PlanetPhysics/Dirac's Delta Distribution

It is widely known that distributions play important roles in Dirac's formulation of [[../QuantumParadox/|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 [[../Bijective/|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

$\delta (x)=0=for=x\ne 0,=and=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\delta (x)dx=1$.

Moreover, for any continuous function $F$:

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\delta (x)F(x)dx=F(0)$

or in $n$ dimensions:

$\underset{{ℝ}^n}{\int }\delta (x-s)f(s)d^{n}s=f(x)$

one could attempt to define the values of $\delta (x)$ via a series of normalized Gaussian functions ([[../LebesgueMeasure/|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 $ℝ$ (or $ℂ$), having the property $\delta [f]=f(0).$ One may consider this as an [[../NormInducedByInnerProduct/|inner product]] $⟨f,\delta ⟩=\underset{0}{\overset{\mathrm{\infty }}{\int }}f(t)\delta (t)dt$ of a function $f$ and another "function" $\delta$, when the well-known formula $\underset{0}{\overset{\mathrm{\infty }}{\int }}f(t)\delta (t)dt=f(0)$ holds.\