The Remarkable Dirac Delta Function

The Remarkable Dirac Delta Function

This is a brief introduction to the Dirac Delta function – named after the legendary Nobel prize winning physicist Paul Dirac. Dirac was one of the founding fathers of the mathematics of quantum mechanics, and is widely regarded as one of the most influential physicists of the 20th Century.  This topic is only recommended for students confident with the idea of limits and was inspired by a Quora post by Arsh Khan.

Dirac defined the delta function as having the following 2 properties:

The first property as defined above is that the delta function is 0 for all values of t, except for t = 0, when it is infinite.

The second property defined above is that the integral of the delta function – and the area of the graph between 2 points (either side of 0) is 1.    We can take the bottom integral where we integrate from negative to positive infinity as this will be more useful later.

The delta function (technically not a function in a normal sense!) can be represented as the following limit:

Whilst this looks a little intimidating, it just means that we take the limit of the function as epsilon (ε) approaches 0.  Given this definition of the delta function we can check that the 2 properties outlined above hold.

For the first limit above we set t not equal to 0.  Then, because it is a continuous function when t is not equal to 0, we can effectively replace epsilon with 0 in the first limit above to get a limit of 0.  In the second limit when t = 0 we get a limit of infinity.  Therefore the first property holds.

To show that the second property holds, we start with the following integral identity from HL Calculus:

Hopefully this will look similar to the function we are interested in.  Let’s play a little fast and loose with the mathematics and ignore the limit of the function and just consider the following integral:

Therefore (using the fact that the graph of arctanx has horizontal asymptotes at positive and negative pi/2 for the final part) :

So we have shown above that the integral of every function of this form will have an integral of 1, regardless of the value of epsilon, thus satisfying our second property.

The use of the Dirac Function

So far so good.  But what is so remarkable about the Dirac function?  Well, it allows objects to be described in terms of a single zero width (and infinitely high) spike, but despite having zero width, this spike still has an area of 1.   This then allows the representation of elementary particles which have zero size but finite mass (and other finite properties such as charge) to be represented mathematically.  With the area under the curve = 1 it can also be thought of in terms of a probability density function – i.e representing the quantum world in terms of probability wave functions.

A graphical representation:

This is easier to understand graphically.  Say for example we choose a value epsilon (ε) and gradually make it smaller (i.e we find the limit as ε approaches 0).  When ε = 5 we have the following:

When ε = 1 we have the following:

When ε = 0.1 we have the following:

When ε = 0.01 we have the following:

You can see that as  ε approaches 0 we get a function which is close to 0 everywhere except for a spike at zero.  The total area under the function remains at 1 for all ε.

Therefore we can represent the Dirac Delta function with the above graph.  In it we have a point with zero width but with infinite height – and still with an area under the curve of 1!