Commutators Definition and 88 Threads

I am trying to understand the following which is proving difficult: It is found that (and the proof here is clear) [P, Jj] anticommutes with Vi Where P = parity operator Jj and Vi are the j th and i th components of the angular momentum vector and an arbitrary vector respectively...

I've been trying to work out some expressions involving commutators of vector operators, and I am hoping some of y'all might know some identities that might make my job a little easier. Specifically, what is \left[\mathbf{\hat A}\cdot\mathbf{\hat B}, \mathbf{\hat C}\right]? Are there any...

A footnote to Griffiths reads "In a deep sense all of the mysteries of quantum mechanics can be traced to the fact that position and momentum do not commute. Indeed, some authors take the canonical commutation relation as an axiom of the theory, and use it to derive [itex]p = (\hbar /...

Homework Statement If we have a harmonic oscillator with creation and annhilation operators a_{-} a_{+} , respectively. The commutation relation is well known: [a_{+},a_{-}] = I However, if we have two independent oscillators with operators a'_{-} a'_{+} As the operators are the...

Hi, I have this question for a problem sheet: Use the unit operator to show that a Hermitian operator A can be written in terms of its orthonormal eigenstates ln> and real eigenvalues a as : A=(sum of) ln>a<nl and hence deduce by induction that A^k = (sum of) ln>a^k<nl I have no...

Homework Statement why we do use commutation? what is physical difference between commutators and Poisson Brackets? Homework Equations The Attempt at a Solution

Commutators on a discrete QM lattice = ? Please let me know if any of the following is unclear: I was thinking about how you could go about doing QM not in a continuous space but instead on a lattice, take 1D for simplicity. Let's use a finite (not countably infinite) number of positions say...

Homework Statement Using the results of the previous problem, find [x,p2 ] and from that determine [x,p2 ]\psi(x) Homework Equations The solution to the previous problem was [A,BC]=[A,B]C+B[A,C] The Attempt at a Solution As I'm suppose to use the results of the previous problem I...

Hi, I have a conceptual question concerning causality and locality in QM. Causality plays a role in second quantization when doing QFT, which one calls "micro-causality"; the commutators between fields disappear when the interval between them is spacelike. However, how does this fit in...

Commutators In quantum...! Recently my class was taught about commutators and their applications in angular momentum operator.Unfortunately due to health reasons I was not able to attend them and now can't get any extra classes and to make things worse the books I was consulting don't have this...

Let T be the set of all matrics of the form AB - BA, where A and B are nxn matrics. Show that span T is not Mnn. 1) does "span T is not Mnn" mean that Mnn does not span T? Thanks

Homework Statement (Introduction to Elementary Particles, David Griffiths. Ch 7 Problem 7.8 (c)) Find the commutator of H with the spin angular momentum, S= \frac{\hbar}{2}\vec{\Sigma}. In other words find [H,S] Homework Equations For the Dirac equation, the Hamiltonian...

Homework Statement Suppose that the commutator between two Hermitian operators â and \hat{}b is [â,\hat{}b]=λ, where λ is a complex number. Show that the real part of λ must vanish. Homework Equations Let A=â B=\hat{}b The Attempt at a Solution AΨ=aΨ BΨ=bΨ...

Hi. I've been trying to calculate a couple of commutators, namely [\Psi(r),H] and [\Psi^{\dagger}(r),H] where H is a free particle hamiltonian in second quantization. I have attached my attempts and I would greatly appreciate if anyone could tell me if I am right or if there is a better way to...

I'm completely lost and need some advice on how to continue. I need to prove the 1st line on the link http://upload.wikimedia.org/math/0/f/8/0f873eaca989ffa1af9a323c6e62f3ed.png

Let a QM system be described in the Heisenberg picture by position variables q_j with corresponding conjugate momenta p_j. We have the equal-time commutators [q_j(t),p_k(t)]=i\hbar \delta_{jk} In quantum field theory, for the Dirac spinor field we have the equal-time commutator...

Help me to be sure I got this right. Two commutating operators: 1)have common eigenstates or the same? 2)can be observe at the same time. thanks

Sorry for spamming the forums, but one last question for today! If \Sigma^k=\frac{i}{2} \epsilon_{kij} [\gamma^i , \gamma^j] where [A,B]=AB-BA Why does {\Sigma^1=2i \gamma^2\gamma^3 (that's what my notes say, anyway) I think it should equal...

1. Types off commutator and their effect on current densit? [b.]2I want to ask how many types of commutators are there in a DC machine and what is their effect on current density..[\b] [b]3.i have been working on it for a long time and came to know tht there are two types Brushed...

Hi. Cohen-Tannoudji has this section in his quantum mechanics book where he derives a bunch of relations which are true for operators having the commutation relation [Q,P]=i\hbar. Is there any special significance to this value of a commutator? Would things be much different if it had the...

Hi all, I've taken a two-course undergrad QM sequence and have been reading Shankar's Principles of Quantum Mechanics. There is some reference to the similarity between the Poisson bracket in Hamiltonian mechanics and the commutator in QM. E.g. \{x, p\} = 1 (PB) [x, p] = i \hbar...

