D.75 Solv­ing the NMR equa­tions

To solve the two cou­pled or­di­nary dif­fer­en­tial equa­tions for the spin up and down prob­a­bil­i­ties, first get rid of the time de­pen­dence of the right-hand-side ma­trix by defin­ing new vari­ables $A$ and $B$ by

$$a=Ae^{{\rm i}\omega t/2}, \quad b=Be^{-{\rm i}\omega t/2}.$$

Then find the eigen­val­ues and eigen­vec­tors of the now con­stant ma­trix. The eigen­val­ues can be writ­ten as $\pm{\rm i}\omega_1$$\raisebox{.5pt}{$/$}$​$f$, where $f$ is the res­o­nance fac­tor given in the main text. The so­lu­tion is then

$$\left(\begin{array}{c}A\\ B\end{array}\right) = C_1 \vec ... ...e^{{\rm i}\omega_1t/f} + C_2 \vec v_2 e^{-{\rm i}\omega_1t/f}$$

where $\vec{v}_1$ and $\vec{v}_2$ are the eigen­vec­tors. To find the con­stants $C_1$ and $C_2$, ap­ply the ini­tial con­di­tions $A(0)$ $\vphantom0\raisebox{1.5pt}{$=$}$ $a(0)$ $\vphantom0\raisebox{1.5pt}{$=$}$ $a_0$ and $B(0)$ $\vphantom0\raisebox{1.5pt}{$=$}$ $b(0)$ $\vphantom0\raisebox{1.5pt}{$=$}$ $b_0$ and clean up as well as pos­si­ble, us­ing the de­f­i­n­i­tion of the res­o­nance fac­tor and the Euler for­mula.

It's a mess.