Commute Definition and 68 Threads

In Peskin P85: It says the Time-ordered exponential is just a notation，in my understanding, it means $$\begin{aligned} &T\left\{ \exp \left[ -i\int_{t_0}^t{d}t^{\prime}H_I\left( t^{\prime} \right) \right] \right\}\\ &\ne T\left\{ 1+(-i)\int_{t_0}^t{d}t_1H_I\left( t_1 \right)...

If ##A## and ##B## are two matrices which commute, then probably ##AB^{n} = B^{n}A## too. Now, I could probably do all the work of proving this with induction, but I feel like there should be a more elegant way to prove this, though I don't know exactly how to avoid writing ellipses. If I start...

My assumption is that knowing potential can lead to knowing the position, but I don't know how this can be.

I want to show that ##[C, a_{r}] = 0##. This means that: $$ Ca_{r} - a_{r}C = \sum_{i,j} g_{ij}a_{i}a_{j}a_{r} - a_{r}\sum_{i,j} g_{ij}a_{i}a_{j} = 0$$ I don't understand what manipulating I can do here. I have tried to rewrite ##g_{ij}## in terms of the structure...

I know that in the case of central potential V(r) the hamiltonian of the system always commutes with l^2 operator. But what happends in this case?

My teacher said me this commutator is zero because the spherical harmonics are eigenfunctions of L^2. Actually, he said that any operator must commute with its eigenfunctions. I tried an example: [L^2,Y_20] expressing L^2 on spherical coordinates and I determined this commutator is not zero...

I am attaching an image from David J. Griffith's "Introduction to Quantum Mechanics; Second Edition" page 205. In the scenario described (the Hamiltonian treats the two particles identically) it follows that $$PH = H, HP = H$$ and so $$HP=PH.$$ My question is: what are the necessary and...

Does the time ordering operator ##\mathcal{T}## commute with the normal ordering operator ##\hat{N}##? i.e. is $$[ \mathcal{T},\hat{N}] =0$$ correct?

While deriving continuity equation in Fluid mechanics, our professor switched the order of taking total time derivative and then applying delta operator to the function without stating any condition to do so(Of course I know it is Physics which alows you to do so) . So,I began to think...

Let's consider Bohm's paradox (explaining as follows). A zero spin particle converts into two half-spin particles which move in the opposite directions. The parent particle had no angular momentum, so total spin of two particles is 0 implying they are in the singlet state. Suppose we measured Sz...

Suppose we have to deal with the question : $$\frac{\partial}{\partial x}\frac{\partial}{\partial y}=?\frac{\partial}{\partial y}\frac{\partial}{\partial x}$$ This seems true for independent variables. But if at the end x and y are linked in some way like $$x=f(t),y=g(t)$$ this is no more the...

...to give a number? https://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2016/lecture-notes/MIT8_04S16_LecNotes5.pdf On page 6, it says, " Matrix mechanics, was worked out in 1925 by Werner Heisenberg and clarified by Max Born and Pascual Jordan. Note that, if we were to write xˆ...

Homework Statement Show that the mean value of a time-independent operator over an energy eigenstate is constant in time. Homework Equations Ehrenfest theorem The Attempt at a Solution I get most of it, I'm just wondering how to say/show that this operator will commute with the Hamiltonian...

In quantum field theory (QFT), the requirement that physics is always causal is implemented by the microcausality condition on commutators of observables ##\mathcal{O}(x)## and ##\mathcal{O}'(y)##, $$\left[\mathcal{O}(x),\mathcal{O}'(y)\right]=0$$ for spacelike separations. Intuitively, I've...

Hi All, Perhaps I am missing something. Schrodinger equation is HPsi=EPsi, where H is hamiltonian = sum of kinetic energy operator and potential energy operator. Kinetic energy operator does not commute with potential energy operator, then how come they share the same wave function Psi? The...

Let ##g(x,t)=\int f(k,x,t)\,dk## Under what conditions is the following true? ##g(x,0)=\int f(k,x,0)\,dk## That is, we can get the value of ##g(x,t)## when ##t=0##, by (1) either substituting ##t=0## into ##g(x,t)## or (2) by first substituting ##t=0## into ##f(k,x,t)## and then integrating...

