The intermediate product of the last two matrices is shown to illustrate the matrix multiplication. Along the diagonal of the resultant matrix o\ you can still see the carbon z magnetization, but the proton z magnetization is gone (compare to Sz). The off-diagonal elements correspond to in-phase proton magnetization on the —y axis (compare Iy).

To calculate the effect of the 1/(2/) delay we need to know the energy differences among the four spin states aias, ai js, jias, and jjs so that the phase factors eMt and e—mt can be applied to the off-diagonal elements of the density matrix. The frequencies for the aIas — jias(1,3) and aj —^ jijs (2,4) transitions differ only by the coupling constant /ch-

and the phase factors are el(w'+nJ)T for the (1,3) transition and el(w'—nJ)T for the (2,4) transition (Fig. 10.37). With a delay of t = 1/(2/) and for an on-resonance peak (mi = 0), these factors become en/2 and e—n/2 or simply i and —i, respectively. The elements above the diagonal have phase factors of e—l(w'+nJ)T for the (1,3) transition and e—l(w'—nJ)t for the (2,4) transition, which become e—n/2 and em/2 or simply —i and i. Thus the density matrix becomes:

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