Commutator Definition and 274 Threads

Homework Statement Prove the following identity: e^{x \hat A} \hat B e^{-x \hat A} = \hat B + [\hat A, \hat B]x + \frac{[\hat A, [\hat A, \hat B]]x^2}{2!}+\frac{[\hat A,[\hat A, [\hat A, \hat B]]]x^3}{3!}+... where A and B are operators and x is some parameter. Homework Equations...

Homework Statement For \begin{align*}H = \frac{\mathbf{p}^2}{2m} + V(\mathbf{r})\end{align*} use the properties of the double commutator \left[{\left[{H},{e^{i \mathbf{k} \cdot \mathbf{r}}}\right]},{e^{-i \mathbf{k} \cdot \mathbf{r}}}\right] to obtain \begin{align*}\sum_n (E_n - E_s)...

I've just done a textbook exersize to calculate (H,x) and ((H,x),x) for H= p^2/2m +V. Having done the manipulation, my next question is what is the significance of this calculation. Where would one use these commutator and double commutator relations?

I am attempting to calculate the commutator [\hat{X}^2,\hat{P}^2] where \hat{X} is position and \hat{P} is momentum and am running into the following problem. The calculation goes as follows...

i have met a problem about the commutator of x and p. [x,p]=ihbar /p> is the eigenstate of momentum operator p. <p/xp-px/p> =<p/xp/p>-<p/px/p> =p<p/x/p>-p<p/x/p> the second term is got by the momentum operator p acting on the left state. =0...

I have two questions that are based on the following example involving the Hermitian operator i[A,B]=iAB-iBA for the case of a plane polarized photon. The observable (Hermitian Matrix) for the plane polarized photon, which Professor Susskind gave in his quantum mechanics lecture, lecture...

If I have H=p^2/2m+V(x), |a'> are energy eigenkets with eigenvalue E_{a'}, isn't the expectation value of [H,x] wrt |a'> not always 0? Don't I have that <a'|[H,x]|a'> = <a'|(Hx-xH)|a'> = <a'|Hx|a'> - <a'|xH|a'> = 0 ? But if I calculate the commutator, I get: <a'|[H,x]|a'> = <a'|-i p \hbar /...

In David Tong's QFT notes (http://www.damtp.cam.ac.uk/user/tong/qft/qft.pdf p. 43, eqn. 2.89) he shows how the commutator of a scalar field \phi(x) and \phi(y) vanishes for spacelike-separated 4-vectors x and y, establishing that the theory is causal. For equal time, x^0=y^0, the commutator is...

Hi, could someone give me a hand with the two long commutators on page 25 of Peskin and Schroeder? I'm not sure how to deal with the gradient in the first and the laplacian in the second. Thanx alot

Some books begin QM by postulating the Schrodinger equation, and arrive at the rest. Some books begin QM by postulating the commutator relations, and arrive at the rest. Which do you feel is more valid? Or are both equally valid? Is one more physical/mathematical than the other? I...

Homework Statement Hi... I'm having something about the Interaction/Dirac picture. The equation of motion, for an observable A that doesn't depend on time in the Schrödinger picture, is given by: \[i\hbar \frac{d{{A}_{I}}}{dt}=\left[ {{A}_{I}},{{H}_{0}} \right]\] where...

Homework Statement Show \left[x,f(p)\right)] = i\hbar\frac{d}{dp}(f(p))\right. Homework Equations I can use \left[x,p^{n}\right)] = i\hbar\\n\right.p^{n}\right. f(p) = \Sigma f_{n}p^{n} (power series expansion) The Attempt at a Solution I started by expanding f(p) to the power...

This is for the Pauli Matrics 0 and 1 are different Hilbert Spaces \left[(I-Z)_{0}\otimes(I-Z)_{1} , Y_{0}\otimes Z_{1}\right] =\left((I-Z)_{0}\otimes(I-Z)_{1}\right)\left(Y_{0}\otimes Z_{1}\right)-\left(Y_{0}\otimes Z_{1}\right)\left((I-Z)_{0}\otimes(I-Z)_{1}\right)...

I'm having difficulty deciphering my notes which 'proove' that the commutor of two real free fields φ(x) and φ(y) (lets call it i∆) ie. i∆=[φ(x),φ(y)] are Lorentz invariant under an orthocronous Lorentz transformation. Not sure if it helps but φ(x)=∫d3k[α(k)e-ikx+α+(k)eikx]. Now, apparently I...

Homework Statement [A,exp(X*B)] = exp(X*B)[A,B]X Is there a name for this relation? Homework Equations The Attempt at a Solution If not, how do you prove it? A(X*B)^n/n! - (X*B)^n/n! * A

Task: The task is to compute the commutator of L^2 with all components of the r-vector. It seems to be an unusual task for I was unable to find it in any book. Known stuff: I know that [L_i,x_j]=i \hbar \epsilon_{ijk} x_k (\epsilon_{ijk} being the Levi-Civita symbol). Now I would go about as...

