Commutation Definition and 220 Threads

Maxwell field commutation relations I'm looking at Aitchison and Hey's QFT book. I see in Chapter 7, (pp. 191-192), they write down the canonical momentum for the Maxwell field A^\mu(x): \pi^0=\partial_\mu A^\mu \\ \pi^i=-\dot{A}^i+\partial^i A^0 and then write down the commutation...

Homework Statement Deduece the commutation relations of position operator (squared) \hat R^2 with angular momentum \hat L Homework Equations [xi,xj]=0, Lj= εijkxjPk, [xi, Pl]=ih, [xi,Lj]=iℏϵijkxk The Attempt at a Solution The previous question related R and L and the result was [\hat R,\hat...

In Quantum Field Theory by Lancaster, equation 3.14 $$ [\hat{a_i},\hat{a_j}^\dagger]=\delta{ij}$$ is introduced as "we define". Yes, example 2.1, where the creation and annihilation operators applied to harmonic operator states, there is a nice simple proof that this is true (although...

Homework Statement Firstly, I'm looking at this: I'm confused because my understanding is that the commutator should be treated like so: $$[a,a^{\dagger}] = aa^{\dagger} - a^{\dagger}a$$ but the working in the above image looks like it only goes as far as $$aa^{\dagger}$$ This surely...

Dear Everyone, A simple question. Do α and β spins commute? In other words, can we say αβ = βα ? Thank you for your help.

I find it difficult to believe that the canonical commutation relations for a complex scalar field are of the form ##[\phi(t,\vec{x}),\pi^{*}(t,\vec{y})]=i\delta^{(3)}(\vec{x}-\vec{y})## ##[\phi^{*}(t,\vec{x}),\pi(t,\vec{y})]=i\delta^{(3)}(\vec{x}-\vec{y})## This seems to imply that the two...

I would like to show that the commutation relations ##[a_{\vec{p}},a_{\vec{q}}]=[a_{\vec{p}}^{\dagger},a_{\vec{q}}^{\dagger}]=0## and ##[a_{\vec{p}},a_{\vec{q}}^{\dagger}]=(2\pi)^{3}\delta^{(3)}(\vec{p}-\vec{q})## imply the commutation relations...

Homework Statement Given the mode expansion of the quantum field ##\phi## and the conjugate field one can derive $$\mathbf P = \int \frac{d^3 \mathbf p}{(2\pi)^3 2 \omega(\mathbf p)} \mathbf p a(\mathbf p)^{\dagger} a(\mathbf p)$$ By writing $$e^X = \text{lim}_{n \rightarrow \infty}...

Homework Statement Need to show that [a,f(a,a^\dagger]=\frac{\partial f}{\partial a^\dagger} Homework Equations [a,a^\dagger]=1 The Attempt at a Solution Need to expand f(a,a^\dagger) in a formal power series. However I don´t know how to do it if the variables don´t commute.

Homework Statement I would like to know how to derive the quantum commutation relations in matrix form, $$i \hbar \partial_t x(t)= [x(t),E]$$ $$i \hbar \partial_t P(t)= [P(t),E]$$ Where X(t), P(t) and E are the position, momentum and the energy of the particle, respectively. 2. Homework...

Hi, friends! I have been struggling to understand the only derivation of Ampère's law from the Biot-Savart law for a tridimensional distribution of current that I have been able to find, i.e. Wikipedia's outline of proof, for more than a month with no result. I have also been looking for a proof...

Homework Statement By the time t = 0, the network was in steady state. At time t = 0, the switch is turned on. Find the voltage on the capacitance C2 immediately after the commutation.[/B]Homework Equations KCL i(-0) = -ic1(+0) - ic2(+0) KVL E-i(-0) * R-Vc1(-0) = 0 Vc1(+0) = Vc2(+0)...

I would like to prove that the angular momentum operators ##\vec{J} = \vec{x} \times \vec{p} = \vec{x} \times (-i\vec{\nabla})## can be used to obtain the commutation relations ##[J_{i},J_{j}]=i\epsilon_{ijk}J_{k}##. Something's gone wrong with my proof below. Can you point out the mistake...

I've heard it said that the commutation relations of the generators of a Lie algebra determine the multiplication laws of the Lie group elements. I would like to prove this statement for ##SO(3)##. I know that the commutation relations are ##[J_{i},J_{j}]=i\epsilon_{ijk}J_{k}##. Can you...

The generators ##(A_{ab})_{st}## of the ##so(n)## Lie algebra are given by: ##(A_{ab})_{st} = -i(\delta_{as}\delta_{bt}-\delta_{at}\delta_{bs}) = -i\delta_{s[a}\delta_{b]t}##, where ##a,b## label the number of the generator, and ##s,t## label the matrix element. Now, I need to prove the...

