Commutator Definition and 266 Threads

Homework Statement Question: (With the following definitions here: ) - Consider ##L_0|x>=0## to show that ##m^2=\frac{1}{\alpha'}## - Consider ##L_1|x>=0 ## to conclude that ## 1+A-2B=0## - Consider ##L_2|x>=0 ## to conclude that ##d-4A-2B=0## - where ##d## is the dimension of the space...

I am reading a proof of why \left[ \hat{L}_x, \hat{L}_y \right ] = i \hbar \hat{L}_z Given a wavefunction \psi, \hat{L}_x, \hat{L}_y \psi = \left( -i\hbar \right)^2 \left( y \frac{\partial}{\partial z} - z \frac {\partial}{\partial y} \right ) \left (z \frac{\partial \psi}{\partial x} -...

Homework Statement (I have dropped the hats on the ##\alpha_{n}^{u}## operators and ##L_{m}##) ##[\alpha_{n}^u, \alpha_m^v]=n\delta_{n+m}\eta^{uv}## ##L_m=\frac{1}{2}\sum\limits_{n=-\infty}^{\infty} : \alpha_{m-n}^u\alpha_{n}^v: \eta_{uv}-\delta_{m,0}## where : denotes normal-ordered. Show...

Hi. I'm afraid I might just be discovering quite a big misunderstanding of mine concerning the meaning of the expectation value of a commutator for a given state. I somehow thought that if the expectation value of the commutator of two observables ##A, B## is zero for a given state...

How would ##[p_x, r]## be expanded? Where ##r=(x,y,z)##, the position operators. Do you do the commutators of ##p_x## with ##x, y,z## individually? So ##[p_x,x]+[p_x,y]+[p_x,z]## for example?

Given the property, [A,BC] = [A,B]C+B[A,C], is it true that, if B=C, then [A,B]C+B[A,C]=[A,B]C+B[A,B]=[A,B](C+B)? I apologize if I have posted in the wrong forum.

Hi I have seen an example of commutator of the Parity operator of the x-coordinate , Px and angular momentum in the z-direction Lz calculated as [ Px , Lz ] ψ(x , y) = -2Lz ψ (-x , y) I have tried to calculate the commutator without operating on a wavefunction and just by expanding...

Hi there I came across this paper. the author defines a covariant derivative in (1.3) ##D_\mu = \partial_\mu - ig A_\mu## He defines in (1.6) ##F_{jk} = i/g [D_j,D_k]## Why is it equal to ##\partial_j A_k - \partial_k A_j - ig [A_j, A_k]##? I suppose that it comes from a property of Lie...

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...

i am having a hard time understanding why do we place the brushes on the Magnetic neutral axis, the textbook simply says, because this is where the current reversal takes place, is the point where emf from different meet is the same as the point of current reversal?

Homework Statement a = √(mω/2ħ)x + i√(1/2ħmω), a† = √(mω/2ħ)x - i√(1/2ħmω), find [a,a†] the solution is given. it should be 1. Homework Equations [a,b] = ab -ba The Attempt at a Solution im guessing there is something I'm missing or I'm not doing something somewhere. I'm just doing what...

Homework Statement Find the resul of [Jx Jy , Jz] where J is the angular momentum operator. Possible answers to this multiple chioce question are A) 0 B) i ħ Jz C) i ħ Jz Jx D) i ħ Jx Jz E) i ħ Jx Jy Homework Equations [AB,C]=A [B,C]+[A,B] B [Ji , Jj]=i ħ εijk Jk where εijk is the Levi-Civita...

Homework Statement Show that \left [ A,B \right ]^{\dagger}=-\left [A,B \right ] Homework Equations \left [ A,B \right ] = AB-BA \left (AB \right)^{\dagger}= B^{\dagger}A^{\dagger} The Attempt at a Solution \left [ A,B \right ]^{\dagger}=\left (AB-BA \right )^{\dagger} =\left (AB...

