Algebra spin operators

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. 

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. 

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. 

As in previous chapters, we do not attempt to provide a rigorous treatment here, but rather develop the concept of a density matrix, which is often unfamiliar to chemists, and show how it may easily be used to understand the behavior of one-spin and two-spin systems. As in the treatment of complex spectra in Chapter 6, we shall see that the density matrix approach can be readily extended to larger spin systems, but with a great deal of algebra and often with little physical insight. However, in the course of treating the simple spin systems, we will notice that some of the results can be obtained more succinctly by certain manipulations of the spin operators Ix, Iy, and Iz, with which we are familiar. 

As we shall see, it is very helpful to be able to compute the behavior of the spin system from the sort of matrix multiplications that we have already carried out. On the other hand, it is often possible to simplify the algebraic expressions by using the corresponding spin operators. In fact, this is the concept of the product operator formalism that we discuss later. Note that from Eqs. 11.35 and 11.36, the (redefined) density matrix at equilibrium can be written in operator form as... 

Actually, the components of the bosonized SB spin operator act as generators of rotations in spin space, i.e., Si(pi ) satisfies the spin algebra. Since and commute with the SB t-J Hamiltonian, the constraints in Eq. 5 and Eq. 6 can be ensured by introducing the time-independent Lagrange multipliers and = consequence, the... 

For four-component wave functions it is the spin operator E that changes sign under time reversal, and by simple extension of the algebra above we can write the four-component operator as... 

The tliree protons in PH are identical aud indistinguishable. Therefore the molecular Hamiltonian will conmuite with any operation that pemuites them, where such a pemiutation interchanges the space and spin coordinates of the protons. Although this is a rather obvious syimnetry, and a proof is hardly necessary, it can be proved by fomial algebra as done in chapter 6 of [1]. 

BalK83 Balasubramanian, K. Operator and algebraic methods for NMR spectroscopy II. NMR projection operations and spin functions. J. Chem. Phys. 78 (1983) 6369-6376. 

In Chapter 1 we have discussed the familiar realization of quantum mechanics in terms of differential operators acting on the space of functions (the Schrodinger wave function formulation, also called wave mechanics ). A different realization can be obtained by means of creation and annihilation operators, leading to an algebraic formulation of quantum mechanics, sometimes called matrix mechanics. For problems with no spin, the formulation is done in terms of boson creation, b (, and annihilation, ba, operators, satisfying the commutation relations... 

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... 

The results of the current section, both the lowering operators and the classification, will come in handy in Section 8.4, where we classify the irreducible representations of so(4). One can apply the classification of the irreducible representations of the Lie algebra sm(2) to the study of intrinsic spin, as an alternative to our analysis of spin in Section 10.4. More generally, raising and lowering operators are widely useful in the study of Lie algebra representations. 

While finding the numerical values of any physical quantity one has to express the operator under consideration in terms of irreducible tensors. In the case of Racah algebra this means that we have to express any physical operator in terms of tensors which transform themselves like spherical functions Y. On the other hand, the wave functions (to be more exact, their spin-angular parts) may be considered as irreducible tensorial operators, as well. Having this in mind, we can apply to them all operations we carry out with tensors. As was already mentioned in the Introduction (formula (4)), spherical functions (harmonics) are defined in the standard phase system. 

In the general case, there is no simple algebraic expression for the first term of (5.42), therefore, for its evaluation one must utilize the tables of numerical values of the submatrix elements of operators Uk. The most complete tables, covering also the case of operators Vkl, may be found in [87], For operators, also depending on spin variables, the analogue of formula (5.41) will have the form... 

C. Dewdney, P. R. Holland, A. Kyprianidis, Z. Marie, and J. P. Vigier, Stochastic physical origin of the quantum operator algebra and phase space interpretation of the Hilbert space formalism the relativistic spin-zero case, Phys. Lett. A 113A(7), 359-364 (1988). 

A systematic investigation of the application of Lie algebra to NMR was presented.29 The symmetry properties of the nuclear spin systems were naturally included in selection of the sets of the basis operators. With this theoretical framework, the existing sets of basis operators used for various specific purposes can be treated in a unified manner and their respective advantages and disadvantages can be evaluated. A number of 2H MAS spectra calculated on the basis of that theoretical framework are shown in Fig. 2. The... 

Well-known realizations of the generators of this Lie algebra are given by the three components of the orbital angular momentum vector L = r x p, the three components of the spin S = a realized in terms of the Pauli spin matrices (Schiff, 1968), or the total one-electron angular momentum J = L + S. The components of each of these vector operators satisfy the defining commutation relations Eq. (4) if we use atomic units. We should also note that the vector cross-product example mentioned earlier also satisfies Eq. (4) if we define E = iey, j = 1, 2, 3. 

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