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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.
C. David Sherrill