Delta function

The delta function is viewed as a functional or generalized function since it has properties that are not considered well behaved. The need for the delta-function arises when dealing with a continuous set of observable values and the need to distinguish between states

and that are arbitrarily close. Clearly, the underlying mathematical formalism encounters problems based on the notion that a state is in an infinitely precise location or moving with an infinitely precise momentum. (e.g. infinitely precise momentum results in a completely undefined position and then a wavefunction that has no proper normalization.) The derivative is a process that similarly deals with these problems and is justified as a limit. The fact that there are an infinite number of points between any two points no matter how close you put them is a quandary that arises when treating any variable as continuous. Just remember that in order to understand what we even mean by continuous one must have an algorithm that reaches the notion of continuity at the end of some sequence.

Thus the definition of the delta function is formally based on the limit of a sequence of functions that approach the delta function behavior. One of these limits that can be found in a standard mathematics book or on the web at wikipedia or Wolfram’s math world is given below.

WIKIPEDIA

or for

In the homework examining particle in a box solutions as . There are two approaches:

1. Examine the problem with the normalization factor included an encounter

which shows that the functions in the limit are orthonormal in the traditional sense.

2. Or we encounter the following integral without the normalization factor and this leads to the delta function.

which based on the mathematics of delta functions is a proper definition for this function. The properties of this function point a) is straightforward L’hopitale’s rule allows one to find the limit as goes to zero and then the limit as a goes to infinity is straightforward but point b) does not appear as obvious. It does become a sinusoidal oscillation with a wavelength of zero. If you integrate a function against an oscillating function with 0 wavelength the contribution will be zero. The integral oscillates so rapidly because two waves that are almost the same frequency and take a very long distance a to become out of phase. Thus as the integrals are performed over a large range the results of the integrals become very different the larger a becomes:

if a is small. The small difference is based of the fact that the sine functions are almost identical and so their integral will be almost identical. If the range is greater than the wavelength then any complete oscillation integrates to zero. Thus the integral can be viewed as due to the fraction of a wavelength left in the last stretch of the range of x. Since two waves that are close in frequency remain almost identical for a long range the difference will be small. However, if then even the smallest difference results in a significant phase difference and thus the integral has a very different value. Thus the dependence of the integral on leads to a rapidly changing result. As this rapid oscillation has an frequency or zero wavelength. If you integrate a convolution of a function of with this rapidly changing sinusoidal function then it must result in a zero contribution if the function is smoothly varying. The rapid oscillation causes additions and subtractions at such a rapid rate that the function hasn’t changed. Thus the impact of the function away from zero b) has the proper behavior.

Another discussion can be found as Normmomentum.doc. It works through the same difference in reaching a normalization for these states but a bit more carefully.