Homework Statement See , in the methods of improving the commtuation , one of the method is using interpoles now what I think that interpoles aids commutation by nullifying the reactance voltage in the conductors. Though this point is mentioned with the working but it is also mentioned that...

Homework Statement 1. Show that the Lorentz algebra generator ##J^{\mu \nu} = i(x^{\mu}\partial^{\nu}-x^{\nu}\partial^{\mu})## lead to the commutation relation ##[J^{\mu \nu}, J^{\rho \sigma}] = i(g^{\nu \rho}J^{\mu \sigma} - g^{\mu \rho}J^{\nu \sigma}-g^{\nu \sigma}J^{\mu \rho}+g^{\mu...

Homework Statement Using the given equations prove that Homework Equations , ,[/B] + (it won't render together in Maple for whatever reason) The Attempt at a Solution So I started with expanding the Jacobi Identity (the third relevant equation) and through tedious algebra arrived at...

Homework Statement Use the formulas given (which have been solved in previous questions) prove that where w_12 is a complex constant. From here, induce that where eps_abc is the fully anti-symmetric symbol Homework Equations The equations given to use are: The Attempt at a...

Hi! If I have understood things correctly, in a multi-electron atom you have that the spin operator ##S## commutes with the orbital angular momentum operator ##L##. However, as these operators act on wavefunctions living in different Hilbert spaces, how is it possible to even calculate the...

Homework Statement See uploaded file. Homework Equations I guess one needs to keep in mind this: https://en.wikipedia.org/wiki/Complete_set_of_commuting_observables The Attempt at a Solution Basically, my question is about the notation: 1) What does the subscript "ee" stand for in H_ee? And...

I am trying to do go over the derivations for the principle of least action, and there seems to be an implicit assumption that I can't seem to justify. For the simple case of particles it is the following equality δ(dq/dt) = d(δq)/dt Where q is some coordinate, and δf is the first variation in...

Hello, I have got a three phase motor connected in delta that is controlled in six step commutation. When I measure the line current of one of the phase under oscilloscope, I got something like this : Can anyone tell me why the amplitude of the current isn't 3.1A and -3.1A? What I got here...

Problem Question My question isn't an entire homework problem, but rather for a certain mathematical step in the problem which I assume to be very simple. The problem is dealing with interacting fermion systems using second quantization formulas. I am essentially following my notes from class...

Homework Statement Consider the Dirac Hamiltonian ##\hat H = c \alpha_i \hat p_i + \beta mc^2## . The operator ##\hat J## is defined as ##\hat J_i = \hat L_i + (\hbar/2) \Sigma_i##, where ##\hat L_i = (r \times p)_i## and ##\Sigma_i = \begin{pmatrix} \sigma_i & 0 \\0 & \sigma_i...

It seems to be implied, but I can't find it explicitly - the order in which linear operators are applied makes a difference. IE given linear operators A,B then AB is NOT necessarily the same as BA ? I thought it was only with rotation operators that the order made a difference? I noticed this...

So the total angular momentum operator J commutes with any scalar operator S. The argument for this is that J is the generator of 'turntable rotations' (by this I mean we rotate the whole object about an axis, along with its orientation) and the expectation value of any scalar operator has to be...

Homework Statement Consider left-handed fermions in two spacetime dimensions ##(t,x)##: ##\psi_L=\frac{1}{2}(1-\gamma_5)\psi_D## with ##J_0^\epsilon(t,x)=\psi_L^+(x+\epsilon)\psi_L(x-\epsilon)##. (a). Use canonical equal-time anti-commutation relations for fermions to compute...

Homework Statement Given a Poincaré transformation, Lorentz+translation, I have to find the Poincaré generators in the scalar field representation and then prove that the commutation relations. I've done the first part but I can't prove the commutation relations. Homework Equations...

Homework Statement Given that [A_i,J_j]=i\hbar\epsilon_{ijk}Ak where A_i is not invariant under rotation Show that [J^2,Ai]=-2i\hbar\epsilon_{ijk}J_jAk-2\hbar^2A_i Homework Equations [AB,C]=A[B,C]+[A,C]B [A,B]=-[B,A]The Attempt at a Solution [J^2,Ai]=[J_x^2,Ai]+[J_y^2,Ai]+[J_z^2,Ai]...

Hi , I need help with the this exercise: a) Work out all of the canonical commutation relations for components of the operators r and p: [x,y] [x,py] [x,px] [py,pz] and so on. Answer: [ri,pj]=−[pi,rj]=iℏδij [ri,rj]=−[pi,pj]=0 , where the indices stand for x, y, or z and rx=x ry=y rz=z where...

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...