Homework Statement A is a Hermitian operator which commutes with the Hamiltonian: \left[A,H\right]=AH-HA=0 To be shown: \frac{d}{dt}A=0 Homework Equations Schrödinger equation: i\hbar\frac{\partial}{\partial t}\psi=H\psi with the Hamilton operator H. The Attempt at a Solution I...

Equation 2.14 of Srednicki is: U(\Lambda)^{-1} M^{\mu \nu} U(\Lambda) = \Lambda^{\mu}_{\,\,\rho} \Lambda^{\nu}_{\, \, \sigma} M^{\rho \sigma} He says that writing \Lambda = 1 + \delta \omega , one obtains the usual commutation relation of the M_{\mu \nu} : [ M^{\mu \nu},M^{\rho...

Homework Statement Show the three components of angular momentum: L_x, L_y and L_z commute with nabla^2 and r^2 = x^2 + y^2 = z^2Homework Equations [A, B] = AB - BA For example: [L_x, \nabla^2] = L_x \nabla^2 - \nabla^2 L_x The Attempt at a Solution L_x \nabla^2 =...

Can I write: [\hat{p^2},\hat{x}]\hat{p} = \hat{p}[\hat{p^2}, \hat{x}] in relation to position and momentum operators?

Hey guys, Tryin to do Q1 in http://members.iinet.net.au/~housewrk/QM/AQM2006.ex.newnotation.pdf and I am having trouble in b.) i get the commutator equal to c * permutation tensor (sigma . p * (xi pk) -xi pk * sigma . p) and i know I am missing some cruical step to recombine this, ie i...

Homework Statement The question is 'show that the commutator [AB,C]=A[B,C]+[A,C]B' Homework Equations I'm not sure, a search for a proof gave the names 'ring theory' and 'Leibniz algebra', but further searching hasn't provided a proof so far and it seems it is just accepted as a...

Why are the commutators in QFT equal-time commutators? I am talking about things like [\phi(x,t),\pi(x',t)]=i\delta (x-x') where pi is the canonically-conjugate momentum density to phi. Shouldn't a relativistic approach treat time and space more equivalently? Something like...

I'm having trouble with commutators. I have to solve them 2 ways. First, using [x,p]=i\hbar and other identities/formulas, and the the second method the "direct way". 1.) x,\hat{H} My work: [x,\hat{H}]\psi &= x\hat{H}\psi - \hat{H}x\psi = x \left ( \frac{p^2}{2m} + V(x) \right )\psi -...

\left[L_{x},L_{y}\right]=\left[yp_{z}-zp_{y},zp_{x}-xp_{z}\right] =\left[yp_{z},zp_{x}\right]-\left[zp_{y},zp_{x}\right]-\left[yp_{z},xp_{z}\right]+\left[zp_{y},xp_{z}\right] How next? My book is not of much help I Tried \left[A,BC\right]=\left[A,B\right]C+B\left[A,C\right] But...

Homework Statement Simplify the commutator [A,B] and give the expectation value of [A,B] in the ground state for an isotropic harmonic oscillator (mass m) that has the energy \hbar \omega /2 when A = xp_x B = y Homework Equations [AB,C] = A[B,C] + [A,C]B [p_i,x_j] =...

OK, I'm a wee bit sleep deprived and cannot recollect some facts about the Dirac quantization of gauge theories. With the quantization of the parametrized nonrelativistics particle, do we still change the Poisson bracket into commutators? More specifically, for the non-relativistic particle...

Quantum Physics -- Calculating Commutators The problem states: Calculate the commutators [x,L[SIZE="1"]x], [y,L[SIZE="1"]x], [z, L[SIZE="1"]x], [x, L[SIZE="1"]y], [y, L[SIZE="1"]y], [z, L[SIZE="1"]y]. Do you see a pattern that will allow you to state the commutators of x, y, z with...

I have a question where I need to calculate commutators, but I just need to check one detail of this. I need to work out [x,xp] What I need to check is the following: [x,xp] = xxp - xpx but does this then mean that I get (x^2 p - X^2 p) or (X^2 p - something else)?? Sorry about...

Hi folks. I've come across a method to determine the controllability of a quantum system that depends on the Lie group generated by the commutator of the skew-Hermetian versions of the field free and interaction (dipole) Hamiltonians. If, for an N dimensional system the dimension of the group...

An interesting question was posed, and since I have many problems of this type I'll just make the question general: Suppose you have operations A and B, if [A, B] != 0, then what can you conclude about a simultaneous measurement of A and B? For example, if A was momentum in the x direction's...

Okay, I'm a geek with a lot of time on my hands, so I'm going through all the problems in Sakuri. The problem: Calculate [x^2,p^2] . Simple enough. There are basically two fundamental attacks to do this. 1. Direct computation. I get that [x^2,p^2]=2i \hbar (xp+px) , which I got both by...

I am not really clear on what is meant by commutators. I know that the commutator of G is ABA^-1B^-1, but I am not sure how to check if a group is solvable by having the commutator eventually equal the trivial group. For example, I know that the Heisenberg group of 3x3 upper triangular...