Dirac delta function
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 a 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