Commutation Definition and 208 Threads

My first question is, does any operator commute with itself? If this is the case, is there a simple proof to show so? If not, what would be a counter-example or a "counter-proof"? My second question has to do with the properties of an eigenvalue problem. If you have an operator Q such that...

Hey guys, i'm stuck (yet again! :) ) I am somewhat confused by Dirac spinors u,\bar{u}. Take the product (where Einstein summation convention is assumed): u^r u^s\bar{u}^s Is this the same as u^s\bar{u}^s u^r? Probably not because u^r is a vector while the other thing is a matrix...

Homework Statement How would you show that T^{abc}S_{b} = S_{b}T^{abc} but T^{abc} S_{bd} \neq S_{bd} T^{abc} in general? The Attempt at a Solution If I write out the sums explicitly, they appear totally the same to me. Any hints or ideas please?

If you have a Grassman number \eta that anticommutes with the creation and annihilation operators, then is the expression: <0|\eta|0> well defined? Because you can write this as: <1|a^{\dagger} \eta a|1>=-<1| \eta a^{\dagger} a|1> =-<1|\eta|1> But if \eta is a constant, then...

Hi all I'm new on this forum. I'm here since I'm working with n-extended susy and R-sym and I don't know how to calculate a commutator. First of all I introsuce my notation: \mu_A T^A is a potential cupled to R-sym generator \mu_{\alpha i} is a superpotential cupled to supercharges...

In Srednicki's book, he discusses quantizing a non-interacting spin-0 field \phi(x) by defining the KG Lagrangian, and then using it to derive the canonical conjugate momentum \pi(x) = \dot{\phi}(x). Then, he states that, by analogy with normal QM, the commutation relations between these fields...

Hey guys, I'm studying some quantum physics at the moment and I'm having some problems with understanding the principles behind the necessary lineair algebra: 1) If two operators do NOT commutate, is it correct to conclude they don't have a similar basis of eigenvectoren? Or are there more...

Hi everyone My question is about QFT I'm reading mandle and shaw in chapter 2 as you know there is a question (2.4) about the commutations between the field and momentum. [ [P]^{}[j], \phi ] as momentum is in integral form I don't know how to prove them! I tried to open the terms...

Does x & y directions commute? Seem trivial! Just wondering whether any measurement made in the x direction affect it's y coordinate.

Can we always calculate the commutation relations of two observables? If so, what’s the commutator of P (momentum) and H (Hamiltonian) in infinite square well, considering that the momentum is not a conserved quantity?

Im reading in a quantum mechanics book and need help to show the following relationship, (please show all the steps): If A,B,C are operators: [A,BC] = B[A,C] + [A,B]C

Hi. So I have learned that this holds for the trace if A and B are two operators: \text{Tr}(AB)=\text{Tr}(BA). Now I take the trace of the commutator between x and p: \text{Tr}(xp)-\text{Tr}(px)=\text{Tr}(xp)-\text{Tr}(xp)=0. But the commutator of x and p is i\hbar. Certainly the trace of...

Homework Statement calculate the following commutation relations [L_{x}L_{y}]= [L_{y}L_{z}]= ][L_{z}L_{x}]= Homework Equations [L_{x},L_{y}]= -\hbar^2[y\frac{\partial}{\partial x}-x\frac{\partial}{\partial y}] where the expression in the parentheses describes L_{z} but i...

I am aware that the commutation relation between conjugate variables shows that one quantity is the Fourier transform of the other, and so to imply the Heisenberg Uncertainty condition. So for example, the commutation relation between x, p (position and momentum respectively) leads to a non-zero...

i am unable to understand dc motor commutation can anybody help please

Just wondering if any1 would by any chance know how to calculate the commutation relations when L = R x P is the angular momentum operator of a system? [Li, P.R]=? P

In Baym's Lectures on Quantum Mechanics he derives the following formula [n.L,L]=ih L x n (Where n is a unit vector) I follow everything until this line: ih(r x (p x n)) + ih((r x n) x p) = ih (r x p) x n I can't seem to get this to work out. What properties is he using here?

Homework Statement It has been shown that the operators (Lx)op and (Ly)op do not commute but satisfy the following equation: (Lx)op(Ly)op - (Ly)op(Lx)op = i(hbar)(Lz)op (a) Use this relation and the two similar equations obtained by cycling the coordinate labels to show that...

