Editing Dirac delta function (section)

==Applications==

===Probability theory===
In [[probability theory]] and [[statistics]], the Dirac delta function is often used to represent a [[discrete distribution]], or a partially discrete, partially [[continuous distribution]], using a [[probability density function]] (which is normally used to represent absolutely continuous distributions).  For example, the probability density function {{math|''f''(''x'')}} of a discrete distribution consisting of points {{math|1='''x''' = {{brace|''x''<sub>1</sub>, ..., ''x<sub>n</sub>''}}}}, with corresponding probabilities {{math|''p''<sub>1</sub>, ..., ''p<sub>n</sub>''}}, can be written as

<math display="block">f(x) = \sum_{i=1}^n p_i \delta(x-x_i).</math>

As another example, consider a distribution in which 6/10 of the time returns a standard [[normal distribution]], and 4/10 of the time returns exactly the value 3.5 (i.e. a partly continuous, partly discrete [[mixture distribution]]).  The density function of this distribution can be written as

<math display="block">f(x) = 0.6 \, \frac {1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} + 0.4 \, \delta(x-3.5).</math>

The delta function is also used to represent the resulting probability density function of a random variable that is transformed by continuously differentiable function. If {{math|1=''Y'' = g(''X'')}} is a continuous differentiable function, then the density of {{mvar|Y}} can be written as

<math display="block">f_Y(y) = \int_{-\infty}^{+\infty} f_X(x) \delta(y-g(x)) \,dx.  </math>

The delta function is also used in a completely different way to represent the [[local time (mathematics)|local time]] of a [[diffusion process]] (like [[Brownian motion]]).  The local time of a stochastic process {{math|''B''(''t'')}} is given by
<math display="block">\ell(x,t) = \int_0^t \delta(x-B(s))\,ds</math>
and represents the amount of time that the process spends at the point {{mvar|x}} in the range of the process.  More precisely, in one dimension this integral can be written
<math display="block">\ell(x,t) = \lim_{\varepsilon\to 0^+}\frac{1}{2\varepsilon}\int_0^t \mathbf{1}_{[x-\varepsilon,x+\varepsilon]}(B(s))\,ds</math>
where <math>\mathbf{1}_{[x-\varepsilon,x+\varepsilon]}</math> is the [[indicator function]] of the interval <math>[x-\varepsilon,x+\varepsilon].</math>

===Quantum mechanics===
The delta function is expedient in [[quantum mechanics]]. The [[wave function]] of a particle gives the [[probability amplitude]] of finding a particle within a given region of space.  Wave functions are assumed to be elements of the Hilbert space {{math|''L''<sup>2</sup>}} of [[square-integrable function]]s, and the total probability of finding a particle within a given interval is the integral of the magnitude of the wave function squared over the interval. A set {{math|{{brace|{{ket|''φ<sub>n</sub>''}}}}}} of wave functions is orthonormal if

<math display="block">\langle\varphi_n \mid \varphi_m\rangle = \delta_{nm},</math>

where {{mvar|δ<sub>nm</sub>}} is the Kronecker delta.  A set of orthonormal wave functions is complete in the space of square-integrable functions if any wave function {{math|{{ket|ψ}}}} can be expressed as a linear combination of the {{math|{{brace|{{ket|''φ<sub>n</sub>''}}}}}} with complex coefficients:

<math display="block"> \psi = \sum c_n \varphi_n, </math>

where {{math|1=''c<sub>n</sub>'' = {{bra-ket|''φ<sub>n</sub>''|''ψ''}}}}.  Complete orthonormal systems of wave functions appear naturally as the [[eigenfunction]]s of the [[Hamiltonian (quantum mechanics)|Hamiltonian]] (of a [[bound state|bound system]]) in quantum mechanics that measures the energy levels, which are called the eigenvalues.  The set of eigenvalues, in this case, is known as the [[Spectrum (functional analysis)|spectrum]] of the Hamiltonian.  In [[bra–ket notation]] this equality implies the [[Borel functional calculus#Resolution of the identity|resolution of the identity]]:

<math display="block">I = \sum |\varphi_n\rangle\langle\varphi_n|.</math>

Here the eigenvalues are assumed to be discrete, but the set of eigenvalues of an [[observable]] can also be continuous.  An example is the [[position operator]], {{math|1=''Qψ''(''x'') = ''x''ψ(''x'')}}.  The spectrum of the position (in one dimension) is the entire real line and is called a [[Spectrum (physical sciences)#In quantum mechanics|continuous spectrum]].  However, unlike the Hamiltonian, the position operator lacks proper eigenfunctions.  The conventional way to overcome this shortcoming is to widen the class of available functions by allowing distributions as well, i.e., to replace the Hilbert space with a [[rigged Hilbert space]].{{sfn|Isham|1995|loc=§6.2}}  In this context, the position operator has a complete set of ''generalized eigenfunctions'',{{sfn|Gelfand|Shilov|1966–1968|loc=Vol. 4, §I.4.1}} labeled by the points {{mvar|y}} of the real line, given by

