(200) | |||

(201) |

where represents wavenumber. However, . Hence, we can also write

where is the momentum-space equivalent to the real-space wavefunction .

At this stage, it is convenient to introduce a useful function called the
*Dirac delta-function*. This function, denoted , was
first devised by Paul Dirac, and has the following rather
unusual properties: is zero for , and is infinite
at . However, the singularity at is such that

(205) |

(206) |

which can also be thought of as an alternative definition of a delta-function.

Suppose that
. It follows from Eqs. (203)
and (207) that

(208) |

(209) |

It turns out that we can just as well formulate quantum mechanics using
momentum-space wavefunctions, , as real-space wavefunctions, . The former scheme is known as the *momentum representation* of quantum mechanics. In the momentum representation,
wavefunctions are the Fourier transforms of the equivalent real-space
wavefunctions, and dynamical variables are represented by *different* operators. Furthermore, by analogy with Eq. (192), the
expectation value of some operator takes the form

Consider momentum. We can write

(212) |

where use has been made of Eq. (202). However, it follows from Eq. (210) that

(213) |

(214) |

Consider displacement. We can write

(216) | |||

Integration by parts yields

(217) |

(218) |

(219) |

Finally, let us consider the normalization of the momentum-space wavefunction . We have

(220) |

Hence, if is properly normalized [see Eq. (140)] then , as defined in Eq. (203), is also properly normalized [see Eq. (215)].

The existence of the momentum representation illustrates an important point:
*i.e.*, that there are many different, but entirely equivalent, ways
of mathematically formulating quantum mechanics. For instance, it
is also possible to represent wavefunctions as row and column vectors, and dynamical
variables
as matrices which act upon these vectors.