Editing Self-adjoint operator (section)

{{Short description|Linear operator equal to its own adjoint}}
In [[mathematics]], a '''self-adjoint operator''' on a [[complex vector space]] ''V'' with [[inner product]] <math>\langle\cdot,\cdot\rangle</math> is a [[linear map]] ''A'' (from ''V'' to itself) that is its own [[Adjoint of an operator|adjoint]]. That is, <math>\langle Ax,y \rangle = \langle x,Ay \rangle</math> for all <math>x, y</math> ∊ ''V''. If ''V'' is [[finite-dimensional]] with a given [[orthonormal basis]], this is equivalent to the condition that the [[matrix (mathematics)|matrix]] of ''A'' is a [[Hermitian matrix]], i.e., equal to its [[conjugate transpose]] ''A''{{sup|∗}}. By the finite-dimensional [[spectral theorem]], ''V'' has an [[orthonormal basis]] such that the matrix of ''A'' relative to this basis is a [[diagonal matrix]] with entries in the [[real number]]s. This article deals with applying generalizations of this concept to operators on [[Hilbert space]]s of arbitrary dimension.

Self-adjoint operators are used in [[functional analysis]] and [[quantum mechanics]]. In quantum mechanics their importance lies in the [[Dirac–von Neumann axioms|Dirac–von Neumann formulation]] of quantum mechanics, in which physical [[observable]]s such as [[position (vector)|position]], [[momentum]], [[angular momentum]] and [[Spin (physics)|spin]] are represented by self-adjoint operators on a Hilbert space. Of particular significance is the [[Hamiltonian (quantum mechanics)|Hamiltonian]] operator <math>\hat{H}</math> defined by
: <math>\hat{H} \psi = -\frac{\hbar^2}{2m} \nabla^2 \psi + V \psi,</math>
which as an observable corresponds to the total [[energy]] of a particle of mass ''m'' in a real [[scalar potential|potential field]] ''V''. [[Differential operator]]s are an important class of [[unbounded operator]]s.

The structure of self-adjoint operators on infinite-dimensional Hilbert spaces essentially resembles the finite-dimensional case. That is to say, operators are self-adjoint if and only if they are [[unitary operator|unitarily]] equivalent to real-valued [[multiplication operator]]s. With suitable modifications, this result can be extended to possibly unbounded operators on infinite-dimensional spaces. Since an everywhere-defined self-adjoint operator is necessarily bounded, one needs to be more attentive to the domain issue in the unbounded case. This is explained below in more detail.