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
⟩
{\displaystyle |x\rangle }
we denote the unitary eigenvector of the position operator corresponding to the eigenvalue
x
{\displaystyle x}
, then,
|
x
⟩
{\displaystyle |x\rangle }
represents the state of the particle in which we know with certainty to find the particle itself at position
x
{\displaystyle x}
.
Therefore, denoting the position operator by the symbol
X
{\displaystyle X}
– in the literature we find also other symbols for the position operator, for instance
Q
{\displaystyle Q}
(from Lagrangian mechanics),
x
^
{\displaystyle {\hat {\mathrm {x} }}}
and so on – we can write
X
|
x
⟩
=
x
|
x
⟩
,
{\displaystyle X|x\rangle =x|x\rangle ,}
for every real position
x
{\displaystyle x}
.
One possible realization of the unitary state with position
x
{\displaystyle x}
is the Dirac delta (function) distribution centered at the position
x
{\displaystyle x}
, often denoted by
δ
x
{\displaystyle \delta _{x}}
.
In quantum mechanics, the ordered (continuous) family of all Dirac distributions, i.e. the family
δ
=
(
δ
x
)
x
∈
R
,
{\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
X
{\displaystyle X}
.
It is fundamental to observe that there exists only one linear continuous endomorphism
X
{\displaystyle X}
on the space of tempered distributions such that
X
(
δ
x
)
=
x
δ
x
,
{\displaystyle X(\delta _{x})=x\delta _{x},}
for every real point
x
{\displaystyle x}
. It's possible to prove that the unique above endomorphism is necessarily defined by
X
(
ψ
)
=
x
ψ
,
{\displaystyle X(\psi )=\mathrm {x} \psi ,}
for every tempered distribution
ψ
{\displaystyle \psi }
, where
x
{\displaystyle \mathrm {x} }
denotes the coordinate function of the position line – defined from the real line into the complex plane by