Matrix and Spin Operators

We saw in previous sections that it is often possible to express the density matrix for a particular situation in a more concise operator form. For example, in Eq. 11.37 we saw that the density matrix at equilibrium is given by (e/2)Iz, and in Eq. 11.57 we showed that application of a nonselective 90 x pulse to a two-spin system generated a density matrix that could be expressed as (e/2)Iy + (es/2)Sy. To make the expressions even simpler, we can drop the normalizing factors and for heteronuclear systems incorporate the relative magnitudes given by the e s into the spin operators themselves. For this example, we then obtain simply Iy + Sr 

Let s now look at the density matrix as it evolves for the pulse sequences that we outlined in Section 11.5. We have illustrated INEPT and related pulse sequences by the two-spin system in which I = H and S = 13C.The equilibrium density matrix can be considered as the sum of two matrices  

But the first matrix of the sum is just Iz (multiplied by 4 to account for the yH/ yc ratio) and the second is Sz. As indicated earlier, we have also dropped the /2 normalizing factors. (The matrices for a two-spin system are given in Appendix D.) So we can write 

Consider now the density matrix at the conclusion of the evolution periods and after the 180 applied to both H and 13C. As shown in Eq. 11.63 (and in the following), the S populations are inverted and can obviously be represented simply by — S. However, the expressions for the I coherences have some, but not all, signs changed from p(2) hence they cannot be represented by Ix or its negative. It turns out that the product IXSZ works  

We saw in Eq. 11.65 that p(5) consists of two 13C coherences of opposite sign, so we expect something similar to the product in Eq. 11.74. Actually, we find 

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