Fourier Transform

To allow wave **functions to extend to infinity**, we will expand the interval used

As we do this we will use the wave number

As , can take on any value, implying we will have a continuous distribution of . Our sum over becomes an integral over .

If we define , we can make the transform come out with the constants we want.

Standard Fourier Series | |

Standard Fourier Series | |

redefine coefficient | |

f stays the same | |

but is rewritten in new A and dk | |

result | |

result |

This is just the extension of the Fourier series to all .

If is normalized, then will also be normalized with this (symmetric) form of the Fourier Transform. Thus, if is a probability amplitude in position-space, can be a probability amplitude (in k-space).

Jim Branson 2013-04-22