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 (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 $${\displaystyle |x\rangle }$$ we denote the unitary eigenvector of the position operator corresponding to the eigenvalue ${\displaystyle x}$, then, ${\displaystyle |x\rangle }$ represents the state of the particle in which we know with certainty to find the particle itself at position ${\displaystyle x}$.

Therefore, denoting the position operator by the symbol ${\displaystyle X}$ – in the literature we find also other symbols for the position operator, for instance ${\displaystyle Q}$ (from Lagrangian mechanics), ${\displaystyle {\hat {\mathrm {x} }}}$ and so on – we can write $${\displaystyle X|x\rangle =x|x\rangle ,}$$ for every real position ${\displaystyle x}$.

One possible realization of the unitary state with position ${\displaystyle x}$ is the Dirac delta (function) distribution centered at the position ${\displaystyle x}$, often denoted by ${\displaystyle \delta _{x}}$.

In quantum mechanics, the ordered (continuous) family of all Dirac distributions, i.e. the family $${\displaystyle \delta =(\delta _{x})_{x\in \mathbb {R} },}$$ is called the (unitary) position basis (in one dimension), just because it is a (unitary) eigenbasis of the position operator ${\displaystyle X}$ 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 ${\displaystyle X}$ on the space of tempered distributions such that $${\displaystyle X(\delta _{x})=x\delta _{x},}$$ for every real point ${\displaystyle x}$. It's possible to prove that the unique above endomorphism is necessarily defined by $${\displaystyle X(\psi )=\mathrm {x} \psi ,}$$ for every tempered distribution ${\displaystyle \psi }$, where ${\displaystyle \mathrm {x} }$ denotes the coordinate function of the position line – defined from the real line into the complex plane by $${\displaystyle \mathrm {x} :\mathbb {R} \to \mathbb {C} :x\mapsto x.}$$