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depending on the value of mT. Note that these matrices correspond to the product operators Ix, Iy, — Ix, and —Iy, which is the expected progression for chemical-shift evolution. If you focus on the numbers at the lower left of each matrix you can "read" the product operators if you associate 1 with x and i with y. This is a lot of work to carry out the equivalent of

Ix(T i>2m)) Iy, but properly programed computers just love this sort of thing and have no trouble keeping track of it all.

For a two-spin system (e.g., 13C-XH, 13C = S, and XH = I) each pair of spins can be represented by a superposition of the four "pure" states aIaS, a^S, fa^S, and faI faS. In the heteronuclear case the energy difference for the S transitions (aIaS ^ aIfaS and faIaS ^ will be different from the energy difference for the I transitions (aIaS ^ faIaS and aIfaS ^ faIfaS). The wave function for one spin pair is thus

^ = ci aa + C2 afa + C3 fai as + C4 fafa and the probabilities and coherences can be represented by a 4 x 4 matrix:

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