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 Xx\rangle =xx\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
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 mr n \ell m\rangle=\langle 100r 100\rangle \\ &=\int r\left\psi_{n \ell m}(r, \theta...
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=xx\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...
As I understand it, Ψ2 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...
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)...
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 :
\begin{equation}
\langle Bx, x\rangle
\end{equation}
when replaced by:\begin{equation}
\langle Bix...
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...
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?
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}...
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...
hi, initially I want to put into words that I looked up the link (http://physics.stackexchange.com/questions/86824/howtogetthepositionoperatorinthemomentumrepresentationfromknowingthe), and I saw that $$\langle p[\hat x,\hat p]\psi \rangle = \langle p\hat x\hat p\psi \rangle ...
Well i am noobie to quantum physics so i matbe totally incorrect so please bear with me.
I had question how is position operator defined mathematically.
I was reading the momentum position commutator from...
Hi, I wish to get position operator in momentum space using Fourier transformation, if I simply start from here,
$$ <x>=\int_{\infty}^{\infty} dx \Psi^* x \Psi $$
I could do the same with the momentum operator, because I had a derivative acting on psi there, but in this case, How may I get...
In quantum mechanics, the position operator(for a single particle moving in one dimension) is defined as Q(\psi)(x)=x\psi(x) , with the domain D(Q)=\{\psi \epsilon L^2(\mathbb R)  Q\psi\epsilon L^2 (\mathbb R) \} . But this means no squareintegrable function in the domain becomes...
Might be simple but I couldn't see. We can easily derive momentum operator for position space by differentiating the plane wave solution. Analogously I want to derive the position operator for momentum space, however I am getting additional minus sign.
By replacing $$k=\frac{p}{\hbar}$$ and...
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...
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...
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...
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...
Why is the position operator of a particle on the xaxis 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?
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> = xx> be correct.
\overset {}{x} = \int <\psix> (\int<x\overset {\wedge}{x}x'><x'\psi> dx')dx = \int <\psix>...
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...
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...
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...
Hi all,
I understand how to transform between position space and momentum space; it's a Fourier transform:
\varphip>=\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...
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}}...
So I've been having a specific major hangup 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...
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...
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...
Hi, Everybody know that eigenfunction of position operator x' is \delta(xx')
But i also knew that integral of square of current state over entire space is 1(probability)
Then, \int_{\infty}^{\infty}\delta(xx')\delta(xx')^{*} dx is 1?
What is conjugate of \delta(xx') ...
Homework Statement
Find an expression for <PXP> in terms of P(x) defined as <xP> (and possibly P*(x) )
Homework Equations
XP> = xP>
Identity operator: integral of x><x dx
The Attempt at a Solution
Ok...<PXP> add the identity
= Integral [ <PXx> <xP> dx ]
= Integral [<Pxx>...
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...
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.