PlanetPhysics/Dirac's Delta Distribution

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

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

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