Dirac's delta distribution
DiracsDeltaDistribution
2010-03-05 12:45:01
2010-03-05 13:07:57
Definition
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
%\usepackage{psfrag}
%\usepackage{graphicx}
%\usepackage{xypic}
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 available \PMlinkexternal{on line.}{http://www.rose-hulman.edu/~rickert/Classes/ma222/Wint0102/dirac.pdf}
The Dirac delta $\delta(x)$ \emph{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
$$ \delta(x) = 0 \;\text{for}\; x \ne 0, \;\text{and}\; \int_{-\infty}^\infty \delta(x) dx = 1 $$.
Moreover, for any continuous function $F$:
$$ \int_{-\infty}^\infty \delta(x) F(x)dx = F(0) $$
or in $n$ dimensions:
$$\int_{\mathbb{R}^n} \delta(x - s)f(s) \, d^ns = f(x)$$
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
\PMlinkexternal{on line.}{http://www.rose-hulman.edu/~rickert/Classes/ma222/Wint0102/dirac.pdf}
The \emph{Dirac delta}, $\delta$, can be, however, correctly defined as a \emph{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
$$\delta[f] \;=\; f(0).$$
One may consider this as an inner product
$$\langle f,\,\delta\rangle \;=\; \int_0^\infty\!f(t)\delta(t)\,dt$$
of a function $f$ and another ``function'' $\delta$, when the well-known \PMlinkescapetext{formula}
$$\int_0^\infty\!f(t)\delta(t)\,dt \;=\; f(0)$$
holds.\
\begin{thebibliography}{9}
\bibitem{SL51}
Schwartz, L. (1950--1951), Théorie des distributions, vols. 1--2, Hermann: Paris.
\bibitem(WR73)
W. Rudin, {\em Functional Analysis},
McGraw-Hill Book Company, 1973.
\bibitem{hormander}
L. H\"ormander, {\em The Analysis of Linear Partial Differential Operators I,
(Distribution theory and Fourier Analysis)}, 2nd ed, Springer-Verlag, 1990.
\bibitem{TDAB}
Originally from The Data Analysis Briefbook
(\PMlinkexternal{http://rkb.home.cern.ch/rkb/titleA.html}{http://rkb.home.cern.ch/rkb/titleA.html})
\end{thebibliography}