# What is Position operator: Definition and 37 Discussions

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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1. ### I What is the Role of the Radial Position Operator in Quantum Mechanics?

While trying to find the expectation value of the radial distance ##r## of an electron in hydrogen atom in ground state the expression is : ##\begin{aligned}\langle r\rangle &=\langle n \ell m|r| n \ell m\rangle=\langle 100|r| 100\rangle \\ &=\int r\left|\psi_{n \ell m}(r, \theta...
2. ### I Adjoint of position operator

I'm trying to find the adjoint of position operator. I've done this: The eigenvalue equation of position operator is ##\hat{x}|x\rangle=x|x\rangle## The adjoint of position operator acts as ##\left\langle x\left|\hat{x}^{\dagger}=x<x\right|\right.## Then using above equation we've...
3. ### Show that the position operator does not preserve H

The attempt ##\int_{-\infty}^{\infty} |ψ^*(x)\, \hat x\,\psi(x)|\, dxˆ## Using ˆxψ(x) ≡ xψ(x) =##\int_{-\infty}^{\infty} |ψ^*(x)\,x\,\psi(x)|\, dxˆ## =##\int_{-\infty}^{\infty} |ψ^*(x)\,\psi(x)\,x|\, dxˆ## =##\int_{-\infty}^{\infty} |x\,ψ^2(x)|\, dxˆ## I'm pretty sure this is not the...
4. ### I Why do we need the position operator?

As I understand it, |Ψ|2 gives us the probability density of the wavefunction, Ψ. And when we integrate it, we get the probability of finding the particle at whichever location we desire, as set by the limits of the integration. But when we use the position operator, we have integrand Ψ*xΨ dx...
5. M

### Finding the position operator in momentum space

Homework Statement Given ##\hat{x} =i \hbar \partial_p##, find the position operator in the position space. Calculate ##\int_{-\infty}^{\infty} \phi^*(p) \hat{x} \phi(p) dp ## by expanding the momentum wave functions through Fourier transforms. Use ##\delta(z) = \int_{\infty}^{\infty}\exp(izy)...
6. S

### A Can imaginary position operators explain real eigenvalues in quantum mechanics?

Hello, some operators seem to "add up" and give real eigenvalues only if they are applied on the imaginary position, ix, rather than the normal position operator, x, in the integral : $$\langle Bx, x\rangle$$ when replaced by: \langle Bix...
7. ### A Transformation of position operator under rotations

In the momentum representation, the position operator acts on the wavefunction as 1) ##X_i = i\frac{\partial}{\partial p_i}## Now we want under rotations $U(R)$ the position operator to transform as ##U(R)^{-1}\mathbf{X}U(R) = R\mathbf{X}## How does one show that the position operator as...
8. ### I Position Operator Action on Wave Function: $\psi(x)$

Would the action of the position operator on a wave function ##\psi(x)## look like this? $$\psi(x) \ =\ <x|\psi>$$ $${\bf \hat x}<x|\psi>$$ Question 2: the position operator can act only on the wave function?
9. ### I Why no position operator for photon?

Apparently, in QM, the photon does not have a position operator. Why is this so? As usual, thanks in advance.
10. ### Position Operator in Momentum Space?

Homework Statement So, I'm doing this problem from Townsend's QM book 6.2[/B] Show that <p|\hat{x}|\psi> = i\hbar \frac{\partial}{\partial p}<p|\psi> Homework Equations |\psi(p)> = \int_\infty^{-\infty} dp |p><p|\psi> The Attempt at a Solution So, <p|\hat{x}|\psi> = <p|\hat{x}...
11. ### Momentum and Position Operator Commutator Levi Civita Form

Homework Statement Prove that ##[L_i,x_j]=i\hbar \epsilon_{ijk}x_k \quad (i, j, k = 1, 2, 3)## where ##L_1=L_x##, ##L_2=L_y## and ##L_3=L_z## and ##x_1=x##, ##x_2=y## and ##x_3=z##. Homework Equations There aren't any given except those in the problem, however I assume we use...

17. ### Matrix elements of position operator in infinite well basis

Homework Statement Find the eigenfunctions of a particle in a infinite well and express the position operator in the basis of said functions.Homework Equations The Attempt at a Solution Tell me if I'm right so far (the |E> are the eigenkets) X_{ij}= \langle E_i \vert \hat{X} \vert E_j \rangle...
18. ### Does position operator have eigen wave function?

I am learning quantum mechics. The hypothesis is: In the quantum mechanics, all operators representing observables are Hermitian, and their eigen functions constitute complete systems. For a system in a state described by wave function ψ(x,t), a measurement of observable F is certain to...
19. ### Eigenvalues of the position operator

I'm new to QM, but I've had a linear algebra course before. However I've never dealt with vector spaces having infinite dimension (as far as I remember). My QM professor said "the eigenvalues of the position operator don't exist". I've googled "eigenvalues of position operator", checked into...
20. ### Help understanding position operator eigenfunction derivation

I'm having trouble understanding the derivation of the the position operator eigenfunction in Griffiths' book : How is it "nothing but the Dirac delta function"?? (which is not even a function). Couldn't g_{y}(x) simply be a function like (for any constant y) g_{y}(x) = 1 | x=y...
21. ### Rationale of the position operator?

