Chemistry - Kronecker delta

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In mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. So, for example, $δ 12 = 0$ , but $δ 33 = 1$ . It is written as the symbol δ_ij, and treated as a notational shorthand rather than as a function.

$\delta_{ij} = \left\{\begin{matrix} 1 & \mbox{if } i=j \\ 0 & \mbox{if } i \ne j \end{matrix}\right.$

Properties of the Delta Function

The Kronecker delta has the so-called sifting property that for $j\in\mathbb Z$ :

$\sum_{i=-\infty}^\infty \delta_{ij} a_i=a_j.$

This property is similar to one of the main properties of the Dirac delta function:

$\int_{-\infty}^\infty \delta(x-y)f(x) dx=f(y),$

and in fact Dirac's delta was named after the Kronecker delta because of this analogous property.

The Kronecker delta is used in many areas of mathematics. For example, in linear algebra, the identity matrix can be written as $\delta_{ij}\,$ while if it is considered as a tensor, the Kronecker tensor, it can be written $\delta^j_i$ with a contravariant index j. This is a more accurate way to notate the identity matrix, considered as a linear mapping.

Extensions of the Delta Function

In the same fashion, we may define an analogous, multi-dimensional function of many variables

$\delta^{j_1 j_2 ... j_N}_{i_1 i_2 ...i_N}:= \Pi_{l \in \mathbb Z}^N \delta^{j_l}_{i_l}$

This function takes the value 1 if and only if all the upper indices match the corresponding lower one, taking the value zero otherwise.

Kronecker delta

Properties of the Delta Function

Extensions of the Delta Function

See also