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
distribution is not a true function
because it is not uniquely defined for all values of the argument
. Somewhat similar to the older Kronecker delta symbol, the notation
stands for
.
Moreover, for any continuous function
:
or in
dimensions:
one could attempt to define the values of
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,
, 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
(or
), having the property
One may consider this as an inner product
of a function
holds.
As of this snapshot date, this entry was owned by bci1.