Global InformationPosition operator information
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.[1]
In one dimension, if by the symbol we denote the unitary eigenvector of the position operator corresponding to the eigenvalue , then, represents the state of the particle in which we know with certainty to find the particle itself at position .
Therefore, denoting the position operator by the symbol – in the literature we find also other symbols for the position operator, for instance (from Lagrangian mechanics), and so on – we can write for every real position .
One possible realization of the unitary state with position is the Dirac delta (function) distribution centered at the position , often denoted by .
In quantum mechanics, the ordered (continuous) family of all Dirac distributions, i.e. the family is called the (unitary) position basis (in one dimension), just because it is a (unitary) eigenbasis of the position operator 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 on the space of tempered distributions such that for every real point . It's possible to prove that the unique above endomorphism is necessarily defined by for every tempered distribution , where denotes the coordinate function of the position line – defined from the real line into the complex plane by
- ^ Atkins, P.W. (1974). Quanta: A handbook of concepts. Oxford University Press. ISBN 0-19-855493-1.