<math display="block">\varphi_y(x) = \delta(x-y).</math>

The generalized eigenfunctions of the position operator are called the ''eigenkets'' and are denoted by {{math|1=''φ<sub>y</sub>'' = {{ket|''y''}}}}.{{sfn|de la Madrid Modino|2001|pp=96,106}}

Similar considerations apply to any other [[Spectral theorem#Unbounded self-adjoint operators|(unbounded) self-adjoint operator]] with continuous spectrum and no degenerate eigenvalues, such as the [[momentum operator]] {{mvar|P}}. In that case, there is a set {{math|Ω}} of real numbers (the spectrum) and a collection of distributions {{mvar|φ<sub>y</sub>}} with {{math|''y'' ∈ Ω}} such that

<math display="block">P\varphi_y = y\varphi_y.</math>

That is, {{mvar|φ<sub>y</sub>}} are the generalized eigenvectors of {{mvar|P}}. If they form an "orthonormal basis" in the distribution sense, that is:

<math display="block">\langle \varphi_y,\varphi_{y'}\rangle = \delta(y-y'),</math>

then for any test function {{mvar|ψ}},

<math display="block"> \psi(x) = \int_\Omega  c(y) \varphi_y(x) \, dy</math>

where {{math|1= ''c''(''y'') = {{angbr|''ψ'', ''φ<sub>y</sub>''}}}}. That is, there is a resolution of the identity

<math display="block">I = \int_\Omega |\varphi_y\rangle\, \langle\varphi_y|\,dy</math>

where the operator-valued integral is again understood in the weak sense.  If the spectrum of {{mvar|P}} has both continuous and discrete parts, then the resolution of the identity involves a summation over the discrete spectrum and an integral over the continuous spectrum.

The delta function also has many more specialized applications in quantum mechanics, such as the [[delta potential]] models for a single and double potential well.

===Structural mechanics===
The delta function can be used in [[structural mechanics]] to describe transient loads or point loads acting on structures. The governing equation of a simple [[Harmonic oscillator|mass–spring system]] excited by a sudden force [[impulse (physics)|impulse]] {{mvar|I}} at time {{math|1=''t'' = 0}} can be written

<math display="block">m \frac{d^2 \xi}{dt^2} + k \xi = I \delta(t),</math>

where {{mvar|m}} is the mass, {{mvar|ξ}} is the deflection, and {{mvar|k}} is the [[spring constant]].

As another example, the equation governing the static deflection of a slender [[beam (structure)|beam]] is, according to [[Euler–Bernoulli beam equation|Euler–Bernoulli theory]],

<math display="block">EI \frac{d^4 w}{dx^4} = q(x),</math>

where {{mvar|EI}} is the [[bending stiffness]] of the beam, {{mvar|w}} is the [[deflection (engineering)|deflection]], {{mvar|x}} is the spatial coordinate, and {{math|''q''(''x'')}} is the load distribution. If a beam is loaded by a point force {{mvar|F}} at {{math|1=''x'' = ''x''<sub>0</sub>}}, the load distribution is written

<math display="block">q(x) = F \delta(x-x_0).</math>

As the integration of the delta function results in the [[Heaviside step function]], it follows that the static deflection of a slender beam subject to multiple point loads is described by a set of piecewise [[polynomial]]s.

Also, a point [[bending moment|moment]] acting on a beam can be described by delta functions. Consider two opposing point forces {{mvar|F}} at a distance {{mvar|d}} apart. They then produce a moment {{math|1=''M'' = ''Fd''}} acting on the beam. Now, let the distance {{mvar|d}} approach the [[Limit of a function|limit]] zero, while {{mvar|M}} is kept constant. The load distribution, assuming a clockwise moment acting at {{math|1=''x'' = 0}}, is written

<math display="block">\begin{align}
q(x) &= \lim_{d \to 0} \Big( F \delta(x) - F \delta(x-d) \Big) \\[4pt]
&= \lim_{d \to 0} \left( \frac{M}{d} \delta(x) - \frac{M}{d} \delta(x-d) \right) \\[4pt]
&= M \lim_{d \to 0} \frac{\delta(x) - \delta(x - d)}{d}\\[4pt]
&= M \delta'(x).
\end{align}</math>

Point moments can thus be represented by the [[derivative]] of the delta function. Integration of the beam equation again results in piecewise [[polynomial]] deflection.