Spins lowering operator

Just like the electron spin-raising and spin-lowering operators defined in Equations 7.15 and 7.16, we have the analogous operators for the nuclear spin I ... 

The wave function for the next spin component of the 6 A state can be easily generated from (7.271) by application of the total spin lowering operator S, given by... 

The raising and lowering operators for spin angular momentum as defined by equations (5.18) are... 

Our next goal is to transform this expression into one based on the total electron spin operator, S = si + s2. The first three terms can be simplified by making use of the identity (derived using raising and lowering operators) ... 

If we expand the hyperfine term of the spin Hamiltonian and write the operators in terms of raising and lowering operators ... 

The standard procedure used to solve Eq. (29) is to transform the spin operators into fermionic operators [61]. Let us define the raising and lowering operators... 

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. 

Analogous to (1.169), the electron-spin raising and lowering operators are... 

The spin-lattice relaxation process is usually exponential. Theoretically, the effect of spin-diffusion, characterized by the coefficient D (order of 1(T12 cm2 s 1), has an influence on T, relaxation times when ix > L2/D, where Lis the diffusion path length. NMR studies of model systems f6r rubber networks, based on a styrene-butadiene-styrene block copolymer (SBSy, in which styrene blocks act as a crosslink for polybutadiene rubber segments of known and uniform length, indicate that spin diffusion operating between PS and PB phases causes a lowering of Tg for the PS component in SBS (as compared to the pure PS) and hindering of the motion of the PB component (as compared to the pure PB)51). 

We can define another type of operator that describes the spin state without reference to the x and y (Cartesian) axes. These are the raising and lowering operators that refer to coherence in terms of the transitions between spin states. For example, 1+ refers to the transition of spin I from the a to the fi state, while the S spin does not change state. For the ll, 13C spin system, this is a transition from the n c state to the /3h c state, or from the n/lc state to the state (Fig. 10.27). Likewise, the S coherence refers to the transition from... 

The symbols we have been using to represent spin states (I, S-y, 2IySz, etc.) of the entire ensemble of spins are actually operators they can operate on a spin state (of a single spin pair in our Ha, Hb system) and spit out another spin state. We already saw this with the raising and lowering operators ... 

PROBLEM 3.5.6. One can define linear ladder operators for angular momentum (orbital or spin) the raising operator + = Lx + iLy and the lowering operator =LX — iLy. (a) Verify that brute-force expansion yields + =... 

Each of the terms A — F in Eq. 7.5 converts one of the basis functions— aa, a(3, (3a, (3(3—to the same or another function and is often said to link such pairs of functions. For example, A converts aa to aa because a is an eigenfunction of Iz. (a) Use the information in Section 2.3 to show the linkages for all four basis functions, (b) Convert the spin operators in term B to raising and lowering operators to demonstrate why this term is often called the flip-flop term. 

Because the raising and lowering operators change spin quantum number by one, the first two terms on the right account for double quantum coherence, and the last two flip-flop terms give zero quantum coherence. Clearly, cross products of any two transverse spin components lead to the same result. Moreover, we can represent two orthogonal forms of pure zero quantum and double quantum coherences by combinations of product operators ... 

The attractiveness of surface/pore characterization via NMR spin-lattice relaxation measurements of pore fluid lies in the potential advantages this technique has as compared to the conventional approaches. These include rapid analysis, lower operating costs, analysis of wet materials, no pore shape assumption, a wide range of pore sizes can be evaluated (0.5 nm to >1 /im), no network/percolation effects and the technique is non-destructive. When determining specific surface areas, NMR analysis does not require out-gassing and has the potential for on-line analysis of slurries. 

This Hamiltonian contains the spin operator and in Table 2.1 above this can be rewritten in terms of the raising and lowering operators as />. = so that... 

The operator product 27, xI2x can be expressed in terms of the raising and lowering operators in the following way (note that separate operators are used for each spin 71 and 72 )... 

For double- and zero-quantum coherence in which spins / and j are active it is convenient to define the following set of operators which represent pure multiple quantum states of given order. The operators can be expressed in terms of the Cartesian or raising and lowering operators. 

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