Commutator Definition and 274 Threads

i've been trying to evaluate this commutator the 'easy' way--that is, without using the definition of the momentum operator. the farthest i got was trying to use this rule.. [A, BC] = [A, B]C + B[A, C] so.. [x, p^2] = [x, p]p + p[x, p] so i guess i get 2ihp. but that doesn't make...

This should be easy, since I'm sure I've misunderstood something here. The task is to find the commutator of the x- and y-components of the angular momentum operator. This operator is, according to physics handbook: -i \hbar \bold r \times \nabla I rewrote this as: i \hbar \nabla \times \bold...

Hi there,... For a derivation of the Ehrenfestequations i found the following commutator relations for the Hamilton-Operator in a book: H = \frac{p_{op}^2}{2m} + V(r,t) and the momentum-operator p_{op} = - i \hbar \nabla respectively the position-operator r in position space: [H,p_{op}]...

Hi all! I worked for hours on this simple commutator of real scalar fields in qft: \left[\Phi\left(x\right),\Phi\left(y\right) \right] = i\Delta\left( x-y \right) where \Delta\left(x\right) = \frac{1}{i}\int {\frac{{d^4 p}} {{\left( {2\pi } \right)^3 }}\delta \left( {p^2 - m^2 }...

Do the Hamiltonian (H) and the z-component of angular momentum (L_z) commute? [H, L_z]=0 H = [(-(hbar)^2/2m) dell^2] + V(r, theta, phi) where dell is the gradient, and V is the potential L_z = -i(hbar)(d/d phi) where d is actually a partial derivative I know how to find a...

Geometric Algebra for Physicists, in equation (4.56) introduces the following notation A * B = \langle AB \rangle as well as (4.57) the commutator product: A \times B = \frac{1}{2}\left(AB - BA\right) I can see the value defining the commutator product since this selects all...

Does the relation [f(\hat{A}),\hat{B}] = df(\hat{A})/d\hat{A} follow when A commutes with [A,B]? or is this only valid when [A,B]=1?

Homework Statement Please confirm that the center of a group always contains the commutator subgroup. I am pretty sure its true. Homework Equations The Attempt at a Solution

Hi, guys..This is my first time to post. and I got to aplogize for my bad English..I`m a novice..;; anyway..here`s my curiosity.. From Paul Dirac`s Principles of Quantum Mechanics..p.153 (section of Motion in a central field of force) It says that The angular momentum L of the ptl...

Suppose I want to bypass the entire Hamiltonian formulation of quantum field theory and define the theory using a path integral. Thus all I can calculate are Green's function which are time ordered products of local operators. Given only these (no expansions of the field in creation...

can anyone help?? in quantum mechanics commutator prove [L^x,L^y] = ihL^z given :L^x =(y^(pz)^-z^(py)^) :L^y =(z^(px)^-x^(pz)^) :L^z =(x^(py)^-y^(px)^) where ^ is just showing its operator prove comutator [L^x,L^y] = ihL^z I am swamped at every hurdle and can't seem to get my...

How do i compute the commutator [L,p]?

given that [x,p]=ih, show that if x=x, p has the representation p=-iħ∂/∂x+f(x) where f(x) is an arbitrary function of x

Is it true that ABC = CBA implies [A,B]=[A,C]=[B,C]=0 ?? The converse is of course true, and I cannot find a counter-exemple (ex: no 2 of the above commutation relation above are sufficient), but how is this proven?? :confused:

Let p1,p2 be two density matrices and M be a real, symmetric matrix. Now, <<p1|[M,p2]>>= <<p1|M*p2>>-<<p1|p2*M>>= Tr{p1*M*p2}-Tr{p1*p2*M}= 2i*Tr{(Im(p1|M*p2))}. Why is it that this works out as simply as (x+iy)-(x-iy)? How is Tr{p1*p2*m}=conjugate(Tr{p1*M*p2})? I can't seem to figure...

Question: If g(p) can be Taylor expanded in polynomials, then prove that: \left[x, g\left(p\right)\right] = i\hbar \frac{dg}{dp} To start, I multiply the wave function \Psi and expand the commutator: \left( xg\left(p\right)-g\left(p\right)x \right)\Psi then expand g(p) using...

The only definition of a free group I have is this: If F is a free group then it must have a subgroup in which every element of F can be written in a unique way as a product of finitely many elements of S and their inverses. Now, is it possible to form a commutator subgroup of F? That is...

Ok, I have a stupid question on pauli matrices here but it is bugging me. In a book I'm reading it gives the equation [\sigma_i , \sigma_j] = 2 I \epsilon_{i,j,k} \sigma_k , I understand how it works and everything but I do have a question, when you have k=i/j and i!=j (like 2,1,2) you get a...

Could someone check if I have done this right. R_1 = x^2\partial_3 - x^3\partial_2 R_2 = x^3\partial_1 - x^1\partial_3 R_3 = x^1\partial_2 - x^2\partial_1 Where x^i are coordinates. I need to calculate the commutator [R_1,R_2]. [R_1,R_2] = x^2\partial_3x^3\partial_1 -...

Hi, I've got a commutator relation I'm trying to figure out here. I don't know what I'm doing wrong, but I don't seem to be able to get it right, so hopefully someone can help me through it. Anyway, here's the problem. We're given the Dirac Hamiltonian H_D = \alpha_j p_j + \beta m, where p_j...

i searched the forum, but nothing came up. My question, how do you prove that [A,B] = iC if A and B are hermitian operators? I understand how C is hermitian as well, but i can't figure out how to prove the equation.

I'm having difficulty trying to figure out which of the following is the correct method to properly evaluate the effect of the operators on f(x). Given that, \hat{A}f(x)=<x|\hat{A}|f> If the polarity operator, \hat{U_p}, and the translation operator, \hat{U_t}(a), act as...

The commutator plays a central roll in quantum mechanics. I guess it is hard to study any aspect of quantum mechanics without running into a commutator. I understand you can accept the fact that commutators of compatible measurables equal zero, while those of incompatible measurements equal...

I'm trying to demonstrate that angular momentum and the Hamiltonian commute provided that V is a function of radius r only. Using L = Lx + Ly + Lz components, the definition of Lx = yPz - zPy, etc, the commutator definition and the fact that H = p^2/2m + V. I can reduce [Lx,H] to the...