Homework Statement Simplify the following commutator involving the creation and annihilation operators. [a^{\dagger}a,a \sqrt{a^\dagger a} ] Homework Equations I know that [a,a^\dagger] = 1. The Attempt at a Solution I think I should be trying to put the creation operators to the left...

Hello there. I am trying to proove in a general way that [Lx2,Lz2]=[Ly2,Lz2]=[Lz2,Lx2] But I am a little bit stuck. I've tried to apply the commutator algebra but I'm not geting very far, and by any means near of a general proof. Any help would be greatly appreciated. Thank you.

Is there a way to determine the group from the commutation relations? For example, the commutation relations: [J_x,J_y]=i\sqrt{2} J_z [J_y,J_z]=\frac{i}{\sqrt{2}} J_x [J_z,J_x]=i\sqrt{2} J_y is actually SO(3), as can be seen by redefining J'_x =\frac{1}{\sqrt{2}} J_x : then J'_x , J_y and...

Hi.. I want an explanation of the commutation relation. According to what I understand if two operators commute then they can be measured simultaneously. If they do not commute then the measurement of one depends on other as per the value of the commutator..I hope this is correct by far. In...

Homework Statement What is the commutation relation between the x and y components of angular momentum L = r X P Homework Equations None. The Attempt at a Solution I do r X p and get the angular momentum componants:L_{x} = (-i \hbar) (y \frac{d}{dz} - z \frac{d}{dy}) L_{y} = (-i \hbar) (z...

Homework Statement The Hermitian operators \hat{A},\hat{B},\hat{C} satisfy the commutation relation[\hat{A},\hat{B}]=c\hat{C}. Show that c is a purely imaginary number. The Attempt at a Solution I don't usually post questions without some attempt at an answer but I am at a loss here.

So lately I've been thinking about whether or not it'd be possible to have the commutation relation [x,p]=i \hbar in a Hilbert Space of finite dimension d. Initially, I was trying to construct a lattice universe and a translation operator that takes a particle from one lattice point to the...

How to resolve these both integrals? http://en.zimagez.com/full/dcd7ca20c1b1ac79817defaa1cf6b7547df3f6b56b66dc1f559cec6c8ec77a892af4951fa22433762c63d0bbe83c93c420e5d904519535ce0b5e698fb7816b2c.php

Hi, what is the physics experiment that leads to the position-momentum commutation relation xpx - px x = i hbar What does it mean to multiply the position and momentum operators of a particle? What is the corresponding physical quantity?

I don't understand why we quantize the field by defining the commutation relation.What's that mean?And what's the difference between the commutation and anticommtation?

We all know that quantum theory is based on the commutation relation and superposition principle. The trouble haunting me long time is that how to "get" the famous commutation relation? Could anybody give me an explanation?

So I understand the commutation laws etc, but one thing I can't get my head around is the fact that L^2 commutes with Lx,y,z but L does not. I mean if you found L^2 couldn't you just take the square root of it and hence know the total angular momentum. It seems completely ridiculous that you...

So I'm trying to show that one choice of representation for the SuSy generators fulfills the SuSy algebra... (one of which is \left\{ Q_{a},\bar{Q_{\dot{b}}} \right\}= 2 \sigma^{\mu}_{a\dot{b}} p_{\mu})... For Q_{a}= \partial_{a} - i σ^{μ}_{a\dot{β}} \bar{θ^{\dot{β}}} \partial_{\mu}...

Under the effect of an electric and magnetic field the momentum in the Hamiltonian becomes the canonical momentum, p-qA where p is the linear momentum and A is the vector potential so H=(1/2m)(p-qA)^2 + qV where V is the scalar potential. I am trying to find [H,(p-qA)]. My main question arises...

We know how to find S_{x} and S_{y} if we used S_{+} and S_{-}, and after finding S_{x} and S_{y}, we can prove that [S_{x}, S_{y}]= i\hbarS_{z} (Equation 1) and [S_{y}, S_{z}]= i\hbarS_{x} (Equation 2) and [S_{z}, S_{x}]= i\hbarS_{y} (Equation 3) but can we, starting from Equations 1...

Hi I regard, $$[\partial_t \Psi, \Psi]=0$$ but \Psi is a field-operator. I don't understand why the commutation of the derrivative of the operator \Psi by itself should be zero? THX

Has anyone tried to make physical theories where the derivatives do not commute? I mean there's a condition on the derivatives of every function for them to commute which is learned in first year calculus. I mean in QM and QFT we grew accustomed to operators that do not commute, so why not...

Hello, I am thinking for some hours about the commutation of the field-Operator/(annihilation-Operator): \Psi and the vector-potential: \vec{A(\vec{r})}. I have noticed in my lecture notes that \vec{A(\vec{r})}\Psi = \Psi\vec{A(\vec{r})}. But I don't understand why they commute...