Position operator
In quantum mechanics, the position operator is the operator that corresponds to the position observable of a particle. When the position operator is considered with a wide enough domain, its eigenvalues are the possible position vectors of the particle. In one dimension, if by the symbol | x ⟩ we denote the unitary eigenvector of the position operator corresponding to the eigenvalue x, then, | x ⟩ represents the state of the particle in which we know with certainty to find the particle itself at position x. Therefore, denoting the position operator by the symbol X  – in the literature we find also other symbols for the position operator, for instance Q, x ^ and so on – we can write X | x ⟩ = x | x ⟩, for every real position x. One possible realization of the unitary state with position x is the Dirac delta (function) distribution centered at the position x, often denoted by δ x. In quantum mechanics, the ordered (continuous) family of all Dirac distributions, i.e. the family δ = x ∈ R, is called the (unitary) position basis, just because it is a (unitary) eigenbasis of the position operator X in the space of distributions dual to the space of wave-functions. It is fundamental to observe that there exists only one linear continuous endomorphism X on the space of tempered distributions such that X = x δ x, for every real point x. It's possible to prove that the unique above endomorphism is necessarily defined by X = x ψ, for every tempered distribution ψ, where x denotes the coordinate function of the position line – defined from the real line into the complex plane by x : R → C : x ↦ x.  :\mathbb {R} \to \mathbb {C} :x\mapsto x.}
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