Editing Dirac delta function (section)

===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.