Operator Definition and 1000 Threads

Consider the Hilbert space H=L^2([0,1],dx) . Now we define the operator P=\frac \hbar i \frac{d}{dx} on this Hilbert space with the domain of definition D(P)=\{ \psi \in H | \psi' \in H \ and \ \psi(0)=0=\psi(1) \} . Then it can be shown that P^\dagger=\frac \hbar i \frac{d}{dx} with...

Homework Statement Let ##H## be a normal subgroup of a group ##(G, \star)##, and define ##G/H## as that set which contains all of the left cosets of ##H## in ##G##. Define the binary operator ##\hat{\star}## acting on the elements of ##G/H## as ##g H \hat{\star} g' K = (g \star g') H##...

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...

Hey guys, The expression \partial_{\mu}\partial^{\nu}\phi is equal to \Box \phi when \mu = \nu. However when they are not equal, is this operator 0? I'm just curious cos this sort of thing has turned up in a calculation of mine...if its 0 I'd be a very happy boy

If you have the product of two Grassman numbers C=AB, and take the conjugate, should it be C*=A*B*, or C*=B*A*? The general rule for operators, whether they are Grassman operators (like the Fermion field operator) or the Bose field operator, I think is (AB)^dagger=B^dagger A^dagger. This...

Hi guys, So I've ended up in a situation where I have \partial_{\mu}\Box\phi. where the box is defined as \partial^{\mu}\partial_{\mu}. I'm just wondering, is this 0 by any chance...? Thanks!

So I had a quiz today, and one of the questions was pretty easy, pretty straight forward. Show that ##t^2e^{9t}## is a solution of ##(D-9)^3y## Foil it out, plug in y, and you're done. Well I tried doing something else, that (at least in my mind) should have worked, but it didn't. I said...

Hello, I have a Hamiltonian that describes a particle in a rotating cylindrical container at angular frequency ω. In the lab frame the Hamiltonian is time-dependent and takes the form (using cylindrical coordinates) \mathcal H_o=\frac{\vec P^2}{2m}+V(r,\theta-\omega t,z), where V(r,\theta,z)...

Homework Statement Hey guys, So here's the problem I'm faced with. I have to show that (\Box + m^{2})<|T(\phi(x)\phi^{\dagger}(y))|>=-i\delta^{(4)}(x-y) , by acting with the quabla (\Box) operator on the following...

Hey guys, How does one compute the following quantity: \Box \theta(x_{0})=\partial_{0}\partial^{0}\theta(x_{0})? I know that \partial_{0}\theta(x_{0})=\delta(x_{0}) which is the Dirac delta, but what about the second derivative? Thanks everyone!

So if I understood well, Normal ordering just comes due to the conmutation relation of a and a⁺? right? Is just a simple and clever simplification. Wick Theorem is analogue to normal ordering because it is related to the a and a⁺ again (so related to normal ordering, indeed). However I do not...

Hi everybody! I'm studying the Fourier integral operators but I can't resolve a pass. I'm considering the following operator: $$Au(x)=\frac{1}{{(2\pi h)}^{n'}}\int_{\mathbb{R}_y^m\times\mathbb{R}_\theta^{n'}} e^{i\Psi(x,y,\theta)/h}a(x,y,\theta,h)u(y)\, dy\, d\theta$$ where $$Au\in C^0...

Hello, I am a beginner in electromagnetism. I am trying to find a vector field whose rotation equals 1 with a curl operator. If I say that the vector field is defined by V(y;2x;0) does it work? As a result, I find (0;0;1), am I right?

I am just wondering why or how we introduce the J operator in analyzing ac circuits. I want more of a proof for this.

Homework Statement Homework Equations \check{T} = BTB^{-1} (eq1) The Attempt at a Solution Ok, so I have a couple of questions here if I may ask... First, I want to be sure I understand the wording of (a) and (b) correctly. Is the following true?: (a) Write the matrix T...

Isn't it amusing ?What could be the probable explanation for this?Also when operated by division operator gives the rest of the number as the quotient (Note only when the divisor is 10)

I have a problem in understanding the quantum operators in grand canonical ensemble. The grand partition function is the trace of the operator: e^{\beta(\mu N-H)} (N is the operator Number of particle) and the trace is taken on the extended phase space: \Gamma_{es}= \Gamma_1 \times \Gamma_2...

