first of all, that the structure of Eq. (325) is very attractive, because it implies a time convolution with a Lindblad form, thereby yielding the condition of positivity that many quantum GME violate. However, if we identify the memory kernel with the correlation function of the 1/2-spin operator ux, assumed to be identical to the dichotomous fluctuation E, studied in Section XIV, we get a reliable result only if this correlation function is exponential. In the non-Poisson case, this equation has the same weakness as the generalized diffusion equation (133). This structure is... 

In Chapter 11 we shall also introduce the product operator formalism, in which the basic ideas of the density matrix are expressed in a simpler algebraic form that resembles the spin operators characteristic of the steady-state quantum mechanical approach. Although there are some limitations in this method, it is the general approach used to describe modern multidimensional NMR experiments. 

Consider now the two-spin system, in which chemical shifts and scalar coupling come into play. In Chapter 6 we discussed the two-spin system in detail, both the weakly coupled AX system and the general case, AB. To illustrate the application of the density matrix, we concentrate first on the AX system and then indicate briefly how the results would be altered for AB. To simplify the notation, we call the nuclei I and S, rather than A and X, and use the common notation in which the spin operators and their components are designated, for example, Ix and Sx, rather than the more cumbersome 4(A) or /. Although the I-S notation is usually applied to heteronuclear spin systems, we use it here to include homonuclear systems (e.g., H-H) as well. 

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