Summary

A coupled system of differential equations was solved by writing the equations in matrix form, diagonalizing the matrix, and changing to the eigenvector basis which decoupled the equations and made them easily solvable. The propagator was easily obtained in the eigenvector basis. The problem is then solved either by (i) finding the components of the initial state in the eigenvector basis, and writing the solution in the eigenvector basis, or (ii) keeping the initial state in the original basis and instead transforming the propagator from the eigenvector basis into the original basis.