In QFT, the commutation relation for the field operator \hat{\phi} and conjugate momentum is [\phi(x,t),\pi(y,t)] = i\delta(x-y) Maybe this is obvious, but what would the commutator of \phi or \pi and, say, e^{i k\cdot x} be?

Homework Statement [/B] For a general operator ## \hat{O}##, let ##\hat{O}_{mn}(t)## be defined as: $$ \hat{O_{mn}}(t) = \int u^{*}_{m}(x,t) \hat{O} u_{n}(x,t) $$ and $$ \hat{O_{mn}} = \int u^{*}_{m}(x) \hat{O} u_{n}(x) $$ ##u_{m}## and ##u_{n}## are energy eigenstates with corresponding...

I am reading through a quantum optics book where they are deriving the equations for a quantized EM field and one of the paragraphs state: "In Section 6.1, the problem has been set in the Hamiltonian form by expressing the total energy (6.55) of the system comprising charges and electromagnetic...

Hey there! 1. Homework Statement I've been given the operators a=\sqrt\frac{mw}{2\hbar}x+i\frac{p}{\sqrt{2m\hbar w}} and a^\dagger=\sqrt\frac{mw}{2\hbar}x-i\frac{p}{\sqrt{2m\hbar w}} without the constants and definition of the momentum operator: a=x+\partial_x and a^\dagger=x-\partial_x with...

Here is the question: By using the equality (for boson) ---------------------------------------- (1) Prove that Background: Currently I'm learning things about second quantization in the book "Advanced Quantum Mechanics"(Franz Schwabl). Given the creation and annihilation operators(), define...

Homework Statement Homework Equations no equations required The Attempt at a Solution so here are my answers, i just want to know if they are correct. also, I am really confused about the function of the commutator in the DC motor, the answer i wrote for part b) was based on research off...

The generators ##\{ L^1, L^2 , L^3 , K^1 , K^2 , K^3 \}## of the Lorentz group satisfy the Lie algebra: \begin{array}{l} [L^i , L^j] = \epsilon^{ij}_{\;\; k} L^k \\ [L^i , K^j] = \epsilon^{ij}_{\;\; k} K^k \\ [K^i , K^j] = \epsilon^{ij}_{\;\; k} L^k \end{array} It has the Casimirs C_1 =...

Hey! :o We have that $D_n=\langle a,s\mid s^n=1=a^2, asa^{-1}=s^{-1}\rangle$. I want to show the following: $s^2\in D_n'$ $D_n'\cong \mathbb{Z}_n$ if $n$ is odd $D_n'\cong \mathbb{Z}_{\frac{n}{2}}$ if $n$ is even $D_n$ is nilpotent if and only if $n=2^k$ for some $k=1,2,\dots $ I...

In the srednicki notes he goes from $$H = \int d^{3}x a^{\dagger}(x)\left( \frac{- \nabla^{2}}{2m}\right) a(x) $$ to $$H = \int d^{3}p\frac{1}{2m}P^{2}\tilde{a}^{\dagger}(p)\tilde{a}(p) $$ Where $$\tilde{a}(p) = \int \frac{d^{3}x}{(2\pi)^{\frac{3}{2}}}e^{-ipx}a(x)$$ Is this as simple as...

If we have Poisson bracket for two dynamical variables u and v, we can write as it is known ... This is for classical mechanics. If we write commutation relation, for instance, for location and momentum, we obtain Heisenberg uncertainty relation. But, what is a pedagogical transfer from...

Homework Statement I'm having a bit of trouble evaluating the following commutator $$ \left[T^{+},T^{-}\right] $$ where T^{+}=\int_{M}d^{3}x\:\bar{\nu}_{L}\gamma^{0}e_{L}=\int_{M}d^{3}x\:\nu_{L}^{\dagger}e_{L} and...

Hi, I've just wierded myself out so time to stop for today, but afore I go ... Angular momentum matrices (also Pauli) anti-commute as $ [J_x, J_y] = iJ_z $ So $ Det\left( [J_x, J_y] \right) = i Det(J_z) $ $\therefore Det(J_xJ_y)-Det(J_yJ_x) = i Det(J_z) $ $\therefore...

