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Dirac's delta distribution (Definition)

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 on line.

The Dirac delta $\delta(x)$ distribution is not a true function because it is not uniquely defined for all values of the argument $x$. Somewhat similar to the older Kronecker delta symbol, the notation $\delta(x)$ stands for

$\displaystyle \delta(x) = 0 \;$for$\displaystyle \; x \ne 0, \;$and$\displaystyle \; \int_{-\infty}^\infty \delta(x) dx = 1 $
.

Moreover, for any continuous function $F$:

$\displaystyle \int_{-\infty}^\infty \delta(x) F(x)dx = F(0) $

or in $n$ dimensions:

$\displaystyle \int_{\mathbb{R}^n} \delta(x - s)f(s) \, d^ns = f(x)$

one could attempt to define the values of $\delta(x)$ via a series of normalized Gaussian functions (normal distributions) in the limit of their width going to zero; however, such a limit of the normalized Gaussian function is still meaningless as a function, even though one sees in engineering textbooks especially such a limit as being written to be equal to the Dirac distribution considered above, which it is not. An example of how the Dirac distribution arises in a physical, classical context is available on line.

The Dirac delta, $\delta$, can be, however, correctly defined as a linear functional, i.e. a linear mapping from a function space, consisting e.g. of certain real functions, to $\mathbb{R}$ (or $\mathbb{C}$), having the property

$\displaystyle \delta[f] \;=\; f(0).$
One may consider this as an inner product

$\displaystyle \langle f,\,\delta\rangle \;=\; \int_0^\infty\!f(t)\delta(t)\,dt$
of a function $f$ and another “function” $\delta$, when the well-known formula

$\displaystyle \int_0^\infty\!f(t)\delta(t)\,dt \;=\; f(0)$
holds.

Bibliography

1
Schwartz, L. (1950–1951), ThÃ�orie des distributions, vols. 1–2, Hermann: Paris.
2
WR73) W. Rudin, Functional Analysis, McGraw-Hill Book Company, 1973.
3
L. Hörmander, The Analysis of Linear Partial Differential Operators I, (Distribution theory and Fourier Analysis), 2nd ed, Springer-Verlag, 1990.
4
Originally from The Data Analysis Briefbook (http://rkb.home.cern.ch/rkb/titleA.html)



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Cross-references: inner product, normal distributions, function, quantum mechanics

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