How the Dirac Delta Function Works

A graph with a horizontal line. At a location of 0, there is a vertical line shown with a magnitude of 1.
Schematic diagram of the Dirac delta function by a line surmounted by an arrow. The height of the arrow is usually used to specify the value of any multiplicative constant, which will give the area under the function. The other convention is to write the area next to the arrowhead. Creative Commons (via Wikimedia Commons)

The Dirac delta function is the name given to a mathematical structure that is intended to represent an idealized point object, such as a point mass or point charge. It has broad applications within quantum mechanics and the rest of quantum physics, as it is usually used within the quantum wavefunction. The delta function is represented with the Greek lowercase symbol delta, written as a function: δ(x).

How the Delta Function Works

This representation is achieved by defining the Dirac delta function so that it has a value of 0 everywhere except at the input value of 0. At that point, it represents a spike that is infinitely high. The integral taken over the entire line is equal to 1. If you've studied calculus, you've likely run into this phenomenon before. Keep in mind that this is a concept that is normally introduced to students after years of college-level study in theoretical physics.

In other words, the results are the following for the most basic delta function δ(x), with a one-dimensional variable x, for some random input values:

  • δ(5) = 0
  • δ(-20) = 0
  • δ(38.4) = 0
  • δ(-12.2) = 0
  • δ(0.11) = 0
  • δ(0) = ∞

You can scale the function up by multiplying it by a constant. Under the rules of calculus, multiplying by a constant value will also increase the value of the integral by that constant factor. Since the integral of δ(x) across all real numbers is 1, then multiplying it by a constant of would have a new integral equal to that constant.

So, for example, 27δ(x) has an integral across all real numbers of 27.

Another useful thing to consider is that since the function has a non-zero value only for an input of 0, then if you're looking at a coordinate grid where your point isn't lined up right at 0, this can be represented with an expression inside the function input.

So if you want to represent the idea that the particle is at a position x = 5, then you would write the Dirac delta function as δ(x - 5) = ∞ [since δ(5 - 5) = ∞]. 

If you then want to use this function to represent a series of point particles within a quantum system, you can do it by adding together various dirac delta functions. For a concrete example, a function with points at x = 5 and x = 8 could be represented as δ(x - 5) + δ(x - 8). If you then took an integral of this function over all numbers, you would get an integral that represents real numbers, even though the functions are 0 at all locations other than the two where there are points. This concept can then be expanded to represent a space with two or three dimensions (instead of the one-dimensional case I used in my examples).

This is an admittedly-brief introduction to a very complex topic. The key thing to realize about it is that the Dirac delta function basically exists for the sole purpose of making the integration of the function make sense. When there is no integral taking place, the presence of the Dirac delta function isn't particularly helpful. But in physics, when you are dealing with going from a region with no particles that suddenly exist at only one point, it's quite helpful.

Source of the Delta Function

In his 1930 book, Principles of Quantum Mechanics, English theoretical physicist Paul Dirac laid out the key elements of quantum mechanics, including the bra-ket notation and also his Dirac delta function. These became standard concepts in the field of quantum mechanics within the Schrodinger equation.

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Jones, Andrew Zimmerman. "How the Dirac Delta Function Works." ThoughtCo, Jan. 31, 2018, Jones, Andrew Zimmerman. (2018, January 31). How the Dirac Delta Function Works. Retrieved from Jones, Andrew Zimmerman. "How the Dirac Delta Function Works." ThoughtCo. (accessed March 24, 2018).