Hi, I'm trying to prove that the conformal weight of the bosonic vertex operator :e^{ik\cdot X}: is \left(\frac{\alpha'k^2}{4},\frac{\alpha'k^2}{4}\right). I've done some algebra but I think I am making some mistake with a factor of 2 somewhere because I get a 1/2 instead of a 1/4. My attempt...

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 In the Pauli theory of the electron, one encounters the expresion: (p - eA)X(p - eA)ψ where ψ is a scalar function, and A is the magnetic vector potential related to the magnetic induction B by B = ∇XA. Given that p = -i∇, show that this expression reduces to ieBψ...

Hello question is: As you see when we do del operator on A vector filed in below example it removes exponential form at the end.why does it remove exponential form finally?

If we are given an operator, say in matrix or outer product form, then how can we check if it is a projection operator? Is idempotence a sufficient condition for an operator to be a projection operator or are there any other conditions?

Homework Statement Because I wanted to practice more of operators, I borrowed a textbook from a library for extra problems...I managed to solve (a) to (e), but not the last question...which is: Write out the operator A2 for A: (f) d2/dx2 - 2xd/dx + 1 for which I keep getting a different...

Hello--I am practicing for the upcoming quiz (as part of quantum physics), but have no idea how to solve the sample problem the teacher gave (w/ solution)... Write out the operator A2 for A = d/dx + x (Hint: INCLUDE f(x) before carrying out the operations) so...I tried: A {df(x)/dx + f(x)} =...

Homework Statement Hi, I'm doing a Quantum chemistry and one of my question is to determine if is hermitian or not? I am learning and new to this subject... Cant figure out how to do this question at all. Please helppp! ^Q= i/x^2 d/dx is hermitian or not? Homework Equations The Attempt at a...

What does mean "identically equal to" in the context of differential equations? In class the prof wrote \mu_x \equiv 0. I asked what it meant and he said "it means identical to". Can someone elaborate, for example what purpose does it surve? If it just means a function always has that value, why...

I've been reading Mermin's book on Quantum Computer Science, and in the section in which he discusses the construction of a QFT using 1-Qbit and 2-Qbit gates, he makes reference to some expressions involving linear operators that I'm not familiar with (at least if I've seen them before I've...

Homework Statement Find the eigenvalues and normalized eigenfuctions of the following Hermitian operator \hat{F}=\alpha\hat{p}+\beta\hat{x} Homework Equations In general: ##\hat{Q}\psi_i = q_i\psi_i## The Attempt at a Solution I'm a little confused here, so for example I don't know...

Homework Statement Given that U upon acting on the creation operator gives a creation operator for the transformed momentum $$U(\Lambda) a_p^\dagger U(\Lambda)^\dagger = a_{\boldsymbol{\Lambda} \mathbf{p}}^\dagger $$ and ##\Lambda ## is a pure boost, that is ## U(\Lambda) = e^{i...

Homework Statement Prove that for any stationary state the average of the commutator of any operator with the Hamiltonian is zero: \langle\left[\hat{A},\hat{H}\right]\rangle = 0. Substitute for \hat{A} the (virial) operator:\hat{A} = \frac{1}{2}\sum\limits_i\left(\hat{p}_ix_i...

In quantum mechanics, the phase of the wavefunction for a physical system is unobservable. Therefore, both ψ = ψ(x) and ψ' = ψ(x)eiθ are valid wavefunctions. For ψ = ψ(x), we have the following: \widehat{x}ψ = xψ \widehat{p}ψ = λψ For ψ' = ψ(x)eiθ, we have the following...

Hi, I was curious how I could turn any expression that looks like: x += (100- x) * 0.01; into a function that could be graphed.

Hi, friends! I find an interesting unproven statement in my functional analysis book saying the image of the closed unit sphere through a compact linear operator, defined on a linear variety of a Banach space ##E##, is compact if ##E## is reflexive. Do anybody know a proof of the statement...

Dear friends, I read that, if ##A## is a bounded linear operator transforming -I think that such a terminology implies that ##A## is surjective because if ##B=A## and ##A## weren't surjective, that would be a counterexample to the theorem; please correct me if I'm wrong- a Banach space ##E##...

