D.76 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

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

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.