Position operator

Article summary

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 (e.g. the space of tempered distributions), its eigenvalues are the possible position vectors of the particle. In one dimension, if by the symbol | x \rangle we denote the unitary eigenvector of the position operator corresponding to the eigenvalue x , then, |x \rangle 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 we can write X| x\rangle = x |x\rangle, 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 \delta_x . In quantum mechanics, the ordered (continuous) family of all Dirac distributions, i.e. the family \delta = (\delta_x)_{x \in \R}, is called the (unitary) position basis, just because it is a (unitary) eigenbasis of the position operator X in the space of tempered distributions. It is fundamental to observe that there exists only one linear continuous operator X on the space of tempered distributions to itself, such that X(\delta_x) = x \delta_x, for every real point x .…

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