Position operator Definition and 38 Threads

Hi, as discussed in this recent thread, for a particle without spin the quantum state of the particle is described by a "point" in the Hilbert space of the (equivalence classes) of ##L^2## square-integrable functions ##|{\psi} \rangle## defined on ##\mathbb R^3##. The square-integrable...

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...

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...

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...

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...

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?

Apparently, in QM, the photon does not have a position operator. Why is this so? As usual, thanks in advance.

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/how-to-get-the-position-operator-in-the-momentum-representation-from-knowing-the), 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 square-integrable 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 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?

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>...

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: \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...

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

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 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...

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...

Does a position operator exist?

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(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') ...

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>...

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)?

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.