Homework Statement Consider a state | l, m \rangle, an eigenstate of both \hat{L}^{2} and \hat{L}_{z}. Express \hat{L}_{x} in terms of the commutator of \hat{L}_{y} and \hat{L}_{z}, and use the result to demonstrate that \langle \hat{L}_{x} \rangle is zero. Homework Equations [...

Can somebody please explain the following? Given the measurements of 2 different physical properties are represented by two different operators, why is it possible to know exactly and simultaneously the values for both of the measured quantities only if the operators commute?

I'm having a course in advanced quantum mechanics, and we're using the book by Sakurai. In his definition of angular momentum he argues from what the classical generator of angular momentum is, and such he defines the generator for infitesimal rotations as...

Homework Statement Let \psi(\vec{r},t) be the wavefunction for a free particle of mass m obeying Schrodinger equation with V=0 in 3 dimensions. At t=0, the particle is in a known initial state \psi_0(\vec{r}). Using Ehrenfest's theorem, show that the expectation value <x^2> in the state...

I am trying to get a grip on the commutation properties of operators. Different authors get to those differently: some start from translator operators, some relate those to Poisson brackets, etc... My objective is to get a good intuitive feeling of what commuting and not commuting observables...

Homework Statement Suppose the operators P and Q satisfy the commutation relation [P,Q]=a, where a is a constant(a number, not an operator). a)Reduced the commutator [P,Q^n] where Q^n means the product of n Qs, to the simplest possible form. b) Reduce the commutator...

Homework Statement I'm working through a bit of group theory (specifically SU(2) commutation relations). I have a question a bout symmetries in the SU(2) group. It's something I'm trying to work through in my lecture notes for particle physics, but it's a bit of a mathsy question so I thought...

There must something wrong with my understanding of this relations because I think the usual way they are derived in many textbooks makes no sense. It goes like this, first assume that to every rotation O(a) in euclidean space there exists a rotation operator R(a) in Hilbert space,second: the...

I'm in chapter two of H. S. Green's Matrix Mechanics and at a sticking point. In section 2.2 he gives the following scenario: An atom emits a photon with angular velocity ω, it has energy Ei before the emission and Ef after, so Ei - Ef = ħω. (That I can understand.) ψi and ψf are...

Homework Statement Show [a+,a-] = -1, Where a+ = 1/((2)^0.5)(X-iP) and a- = 1/((2)^0.5)(X+iP) and X = ((mw/hbar)^0.5)x P = (-i(hbar/mw)^0.5)(d/dx)2. The attempt at a solution It would take forever to write it all up, but in summary: I said: [a+,a-] = (a+a- - a-a+) then...

Homework Statement None of the operators Lz, Ly and Lx commute. Show that [Lx,Ly] = i(h-cross)Lz Homework Equations Where Lx is defined as Lx=-ih ( y delta/delta x - z delta/delta y) Where Ly is defined as Ly=-ih ( z delta/delta x - x delta/delta z) Where Lz is defined as Lz=-ih (...

Hello, I am still having a hard time with tensors... The answer is probably obvious, but is it always the case (for an arbitrary metric tensor g_{\mu \nu} that g_{ab}g_{cd}=g_{cd}g_{ab} ? I was trying to find a formal proof for that, and was wondering if we could use the relations: (1)...

How do you work out the commutator of two operators, A and B, which have been written in bra - ket notation? alpha = a beta = b A = 2|a><a| + |a><b| + 3|b><a| B = |a><a| + 3|a><b| + 5|b><a| - 2|b><b| The answer is a 4x4 matrix according to my lecturer... Any help much appreciated...

I am reading the book by J.J.Sakurai, in chapter 3, there is a relation given as \langle \alpha', jm|J_z A |\alpha, jm\rangle Here, j is the quantum number of total angular momentum, m the component along z direction, \alpha is the third quantum number. J_z is angular momentum operator, A...

Homework Statement Suppose that two operators P and Q satisfy the commutation relation: [P,Q]=P. Suppose that psi is an eigenfunction of the operator P with eigenvalue p. Show that Qpsi is also an eigenfunction of P, and find its eigenvalue. Homework Equations The Attempt at a...

Homework Statement Show, by series expansion, that if A and B are two matrices which do not commute, then e^{A+B} \ne e^Ae^B, but if they do commute then the relation holds. Homework Equations e^A=1+A e^B=1+B e^{A+B}=1+(A+B) The Attempt at a Solution Assuming that the first 2...