By commutative, we know that ##ab = ba## for all a,b in G. Thus, why do we need to prove separately that ##a^n b^m = b^ma^n##? Isn't it the case that ##a^n## and ##b^m## are in fact elements of the group? So shouldn't the fact that they commute automatically be implied?

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...

In general I'm wondering if \lim_{x\to0} \left[\frac{d}{dy} \frac{d}{dx} f(x,y)\right] = \frac{d}{dy} \left[\lim_{x\to0} \frac{d}{dx} f(x,y)\right] holds true for all f(x,y). Thanks.

Is it possible to find matrices that commute but eigenvectors of one matrix are not the eigenvectors of the other one. Could you give me example for it?

Hello. I read the textbook and found that common eigenfunctions are even possible for degenerate eigenvalues. Let's say operators A and B commutes and eigenvalue a of operator A is N-fold degenerate, means that there are N linearly independent eigenfunctions having same eigenvalue a. These...

I'm confused about the statement that if operators commute then eigenstates are shared. My main confusion is this one: ##L^2## commutes with ##L_i##. Then these two share eigenstates. But ##L_x## and ##L_y## do not commute, so they don't share eigenstates. Isn't this violating some type of...

I am posting this here instead of "Science and math textbooks" because I am looking for suggestions on an easy reading but educational book, not a workbook. I am by no means an expert, but I have a solid backgound in math and physics from college (Engineering), as well as self-study with...

Hey! :o Let $G$ be a finite group. I want to show that the probability that two elements of $G$ commute is $\frac{m}{|G|}$, where $m$ is the number of conjugacy classes of $G$. A conjugacy class is $O_x=\{g*x\mid g\in G\}=\{g^{-1}xg\mid g\in G\}$, right? (Wondering) Do we maybe take $x$...

Hi. We can write a polarised photon as ##\left|\alpha\right\rangle=\cos(\alpha)\left|\updownarrow\right\rangle+\sin(\alpha)\left|\leftrightarrow\right\rangle##. Trigonometry gives us $$\left\langle\alpha | \beta\right\rangle=\cos(\alpha)\cos(\beta)+\sin(\alpha)\sin(\beta)=\cos(\alpha-\beta)$$...

For example: [itex] D_α D_β D^β F_ab= D_β D^β D_α F_ab is true or not? Are there any books sources?

Say I have a hamiltonian with fermion creation / annihilation operators like this: \sum_{k_1,k_2,k_3,k_4} c_{k_1,\uparrow}^{\dagger} c_{k_2,\downarrow}^{\dagger} c_{k_3,\downarrow} c_{k_4,\uparrow} where the k's are momenta and the arrows indicate spin up / spin down. Can I commute operators...

Hi all, I'm attempting to prove that i \frac{d \xi (t)}{dt}=[\xi(t),H(p,q ; t)] where the Hamiltonian is explicitly time-dependent, in general. We also have some unitary U(t) which generates time-evolution. I wrote up a quick proof but realized afterward that I had assumed that H and...

Homework Statement Determine whether or not the following pairs of operators commute...and there was one I could not solve...according to the back of the textbook, I do understand 14.c does NOT commute, but I don't understand... (14)c. A = SQR B = SQRT Homework Equations ABf(x) - BAf(x) = 0...

Hey! :o I got stuck at the following exercise: If $x \in G$ has order $mn$ with $ (m,n)=1 $, show that there are $y,z$ with $ x=yz $ such that $y$,$z$ commute and they have order $m$ and $n$ respectively. Could you give me some hints?? (Wondering)

This morning, instead of my normal short ride on a single bus form a local coffee shop to work, I had to take a circuitous route involving four buses. http://www.princegeorgecitizen.com/news/local/university-way-closed-1.912351

In K&K's Intro to Mechanics, they kick off the topic of rotation by trying to turn rotations into vector quantities in analogy with position vectors. It's quickly shown, however, that rotations do not commute, making them rather poor vectors. They then show, however, that infinitesimal rotations...

Hi All, I just found this site and this is my first post here. I am working on getting my masters in polymer chemistry and started taking a class this semester which is pretty much all calculus and linear algebra and I just have a hard time with these subjects. I got a homework problem that I...

