(32) |

This leads to the time-dependent Schrödinger equation (where we will suppress variables and

(33) |

How do we solve this equation? It's a

= | (34) | ||

= | (35) |

and the time-dependent equation becomes

(36) |

Furthermore, since the eigenvectors of a Hermitian operator are or can be made orthogonal, we can multiply by and and integrate over

= | c_{1} E_{1} |
(37) | |

= | c_{2} E_{2} |
(38) |

which are simple first-order differential equations solved by

as you can verify by substituting and differentiating.

But what if our original wavefunction
is *not* given
as a linear combination of eigenfunctions? A good strategy is to *re-write*
it so that it is! In the coordinate representation (i.e.,
space),
we can get the coefficients *c*_{i}(0) in an expansion over
orthogonal eigenfunctions
simply as

(41) |

The other strategy would be to try to re-write the propagator in the original basis set. In the problems we do, we will usually use the first approach.