Homework Statement Show that any linear operator \hat{L} can be written as \hat{L} = \hat{A} + i\hat{B}, where \hat{A} and \hat{B} are Hermitian operators. Homework Equations The properties of hermitian operators. The Attempt at a Solution I am not sure where to start with this...

A cashier distributes change using the maximum number of five dollar bills, followed by one dollar bills. For example, 19 yields 3 fives and 4 ones. Write a single statement that assigns the number of distributed 1 dollar bills to variable numOnes, given amountToChange. Hint: Use %. Sample...

Is there a current density operator or something equivalent? If so, how does it relate to other operators like momentum and angular momentum? Basically, the classical picture of a magnetic moment is a little loop of current, I would like to understand the quantum analog.

Dear friends, I have been trying in vain for a long time to understand the proof given in Kolmogorov and Fomin's of Banach's theorem of the inverse operator. At p. 230 it is said that M_N is dense in P_0 because M_n is dense in P. I am only able to see the proof that (P\cap M_n)-y_0 \subset...

Hi! I have a question for you. At the end of the post there's a link. There's the homework which I have to do for an exam. I have to study the Fourier Integral Operator that there is at the begin of the paper. I did almost all the homework but I can't do a couple of things. First: at the point...

Hi Guys, I am facing a problem playing around with some operators and Kets, would like some help! I have \langle \Psi | A+A^\dagger | \Psi \rangle .A Could someone simplify it? Especially is there a way to change the last operator A into A^\dagger? The way I thought about this is...

Is there any difference between Hamiltonian operator and E? Or do we describe H as an operation that is performed over (psi) to give us E as a function of (psi)??

Homework Statement I'm working on this problem: Let \hat{U} an unitary operator defined by: \hat{U}=\frac{I+i\hat{G}}{I-i\hat{G}} with \hat{G} hermitian. Show that \hat{U} can be written as: \hat{U}=Exp[i\hat{K}] where \hat{K} is hermitian. Homework Equations...

Homework Statement a+(k) creates particle with wave number vector k, a(k) annihilates the same; then the Klein-Gordon field operators are defined as ψ+(x) = ∑_k f(k) a(k) e^-ikx and ψ-(x) = ∑_k f(k) a+(k) e^ikx; the factor f contains constants and the ω(k). x is a Lorentz four vector, k is a...

Could anyone help me solve this problem? Let A,B be two subspace of V, a \in A, b \in B. Show that the following operation is linear and bijective: (A + B)/B → A/(A \cap B): a + b + B → a + A \cap B I really couldn't understand how the oparation itself works, i.e, what F(v) really is in...

Homework Statement Consider the following set of eigenstates of a spin-J particle: |j,j > , ... , |j,m > , ... | j , -j > where \hbar^2 j(j+1) , \hbar m are the eigenvalues of J^2 and Jz, respectively. Is it always possible to rotate these states into each other? i.e. given |j,m> and...

Given a system of two identical particles (let's say electrons), of (max) spin 1/2 (which means the magnetic quantum number of each of the electrons can be either 1/2 or -1/2), how can we write the operators (total angular momentum, z-component of the total angular momentum etc.) (a) in the...

Homework Statement I need some help regarding the gradient operator. I recently came across this statement while reading Griffith's Electrodynamics "The gradient ∇T points in the direction of maximum increase of the function T." Wolfram Alpha also states that "The direction of ∇f is the...

If the parity operator ##\hat{P}## is hermitian, then: ##\langle \phi | \hat{P} | \psi \rangle = (\langle \psi | \hat{P} | \phi \rangle)^*## Let us see if the above equation is true. The left hand side of the above equation is: ## \langle \phi | \hat{P} | \psi \rangle =...

Hi, I have real problems with the indices here, can someone give me a step by step explanation how to compute stuff with this formula? *\omega = \frac{\sqrt{|g|}}{r!(m-r)} \omega_{\mu_{1}\mu_{2}...\mu_{r}}\epsilon^{\mu_{1}\mu_{2}...\mu_{r}}_{v_{r+1}...v_{m}}dx^{v_{r+1}}\wedge...\wedge...

I have an operator which isnot Hermitian is there any way to make it hermitian ?