(42) |

Now, just as in eq. 3, we can see that if

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) | ||

= | |||

= | E_{1} |

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

= | (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,

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) |