I am reading Dummit and Foote Section 10.5 on Exact Sequences. I am trying to understand Example 1 as given at the bottom of page 381 and continued at the top of page 382 - please see attachment for the diagram and explanantion of the example. The example, as you can no doubt see, requires...

I am reading Dummit and Foote on Exact Sequences and some of the 'diagrams that commute' have vertical arrows. Can someone please help me with the LaTex for these diagrams. I have given an example in the attachment "Exact Sequences - Diagrams with Vertical Arrows" - where I also frame my...

Suppose ##A## is a ## n \times n## matrix. Define the set ## V = \{ B | AB = BA, B \in M_{n \times n}( \mathbb{F}) \} ## I know that ##V## is a subspace of ##M_{n \times n}( \mathbb{F}) ## but how might I go about finding the dimension of ##V##? Is this even possible? It seems like an...

I have a question about the HUP. As I understand the HUP, it only applies to conjugate attributes that do not commute, such as position and momentum. However, many good quantum numbers do commute, so does this mean that the HUP does not apply to simultaneous measurement of such good quantum...

Homework Statement What are the characteristics of a matrix that commutes with a matrix of ones? Homework Equations None. The Attempt at a Solution I'm helping a buddy with his homework and I can't figure out this problem.

I have bumped into a term a^\dagger \hat{O}_S | \psi \rangle I would really like to operate on the slater determinant \psi directly, but I fear I cannot. Is there any easy manipulation I can perform?

I just have two questions relating to what I have been studying recently. 1) I know that the total energy and momentum operators don't commute, while the kinetic energy and momentum operators do. Why is this the case? (explanation rather than mathematically). 2) One form of the HUP says that...

In classical physics, all observables commute and the commutator would be zero. However this is not true in Quantum Mechanics, observables like position and momentum (time and frequency/energy) don't commute. Why? Is it because the (probability) wave functions/forms of position and momentum...

Homework Statement F:P2->R5 F(xn) = en+1 Consider the linear function D:P4 -> P4 p(x) -> p'(x) Find the matrix of the linear function T:R5 -> R5 such thatHomework Equations ( T ° F ) p(x) = ( F ° D ) ( p(x) )The Attempt at a Solution T ° F ° F-1 = F ° D ° F-1 T = F ° D ° F-1 then what should...

Homework Statement Do the derivatives del and d/dt commute? Or in other words, is it true that: del(d/dt)X = (d/dt)del_X Homework Equations ? The Attempt at a Solution nm, I think I know how to show it now..

I'm preparing for a qualifying exam and this problem came up on a previous qual: Let A and B be nxn matrices. Show that if A + B = AB then AB=BA.

I have two matrices which commute, one of which is definitely diagonal: \textbf{B}diag\{\underline{\lambda}\} = diag\{\underline{\lambda}\}\textbf{B} and I want to know what I can say about \textbf{B} and/or \underline{\lambda}. Specifically, I feel that either one or both of the following...

Prove that exp[A].exp = exp[A+B] only if A and B commute. Homework Statement Prove that eA.eB = e(A+B) only if A and B commute. [b]The attempt at a solution I expanded both sides of the equation. (1+A+A2/2!...)(1+B+B2/2!+..) = (1+(A+B)+(A+B)2/2!+..) Now how to proceed ?

Homework Statement Suppose we had an operator A that multiplied a vector by it's norm: A \mid \psi \rangle = \langle \psi \mid \psi \rangle \mid \psi \rangle I wanted to know what it's commutator with a constant would be. Homework Equations \left[A,B\right] = AB - BA The Attempt at...

Hey guys There are those vectors made of Pauli matrices like \bar{\sigma}^\mu and {\sigma}^\mu. So if I have the product \bar{\sigma}^\mu {\sigma}^\nu I wonder if it is commutative? And if not, what is the commutator? Cheers, earth2

Homework Statement A system with two spins of magnitude 1/2 have spin operators S1 and S2 and total spin S = S1 + S2 B is a B-field in the z direction (0,0,B) The Hamiltonian for the system is given by H = m S1 . S2 + c B.S where m,c are constants. By writing the Hamiltonian in...

I thought the dot product was commutative but there must be something about it that I don't understand. Perhaps the dot product is commutative only for vectors and not for tensors generally? In Kusse and Westwig, p70, it says that the order of terms matters because, in general, \hat{e}_j...