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The effect of a radio frequency pulse is to "mix" the a and f states so that a superposition of states is obtained:

For a single spin, the probability of finding it in a given "pure" energy state is found by multiplying the coefficient of that state by its complex conjugate: the probability of finding the spin in the a state is thus ci-c*, and the probability that the spin will be in the f state is c2-c2. Since a spin can only be in either the a or the f state, the sum of these two probabilities (c1-c* + c2-c2) is 1 (a certainty that it will be in one state or the other). When these probabilities are averaged over the entire population of spins we have the populations of the a and f states. The "cross products" c*-c 2 and c2-c* represent the amount of mixing or superposition of the a and f states. When these are averaged over the entire population of spins we have the single-quantum coherence of the sample, which is the measurable magnetization that we represent as a vector in the x'-y' plane in the vector model. The state of the entire system can be described by the averages of these four products over the whole sample, and these are conveniently represented as a matrix:

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