### 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 and by

Then find the eigenvalues and eigenvectors of the now constant matrix. The eigenvalues can be written as , where is the resonance factor given in the main text. The solution is then

where and are the eigenvectors. To find the constants and , apply the initial conditions and and clean up as well as possible, using the definition of the resonance factor and the Euler formula.

It's a mess.