Fourier's theorerm, applied to one-dimensional wavefunctions, yields
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 x ≠ 0, and is infinite at x = 0. However, the singularity at x = 0 is such that
The delta-function is an example of what is known as a generalized function: i.e., its value is not well-defined at all x, but its integral is well-defined. Consider the integral
Since is only non-zero infinitesimally close to x = 0, we can safely replace f(x) by f(0) in the above integral (assuming f(x) is well behaved at x = 0), to give
where use has been made of Eq. (204). A simple generalization of this result yields
which can also be thought of as an alternative definition of a delta-function.
Suppose that It follows from Eqs. (203) and (207) that
Hence, Eq. (202) yields the important result
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 O(p) takes the form
Consider momentum. We can write
where use has been made of Eq. (202). However, it follows from Eq. (210) that
Hence, using Eq. (207), we obtain
Evidently, momentum is represented by the operator P in the momentum representation. The above expression also strongly suggests [by comparison with Eq. (158)] that can be interpreted as the probability density of a measurement of momentum yielding the value P at time t. It follows that must satisfy an analogous normalization condition to Eq. (140): i.e.,
Consider displacement. We can write
Integration by parts yields
Hence, making use of Eqs. (210) and (207), we obtain
Evidently, displacement is represented by the operator
in the momentum representation.
Finally, let us consider the normalization of the momentum-space wavefunction . We have
Thus, it follows from Eqs. (207) and (210) that
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.