Dirac delta (function)

The Dirac delta is a mathematical object which may be thought of as the continuous counterpart to the discrete Kronecker delta. It is often defined by the properties

0 = 𝛿 (𝑥) \forall 𝑥 \neq 0 1 = \int \infty - \infty 𝛿 (𝑥) 𝑑 𝑥

While often treated like a function, the Dirac delta is actually an example of a Generalised function or distribution. It is associated with many abuses of notation.

warning

This Zettel is neither complete nor rigorous, as is the case with many treatments of the Dirac delta

In quantum mechanics and indeed many of its analytical applications, the Dirac delta is more useful for representing the linear forms for evaluation of a function at a specific point.¹

\int \infty - \infty 𝛿 (𝑥 - 𝑥') 𝑓 (𝑥) 𝑑 𝑥 = 𝑓 (𝑥')

This motivates the Fourier transform representation of the delta function:

𝛿 (𝑥) = 1 2 𝜋 \int \infty - \infty 𝑒 - 𝑖 𝜔 𝑥 𝑑 𝜔

Properties

$𝑓 (𝑥) 𝛿 (𝑥 - 𝑥') = 𝑓 (𝑥') 𝛿 (𝑥)$
$𝛿 (𝑘 𝑥) = 1 | 𝑘 | 𝛿 (𝑥)$
$𝛿 (𝑥) = 1 2 𝜋 \int \infty - \infty 𝑒 𝑖 𝑘 𝑥 𝑑 𝑘$

Higher dimension

The Dirac delta can be generalised to inputs in $ℝ 𝑛$ with

0 = 𝛿 𝑛 (⃗ 𝐯) \forall ⃗ 𝐯 \neq ⃗ 𝟎 1 = \int ℝ 𝑛 𝛿 𝑛 (𝑥) 𝑑 𝑅

Dirac delta (function)

Properties

Higher dimension

See also

Footnotes