Homework Statement I was reading this textbook: https://books.google.com/books?id=sHJRFHz1rYsC&lpg=PA317&ots=RpEYQhecTX&dq=orbit%20center%20operators&pg=PA310#v=onepage&q=orbit%20center%20operators&f=false Homework Equations In the equation of the page (unlabeled), we have $$...

Can someone tell me if the parts of what I've done are right, and explain the questions I've missed. We have just done a lab on split-ring commutators and I'm currently doing a lab report. "Explain what is meant by the torque of an electric motor. Use a diagram" I've said that 'The torque of an...

If I'm not mistaking , all DC and also universal motors have a brushed commutator not a slip ring commutator because the rotor wires need to keep the same current direction as they rotate pass the same magnetic stator pole. so the only DC motor which could operate on slip rings is a homopolar...

Consider two self-adjoint operators A and B with commutator [A,B]=C such that [A,C]=0. Now I consider an operator which is a function of A and is defined by the series ## F(A)=\sum_n a_n A^n ## and try to calculate its commutator with B: ## [F(A),B]=[\sum_n a_n A^n,B]= \\ \sum_n a_n...

Homework Statement Let J-hat be a quantum mechanial angular momentum operator. The commutator [J^hat_x J^hat_y,J^hat_z] is equivalent to which of the following Homework Equations [J^hat_x,J^hat_y]=iħJ^hat_z [J^hat_y,J^hat_z]=iħJ^hat_x [J^hat_z,J^hat_x]=iħJ^hat_y [A,B]=[AB-BA] The Attempt at...

Homework Statement Consider the quantum mechanical Hamiltonian ##H##. Using the commutation relations of the fields and conjugate momenta , show that if ##F## is a polynomial of the fields##\Phi## and ##\Pi## then ##[H,F]-i \partial_0 F## Homework Equations For KG we have: ##H=\frac{1}{2} \int...

Hi, so i want to ask what's the function of the split between the commutator? my guess is to temporarily separate the opposite electrons flow between the coil and external circuit after the half cycle? so that's why the current is reversed? please i really need someone to explain how the...

Hello all, I'm trying to calculate a commutator of two covariant derivatives, as it was done in Caroll, on page 122. The problem is, I don't get the terms he does :-/ If ##\nabla_{\mu}, \nabla_{\nu}## denote two covariant derivatives and ##V^{\rho}## is a vector field, i need to compute...

Hi, This is a question regarding Example 3.6 in Section 3.5 (p.35) of 'QFT for the Gifted Amateur' by Lancaster & Blundell. Given, [a^{\dagger}_\textbf{p}, a_\textbf{p'}] = \delta^{(3)}(\textbf{p} - \textbf{p'}) . This I understand. The operators create/destroy particles in the momentum state...

For particle in the box wave function, it is the eigenfunction of kinetic energy operator but not the eigenfunction of momentum operator. So, do these two operators commute? (or it has nothing to do with commutator stuff?) How about for free particle? For free particle, the wave function is...

Homework Statement So I've been trying to derive field strength tensor. What to do with the last 2 parts ? They obviously don't cancel (or do they?) Homework EquationsThe Attempt at a Solution [D_{\mu},D_{\nu}] = (\partial_{\mu} + A_{\mu})(\partial_{\nu} + A_{\nu}) - (\mu <-> \nu) =...

Homework Statement commutator of [x,p e^(-p) ] Homework EquationsThe Attempt at a Solution answer is i - i.e^(-p)

Homework Statement To show [J2, J+] = 0 2. Homework Equations J+ = Jx + i Jy [J2, Jx ] = 0 [J2, Jy ] = 0The Attempt at a Solution Step 1: L.H.S. = [J2, J+] Step 2: L.H.S. = [J2, Jx + i Jy ] Step 3: L.H.S. = [J2, Jx ] + i [J2, Jy ] Step 4: L.H.S. = 0 + 0 Step...

Homework Statement Suppose A^ and B^ are linear quantum operators representing two observables A and B of a physical system. What must be true of the commutator [A^,B^] so that the system can have definite values of A and B simultaneously? Homework Equations I will use the bra-ket notation for...

