Dirac delta function: Difference between revisions
imported>Paul Wormer (New page: {{subpages}} In physics, the '''Dirac delta function''' is a function introduced by P.A.M. Dirac in his seminal 1930 book on quantum mechanics.<ref>P.AM. ...) |
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The Dirac delta function is not an ordinary well-behaved map <font style="vertical-align: 12%"><math>\mathbb{R} \rightarrow \mathbb{R}</math></font>, but a [[distribution (mathematics)|distribution]], also known as an ''improper'' or ''generalized function''. | The Dirac delta function is not an ordinary well-behaved map <font style="vertical-align: 12%"><math>\mathbb{R} \rightarrow \mathbb{R}</math></font>, but a [[distribution (mathematics)|distribution]], also known as an ''improper'' or ''generalized function''. Physicists express its special character by stating that the Dirac delta function makes only sense as a factor in an integrand. Mathematicians say that the delta function is linear functional on a space of test functions. | ||
==Properties== | |||
Most commonly one takes the lower and the upper bound in the definition of the delta function equal to <math>-\infty</math> and <math> \infty</math>, respectively. From here on this is assumed. | |||
:<math> | |||
\begin{align} | |||
\delta(x-a) &= \delta(a-x), \\ | |||
(x-a)\delta(x-a) &= 0, \\ | |||
\delta(ax) &= |a|^{-1} \delta(x) \quad (a \ne 0), \\ | |||
f(x) \delta(x-a) &= f(a) \delta(x-a), \\ | |||
\int_{-\infty}^{\infty} \delta(x-y)\delta(y-a)\mathrm{d}y &= \delta(x-a) | |||
\end{align} | |||
</math> | |||
The physicist's proof of these properties proceeds by making proper substitutions into the integral and using the ordinary rules of integral calculus. |
Revision as of 10:55, 20 December 2008
In physics, the Dirac delta function is a function introduced by P.A.M. Dirac in his seminal 1930 book on quantum mechanics.[1] Heuristically, the function can be seen as an extension of the Kronecker delta from discrete to continuous indices. The Kronecker delta acts as a "filter" in a summation:
Similarly, the Dirac delta function δ(x−a) may be defined by (replace i by x and the summation over i by an integration over x),
The Dirac delta function is not an ordinary well-behaved map , but a distribution, also known as an improper or generalized function. Physicists express its special character by stating that the Dirac delta function makes only sense as a factor in an integrand. Mathematicians say that the delta function is linear functional on a space of test functions.
Properties
Most commonly one takes the lower and the upper bound in the definition of the delta function equal to and , respectively. From here on this is assumed.
The physicist's proof of these properties proceeds by making proper substitutions into the integral and using the ordinary rules of integral calculus.
- ↑ P.AM. Dirac, The Principles of Quantum Mechanics, Oxford University Press (1930). Fourth edition 1958. Paperback 1981, p. 58