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





|

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





x

:

R

→

C

:
x
↦
x
.


{\displaystyle \mathrm {x} :\mathbb {R} \to \mathbb {C} :x\mapsto x.}

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