# What is Rabi nutation in NMR?

What is the mathematical and physical significance of Rabi Nutation in terms of NMR?

Say we apply a constant magnetic field to a sample of hydrogen nuclei along the z direction, the Hamiltonian is given by: $H=-m \cdot B$. The quantized energy states of an individual proton nuleus would be $\pm \frac{g_n u_N B}{2}$.
So let's say we also apply a time dependent magnetic field in the x direction (RF and about 15Mhz for this example): $$B=B_1cos(\omega t) \hat{i}$$ Our Schrö­din­ger equa­tion is : $$i\frac{d}{dt}\left(\begin{array}{cc} a(t)e^{-iw_+t} \\ b(t)e^{-iw_-t} \end{array} \right) = \left( \begin{array}{cc} \omega_+ & \frac{\omega_R}{2}(e^{-i\omega t}+e^{i\omega t} \\ \frac{\omega_R}{2}(e^{-i\omega t}+e^{i\omega t}) & \omega_-\\ \end{array} \right) \left( \begin{array}{cc} a(t)e^{-iw_+t} \\ b(t)e^{-iw_-t} \\ \end{array} \right)$$ Where $\omega_R$ is the Rabi frequency and $\omega_\pm$ are the original energy eigenvalues divided by $\hbar$. Now the idea with NMR is to tune the magnetic field frequency to $\omega_+-\omega_-$ (the energy spacing$/\hbar$) of the original system. If you can set up your experiment to (roughly) do that, (and solve the DE's with a Rotating Wave approximation) you get these results for the time dependence of the samples' population:
$a(t)=cos(\frac{\omega_R t}{2})$ and $b(t)=isin(\frac{\omega_R t}{2})$
So the population of the two states will oscillate at half the rabi frequency. Knowing this frequency we can control the bulk magnetization by applying timed pulses. For example, by applying a $\frac{\pi}{2}$ pulse followed by a $\pi$ pulse with the right timing, (see Spin Echo) you can get the characteristic relaxation time of the sample, which is the useful information used in MRI's.