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## Matrix Version

We can use matrix notation to re-do the problem above. The time-dependent Schrödinger equation in the original basis becomes
 (42)

Now, just as in eq. 3, we can see that if H was a diagonal matrix, then the equations for c1 and c2 would become decoupled. Again, we can make H diagonal by going to the special basis made of the eigenvectors of H. In this new basis, we will denote vector coefficients and matrix elements with tildes as a reminder that the basis set has changed, and we obtain:

 (43)

Now remember that we've gone to the eigenvector basis (which we'll also denote with tildes to distinguish it from the original basis), so we know that
 = (44) = (45)

Thus we can further simplify the diagonal elements as
 = (46) = = E1

for normalized basis functions. Likewise of course . Hence, we can expand our matrix equation 43 as
 = (47) = (48)

which is the same thing we got before when we assumed the given functions were orthonormal. The only difference is that here we emphasized the diagonalization of H rather than getting the eigenvectors, but of course it is the same process. These decoupled equations can be solved the same way as before to give
 = (49) = (50)

which we could write back in matrix notation (still in the eigenvector or tilde basis) as
 = (51)

We can identify the matrix as the propagator,

 = (52)

in the eigenvector basis.

In the vector/matrix representation, we can go from our original to our tilde coefficients and back as

 = (53) = (54)

where is the matrix made by making each column an eigenvector of in the original basis. We could transform our propagator from the eigenvector basis to the original basis by

 (55)

Next: Dirac Notation Version Up: Decoupling of Equations in Previous: Basis Functions in Coordinate
C. David Sherrill
2000-05-02