Dirac delta function
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 L of square-integrable functions, 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 { | φ ⟩ } of wave functions is orthonormal if

⟨ φ n ∣ φ m ⟩ = δ n m, {\displaystyle \langle \varphi _{n}\mid \varphi _{m}\rangle =\delta _{nm},}

where δ is the Kronecker delta. A set of orthonormal wave functions is complete in the space of square-integrable functions if any wave function | ψ ⟩ can be expressed as a linear combination of the { | φ ⟩ } with complex coefficients:

ψ = ∑ c n φ n, {\displaystyle \psi =\sum c_{n}\varphi _{n},}

where c = ⟨ φ | ψ ⟩. Complete orthonormal systems of wave functions appear naturally as the eigenfunctions of the Hamiltonian (of a 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 of the Hamiltonian. In bra–ket notation this equality implies the resolution of the identity:

I = ∑ | φ n ⟩ ⟨ φ n |. {\displaystyle I=\sum |\varphi _{n}\rangle \langle \varphi _{n}|.}

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, Qψ (x) = x ψ(x). The spectrum of the position (in one dimension) is the entire real line and is called a 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. In this context, the position operator has a complete set of "generalized eigenfunctions", labeled by the points y of the real line, given by

φ y (x) = δ (x − y). {\displaystyle \varphi _{y}(x)=\delta (x-y).}

The eigenfunctions of the position operator are called the eigenstates and are denoted by φ = | y ⟩.

Similar considerations apply to the eigenstates of the momentum operator, or indeed any other (unbounded) self-adjoint operator P on the Hilbert space, provided the spectrum of P is continuous and there are no degenerate eigenvalues. In that case, there is a set Ω of real numbers (the spectrum) and a collection of distributions φ with y ∈ Ω such that

P φ y = y φ y. {\displaystyle P\varphi _{y}=y\varphi _{y}.}

That is, φ are the generalized eigenvectors of P. If they form an "orthonormal basis" in the distribution sense, that is:

⟨ φ y, φ y ′ ⟩ = δ (y − y ′), {\displaystyle \langle \varphi _{y},\varphi _{y'}\rangle =\delta (y-y'),}

then for any test function ψ,

ψ (x) = ∫ Ω c (y) φ y (x) d y {\displaystyle \psi (x)=\int _{\Omega }c(y)\varphi _{y}(x)\,dy}

where c (y) = ⟨ ψ, φ ⟩. That is, there is a resolution of the identity

I = ∫ Ω | φ y ⟩ ⟨ φ y | d y {\displaystyle I=\int _{\Omega }|\varphi _{y}\rangle \,\langle \varphi _{y}|\,dy}

where the operator-valued integral is again understood in the weak sense. If the spectrum of 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.
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