The Dirac delta function is an entity of particular interest for two reasons: It is of a class of tools in physics that presents possible conflicts with the simplistic mathematical dreams of earlier physics, and it exists in possible correspondence with a broad class of natural phenomenon.
It is a mathematical anomaly in that it doesn’t have the quality of continuity that physics often takes as a given. Supposedly, it’s precisely 0 everywhere except at the origin, where it is infinity, and the “area” under this infinite point is 1, even though the definition of area is that is that it’s a collection of lines and has to have more than just one line¹. This isn’t very mathy – it seems like most functions in physics are completely continuous and we can do all sorts of integration and derivation on them. But it doesn’t have to be mathy at all – it just has to be of use in some cases and have enough of a framework to produce something of interest, namely how to treat point charges and impulse functions.
If you have a case where there is some section of something which is absolutely zero (or some other constant) for anything but a point or a number (or infinitely many) of disconnected points then it must be zero (or equal to the same constant) everywhere or else there is a discontinuity. This is, I think, how people tend to imagine real objects – at a given point or region in space, an object either exists or extends into or it doesn’t. If you were to plot a graph of the presence of some object as it rests on a line, you would have something like a binary function of ‘no’ (or 0) where an object isn’t and ‘yes’ (say, 1) where it is. But such a discrete binary function is clearly discontinuous – it goes immediately from 0 to 1, and at no point is it something in between. The so-called “heavy side step function” is a version of this, and the Dirac delta function represents its derivative, a quality physicists are oft concerned with.
I graph it because one of the ways to gain an understanding of it is to treat it as a limit in a sequence of functions which are continuous (or at least piece-wise integrable) but as you increase some variable towards infinity become ever more spiky and point-like, and eventually, as the variable reaches infinity, you are almost left with something exactly like the Dirac delta function. The Dirac delta function represents the instantaneous change, or alteration or creation or destruction of some amount of something. It ties in with the discrete and represents a balance between the laws of physics as we see them.
The animated graph above shows two functions which approximate the Dirac delta function in the limit as a particular constant goes to infinity. One is a simple box function, such that the area of the box is always 1, and the other is basically a squared-sine function with some other stuff thrown in to keep the area 1 as well. They are displayed with a changing scale so that their height appears to be the same – I could have shown you their actual height, but unless the spike is very tight the average value of the function is very low, and that fact makes it difficult to graph (as the spike gets incredibly high very quickly.)
1. Another way to make this point is to say that except at the infinite limit the area under any point on the graph is zero – if you take the integral of a function where the bounds of integration are the same point you usually expect as much.
Over the course of roughly 48 hours, the two visiting professors drank:
The remaining third of my liter bottle of Jack Daniel’s
A 375ml bottle of Jack Daniel’s (that they bought to replace the whiskey of mine they had drunk)
Another liter bottle of Jack Daniel’s, as well as two Victory Storm King Stouts
Three Victory Hop Devils