Representations of the Dirac Deltafunction

The "deltafunction" was invented by P. A. M. Dirac around 1930 in order to compactly express the completeness relation in quantum mechanics. (Essentially equivalent definitions appear in earlier works of Fourier, Kirchhoff, and Heaviside.) The deltafunction is the limit of a function that grows infinitely large in an infinitesimally small region, while its integral remains normalized to 1. The deltafunction is too singular to be considered a function in the usual sense. Mathematicians have, however, accepted it as a linear functional, a "generalized function", or "distribution".
The deltafunction has computational significance only when it appears under an integral sign. Its defining relation can, in fact, be written or, more generally, . There are a number of representations of the deltafunction based on limits of a family of functions as some parameter approaches infinity (or zero). In this Demonstration, five of these representations are illustrated. You can select the parameter to take values from 1 to 10, on its way toward infinity.
The most rudimentary representation is a rectangular pulse: for . Then as the rectangle becomes higher and narrower.
The most commonly cited representation is based on the normalized Gaussian distribution (bell-shaped curve): . The limit is equivalent to , where is the standard deviation.
Closely analogous is the Lorentzian representation: . The Lorentzian function is proportional to the derivative of the arctangent, shown as an inset. In the limit as , the arctangent approaches the unit step function (Heaviside function). Thus the deltafunction represents the derivative of a step function.
In quantum mechanics, one frequently encounters the representation . The rapidly oscillating normalized sinc function gives an effective contribution of zero when .
The Fourier series for the deltafunction contains unit contributions from all frequencies. A deltafunction is approached as the number of terms in the following expansion increases: . This representation applies only in a neighborhood of the origin. More correctly, this Fourier series represents a delta comb function or shah function (after the Russian letter ).


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


Snapshot 1: Gaussian representation for
Snapshot 2: Lorentzian representation
Snapshot 3: Fourier expansion to
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+