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.

In one dimension, if by the symbol

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

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

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

for every real point ${\displaystyle x}$. It's possible to prove that the unique above endomorphism is necessarily defined by

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