When substituted into the time-dependent Schrödinger equation, this gives

= | (57) | ||

= |

Now multiply on the left by and , respectively, and use to obtain

= | E_{1} c_{1}(t) |
(58) | |

= | E_{2} c_{2}(t), |
(59) |

just as before, with solutions once again

c_{1}(t) |
= | (60) | |

c_{2}(t) |
= | (61) |

But what if we are given
in a form that looks different
from that of eq. 56? Since the eigenvector basis must
be complete (although it will usually have more than two basis vectors,
as in this example!), we can always rewrite our state vector in this form,
and the coefficients can always be computed as

(62) |

Note that in this subsection we aren't assuming anything about whether we are working in coordinate () space or momentum () space or some other space. However, if we were working in coordinate space, we could insert the resolution of the identity

(63) |

to obtain

c_{i}(0) |
= | ||

= | (64) |

completely consistent with everything above. The propagator may be written as

(65) |

again with and here representing eigenfunctions of with eigenvalues