Homework Statement Two quantum mechanical operators obey the following commutation relation. [\hat{A},\hat{B}]=i Given this commutation relation which of the following are true or false? Justify your answers. a) The two observables are simultaneously diagonalizable. b) The two satisfy a...

If f_1,f_2,f_3,\ldots and g_1,g_2,g_3,\ldots are some arbitrary real sequences, is it true that \underset{n\to\infty}{\textrm{lim inf}}\;(f_n g_n) = (\underset{n\to\infty}{\textrm{lim inf}}\; f_n) (\underset{n\to\infty}{\textrm{lim inf}}\; g_n)? For arbitrary \epsilon >0 there exists...

[SOLVED] commutation of observables Homework Statement Prove: If the observables (operators) Q1 and Q2 are both constant of the motion for some Hamiltonian H, then the commutator [Q1, Q2] is also a constant of the motion. okay, first question.. am i being asked to prove that [[Q1, Q2], H] =...

Let D = [d11 d12] [d21 d22] be a 2x2 matrix. Prove that D commutes with all other 2x2 matrices if and only if d12 = d21 = 0 and d11 = d22. I know if we can prove for every A, AD=DA should be true, but I really don't know how to proceed from there. I tried equating elements of AD with...

Given the Hamiltonian H = \vec{\alpha} \cdot \vec{p} c + \beta mc^2, How should one interpret the commutator [\vec{x}, H] which is supposedly related to the velocity of the Dirac particle? \vec{x} is a 3-vector whereas H is a vector so how do we commute them. Is some sort of tensor product in...

Hi, just wondering whether the commutation relation [\phi,L_3]=i\hbar holds and similar uncertainty relation such as involving X and Px can be derived ? thanks

Homework Statement Check that the spin matrices (Equations 4.145 and 4.147) obey the fundamental commutation relations for angular momentum, Equation 4.134.Homework Equations Eq. 4.147a --> S_{x} = \frac{\hbar}{2}\begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix} Eq. 4.147b --> S_{y} =...

In reviewing the derivation of the quantization of angular momentum-like operators from their commutation relations, I noticed that there is nothing a priori from which you can deduce the degeneracy of the eigenstates. While this is not a problem for angular momentum, in which other constraints...

Let the translation operator be: F (\textbf {l} ) = exp \left( \frac{-i \textbf{p} \cdot \textbf{l}}{\hbar} \right) where p is the momentum operator and l is some finite spatial displacement I need to find [x_i , F (\textbf {l} )] let me start with a fundamental commutation relation...

i need to find the commutation relation for: [x_i , p_i ^n p_j^m p_k^l] I could apply a test function g(x,y,z) to this and get: =x_i p_i ^n p_j^m p_k^l g - p_i ^n p_j^m p_k^l x_i g but from here I'm not sure where to go. any help would be appreciated.

I have a problem with deriving another result. Sorry I am new to this field. Please see the attached PDF - everything is there.

what is it? i need to know everything about it. i know it encompasses a lot of different stuff but yea if someone could point me to a book or webpage that explains it thoroughly. additionally what does this equal \sigma_{\mu}\sigma_{\alpha}\sigma_{\alpha}\sigma_{\mu} those are pauli...

Homework Statement I need to show the commutation between the spin operator and a uniform magnetic field will produce the same result as the cross product between them. Does this make sense? I don't see how it can be possible. Homework Equations [s,B] (The s should also have a hat...

Is there a book that explain in a formal way the deduction of symmetry/antisymmetry of bosonic/fermionic wave equation e/o commutation relation? I've often noticed that some people use examples for the introcution, but is there an axiomatic deduction?

Homework Statement I've just initiated a self-study on quantum mechanics and am in need of a little help. The position and momentum operators do not commute. According to my book which attemps to demonstrate this property, (1) \hat{p} \hat{x} \psi = \hat{p} x \psi = -i \hbar...

Homework Statement The canonical commutation relations for a particl moving in 3D are [\hat{x},\hat{p_{x}}]= i\hbar [\hat{y},\hat{p_{y}}]= i\hbar [\hat{z},\hat{p_{z}}]= i\hbar and all other commutators involving x, px, y ,py, z , pz (they should all have a hat on eahc of them signifying...