It is widely known that distributions play important roles in Dirac’s formulation of quantum mechanics. An example of how the Dirac distribution arises in a physical, classical context is also available online.

The Dirac delta δ(x) is a distribution, not an ordinary function. Its familiar symbolic properties are

together with the normalization

More generally, for a suitable test function F,

In n dimensions, the corresponding sifting property is

One can approximate the Dirac delta distribution by a family of normalized Gaussian functions whose widths tend to zero. The limiting object is not an ordinary pointwise-defined function; rather, the convergence is understood in the sense of distributions.

The Dirac delta, δ, can be defined rigorously as a linear functional on a suitable space of test functions. Its action is

Equivalently, using pairing notation,

In the common formal notation this is written as

References