Effect of Free Precession

We now look at the evolution of the density matrix for the one spin system as the magnetization precesses in the rotating frame. Once more, we apply Eq. 11.16 in the same manner we did in Section 11.2 to take into account the effect of the rotating frame. In this case we obtain 

Keeping in mind the points noted in setting up the matrices in Eq. 11.39. We obtain the following relation  

The variation of the magnetization with time can be found by applying Eq. 11.6 to each of the components of M. (A ) remains zero, because the product matrix Mzp(t) is easily shown to be traceless. Rather than computing [Mx) and (My) separately, it is easier and more informative to find (Mxy), where 

The complex notation is consistent with our discussion of phase-sensitive detection in Section 3.4, where the real component represents the projection along the x axis and the imaginary component that along the y axis. 

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