Why is the position operator of a particle on the x-axis defined by x multiplied by the wave function? Is there an intuitive basis for this or is it merely something that simply works in QM?
22. ### Eigenvalue of position operator and delta function.

I'd like to show that if there exists some operator \overset {\wedge}{x} which satisfies \overset {-}{x} = <\psi|\overset {\wedge}{x}|\psi> , \overset {\wedge}{x}|x> = x|x> be correct. \overset {-}{x} = \int <\psi|x> (\int<x|\overset {\wedge}{x}|x'><x'|\psi> dx')dx = \int <\psi|x>...
23. ### Position operator for a system of coupled harmonic oscillators

Hi I would like to know if I have a system of coupled harmonic oscillators whether the standard position operator for an uncoupled oscillator is valid, i.e. x_i = \frac{1}{\sqrt{2}} (a_i+a_i^\dagger)? where i labels the ions. To give some context I am looking at a problem involving a...
24. ### Do the eigenfunctions for the position operator form an orthogonal set?

Starting with, \hat{X}\psi = x\psi then, x\psi = x\psi \psi = \psi So the eigenfunctions for this operator can equal anything (as long as they keep \hat{X} linear and Hermitian), right? Well, McQuarrie says that "the eigenfunctions of a Hermitian operator are orthogonal", which can be...
25. ### Commutator of square angular momentum operator and position operator

can someone please help me with this. it's killing me. Homework Statement to show \left[\vec{L}^{2}\left[\vec{L}^{2},\vec{r}\right]\right]=2\hbar^{2}(\vec{r}\vec{L}^{2}+\vec{L}^{2}\vec{r})Homework Equations I have already established a result (from the hint of the question) that...
26. ### Position operator in momentum space (and vice-versa)

Hi all, I understand how to transform between position space and momentum space; it's a Fourier transform: \varphi|p>=\frac{1}sqrt{2\hbar\pi}\int_{\infty}^{\infty} <x|\varphi> exp(-ipx/\hbar)dx But I can't figure out how to transform the operators. I know what they transform into (e.g...
27. ### Matrix element of Position Operator For Hydrogen Atom

Find <nlm|(1/R)|nlm> for the hydrogen atom (nlm = 211), R is just the radial position Operator (X2 + Y2 + Z2)½.
28. ### Matrix Element of Position Operator

Homework Statement Calculate the general matrix element of the position operator in the basis of the eigenstates of the infinite square well. Homework Equations |\psi\rangle =\sqrt{\frac{2}{a}}\sin{\frac{n \pi x}{a}}...
29. ### What is a useful way to talk about eigenstates of the position operator

So I've been having a specific major hang-up when it comes to understanding basic quantum mechanics, which is the position operator. For the SHO, the time independent Schroedinger's equation looks like E\psi = \frac{\hat{p}^2}{2m}\psi + \frac{1}{2}mw^2\hat{x}^2\psi Except that...
30. ### Position operator is it communitative

I was asked to show how the position operator is not communitative in the Shrodinger Wave equation. I thought it was as it is simply mulitplication [x]=integral from negative to positive infinite over f*(x,t) x f(x,t) dx Can anyone help shed some light on this. I may have misunderstood the...
31. ### Does a position operator exist?

Does a position operator exist?
32. ### Position Operator: What's The Catch?

Usually in QM we say that a wavefunction psi is an eigenfunction of some operator if that operator acting on psi gives eigenvalue * psi. The position operator is just "multiply by x". So any psi would seem to fit the above description of an eigenfunction of the position operator with...
33. ### [Q]eigenfunction of position operator and negative energy

Hi, Everybody know that eigenfunction of position operator x' is \delta(x-x') But i also knew that integral of square of current state over entire space is 1(probability) Then, \int_{-\infty}^{\infty}\delta(x-x')\delta(x-x')^{*} dx is 1? What is conjugate of \delta(x-x') ...
34. ### Position operator in infinte vector space

Homework Statement Find an expression for <P|X|P> in terms of P(x) defined as <x|P> (and possibly P*(x) ) Homework Equations X|P> = x|P> Identity operator: integral of |x><x| dx The Attempt at a Solution Ok...<P|X|P> add the identity = Integral [ <P|X|x> <x|P> dx ] = Integral [<P|x|x>...
35. ### Position Operator: f(\hat{x})=f(x)? Effects on g(x)

is it true that: f(\hat{x})=f(x)? What will happen if f(\hat{x})=\frac{\hat{x}}{\hat{x}+1} act on g(x)?
36. ### Fourier Integral involving a position operator

Hello. I'm teaching myself quantum mechanics. I want to understand the meaning of the following integral representation: q^{-\frac{1}{2}} = \kappa \int_{-\infty}^{\infty}\frac{dt}{\sqrt{t}}exp(itq) where q is the quantum mechanical position operator. I know that this is a Fourier Integral...
37. ### What is the physical meaning of the position operator in QFT?

I'm reading some QFT and have been puzzled by the following question: What's the physical meaning of the position OPERATOR X_\mu in QFT? whose position does it measure?:confused: Thanks for any help.