Hi, For SU(2) I can have that all Noether charges commute with one of the charges as one of the generators of the Lie algebra is the identity. Can somebody explain me how this is related to the properties of SU(2)? Charges can be considered to be generators of the transformation. So if this...

Consider the rotation group ##SO(3)##. I know that ##R_{x}(\phi) R_{z}(\theta) - R_{z}(\theta) R_{x} (\phi)## is a commutator? But can this be called a commutator ##R_{z}(\delta \theta) R_{x}(\delta \phi) R_{z}^{-1}(\delta \theta) R_{x}^{-1} (\delta \phi)##?

Homework Statement I am trying to show that ##a(x)[u(x),D^{3}]=-au_{xxx}-3au_{xx}D-3au_{x}D^{2}##, where ##D=d/dx##, ##D^{2}=d^{2}/dx^{2} ## etc.Homework Equations [/B] I have the known results : ##[D,u]=u_{x}## ##[D^{2},u]=u_{xx}+2u_{x}D## The property: ##[A,BC]=[A,B]C+B[A,C] ##*The...

Homework Statement Let the commutator [A,B] = cI, I the identity matrix and c some arbitrary constant. Show [A,Bn] = cnBn-1 Homework Equations [A,B] = AB - BA The Attempt at a Solution So I have started off like this: [A,Bn] = ABn - BnA = cI I'm not sure where to go from here.

Homework Statement Let A and B be two observables that both commute with their commutator [A,B]. a) Show, e.g., by induction, that [A,Bn]=nBn-1 [A,B].The Attempt at a Solution Prove for n=1 [A,B1]=1B1-1 [A,B]. [A,B]=B0[A,B]=[A,B] Show that it is true for n+1 [A,Bn+1]=[A,BnB]=Bn[A,B]+[A,Bn]B...

Hi all! I was wondering what the necessary condition is for two arbitrary matrices, say A and B, to commute: AB = BA. I know of several sufficient conditions (e.g. that A, B be diagonal, that they are symmetric and their product is symmetric etc), but I can't think of a necessary one. Thanks...

Could someone explain to me how the author goes from 2nd to 3rd step I think the intermediate step between 2 and 3 is basically to split up the commutator as [y p_z, z p_x] - [y p_z,x p_z] - [z p_y,z p_x] + [z p_y, x p_z] 2nd term = 0 3rd term = 0 so leftover is [L_x, L_y] = [y p_z, z p_x]...

Homework Statement Prove that ## [L_a,L_b L_b] =0 ## using Einstein summation convention.Homework Equations [/B] ## (1) [L_a,L_b] = i \hbar \epsilon_{abc} L_c ## ## (2) \epsilon_{abc} \epsilon_{auv} = \delta_{bu} \delta_{cv}- \delta_{bv} \delta_{cu}## ## (3) \epsilon_{abc} = \epsilon_{bca}...

Homework Statement Prove that ## [L_a,L_b] = i \hbar \epsilon_{abc} L_c ## using Einstein summation convention. I think I have achieved the solution but I am not sure of my last steps, since this is one of my first excersises using this convention. Homework Equations [/B] ## (1)...

The book calculates the commutator $[H,x_i]$ as $$[H,x_i] = \left[ \sum_j \frac{p_j^2}{2m}, x_i \right] = \frac{2}{2m} \sum_j p_j \frac{\hbar}{i} \delta_{ij} = - \frac{i \hbar p_i}{m},$$ where the hamiltonian operator $H$ is $$H = \sum_j \frac{{\mathbf p}_j^2}{2m_j} + V({\mathbf x}).$$ The book...

Homework Statement Given that the function f can be expanded in a power series of a and a^\dagger, show that: [a,f(a,a^{\dagger})]=\frac{\partial f }{\partial a^\dagger} and that [a,e^{-\alpha a^\dagger a}] = (e^{-\alpha}-1)e^{-\alpha a^{\dagger} a}aThe Attempt at a Solution I've tied using...