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In order to probe the transitions of a nuclear spin $s$ from one of itssubstate quantum numbers $m$ to another substate $m'$, the experimenter appliesa magnetic field ${\bm B}_0$ in some particular direction, such along $\hat{\bmz}$, and then applies an weaker field ${\bm B}_1(t)$ that is oscillatory intime with the angular frequency $\omega$, and is normally perpendicular to${\bm B}_0$, such as ${\bm B}_1(t)=B_1\hat{\bm x}\cos(\omega t)$. In therotating wave approximation, ${\bm B}_1(t)=B_1[\hat{\bm x}\cos(\omegat)+\hat{\bm y}\sin(\omega t)]$. Although this problem is solved for spin$\frac{1}{2}$ in every quantum mechanics textbook, for the general spin $s$case, its general solution has been published only for the overall probabilityof a transition between the states, but the time dependence of the probabilityof finding the nucleus in each of the substates has not previously beenpublished. Here we present an elementary method to solve this problem exactly,and present figures for the time dependencies of the various substates statesfor a variety of initial substate probabilities for a variety of $s$ values. Wefound a new result: unlike the $s=\frac{1}{2}$ case, for which if the initialprobability of finding the particle in one of the substates was 1, and the timedependence of the probabilities of each of the substates oscillates between 0and 1, for higher spin values, the time dependencies of the probabilitiesfinding the particle in each of its substates, which periodic, is considerablymore complicated.

Nuclear Magnetic Resonance for Arbitrary Spin Values in the Rotating Wave Approximation