Homework Statement Find the commutator \left[\hat{p_{x}},\hat{p_{y}}\right] Homework Equations \hat{p_{x}}=\frac{\hbar}{i}\frac{\partial}{\partial x} \hat{p_{y}}=\frac{\hbar}{i}\frac{\partial}{\partial y} The Attempt at a Solution [\hat{p}_{x}...

Evaluate the commutator [H,x], where H is Hamiltonian operator (including terms for kinetic and potential energy). How does it relate to p_x, momentum operator (-ih_bar d/dx)?

Homework Statement Determine \left[\hat{x},\hat{H}\right] Homework Equations The Attempt at a Solution =x\left(-\frac{\hbar^2}{2m}\frac{\delta^2}{\delta{x^2}}+V\right)\Psi-\left(-\frac{\hbar^2}{2m}\frac{\delta^2}{\delta{x^2}}+V\right)x\psi...

Hello, Is it generally the case that [J, H] = dJ/dt? I saw this appear in a problem involving a spin 1/2 system interacting with a magnetic field. If so, why?This seems like a very basic relation but I'm having a bit of brain freeze and can't see the answer right now.

If we define: A_{j}=\omega \hat{x}_{j}+i \hat{p}_{j} and A^{+}_{j}=\omega \hat{x}_{j}-i \hat{p}_{j} Would it be true to say: [A_k , (A^{+}_{i}+A_i)(A^{+}_{j}-A_j)]=0 My reasoning is that, because [\hat{x}_{j}, \hat{p}_{i}]=0 the the ordering of the contents of commutation...

consider a general one dimensional potential v(x) drive an expression for the commutator [H,P] where h is hamiltonian operator and momentum operator. i keep getting zero and i don't think i should. since next part of homework question sais what condition must v(x) satisfy so that momentum will...

I just came across the following claim: \lim_{\hbar\rightarrow 0}[\frac{1}{\hbar}(AB-BA)] (which approaches the classical Poincare commutator) is a derivative with respect to \hbar. I know it looks like derivative, but is it really? Please elaborate.

Homework Statement Prove that if A is an operator which commutes with two components of the rotation generator operator, J, then it commute with its third component. Homework Equations [A_{\alpha},J_{\beta}]=i \hbar \epsilon_{\alpha \beta \gamma} A_{\gamma} (not sure about the sign of...

Homework Statement Part of a much larger problem dealing with the Heisenberg picture. I am not remembering how to start evaluating the following commutator: \left [ a_k(t),\left(\sum_{k,\ell}a_k^\dagger <k|h|\ell>a_\ell\right)\right] Homework Equations See (a) The Attempt at a...

Homework Statement It's a part of a bigger problem, but what I need help with is finding the commutator between x(t) and x(t) at a different time. So basically I need [x(t1),x(t2)] Homework Equations A(t)=e^{iHt/\hbar}Ae^{-iHt/\hbar} The Attempt at a Solution The farthest I can...

I am trying to prove: L_xL_y - LyLx = ihL_z Unfortunately I keep getting L_xL_y - L_yL_x = -ihL_z and I was hoping someone could spot the error in my calculations: L_xL_y - L_yL_x = ( yp_z - zp_y )( zp_x - xp_z ) - ( zp_x - xp_z )( yp_z - zp_y ) = yp_zzp_x - yp_zxp_z - zp_yzp_x +...

i've never really done a proof by induction but i would like to prove a statement about commutator relations so can you please check my proof: claim: [A,B^n]=nB^{n-1}[A,B] if [A,B]=k\cdot I where A,B are operators, I is the identity and k is any scalar. proof: [A,B^2] = [A,B]B+B[A,B] =...

Hello all! I hope some of you are more proficient in juggling with bra-kets... I am wondering if/when the density operator commutes with other operators, especially with unitaries and observables. 1. My guess is, that it commutes with unitaries, but I am not sure if my thinking is correct...

This is not homework, but is not general discussion, so not sure where this would go. In class we were deriving with the radial equations of a hydrogen atom, and in one of the equations was the commutator term: \left[ \frac{d}{d\rho}, \frac{1}{\rho}\right] my attempt was: \left[...

Hi all. I found the following identity in a textbook on second quantization: ([a_1^{\dagger},a_2^{\dagger}]_{\mp})^{\dagger}=[a_1,a_2]_{\mp} but why? ([a_1^{\dagger},a_2^{\dagger}]_{\mp})^{\dagger}=(a_1^{\dagger}a_2^{\dagger}\mp a_2^{\dagger}a_1^{\dagger})^{\dagger}=a_2a_1\mp a_1a_2...

Hi, I am working with the Dirac picture in the second quantification. An operator in this picture is defined as (where some constants are 1) O_I=e^{iH_0t}Oe^{-iH_0t}. Now, it is evident that the hamiltonian H_0 = T + V is the same in Heisenberg or Dirac picture since the exponential...

Okay, I *know* that E and x are supposed to commute, but I'm stuck on one tiny portion when I work through this commutator... So, here's my work. Feel free to point out my error(s): [E,x]\Psi=(i\hbar\frac{\partial}{\partial t}x-xi\hbar\frac{\partial}{\partial t})\Psi ...which...

This is a question about simple non-relativistic quantum mechanics in one dimension. If the energy operator is \imath \frac{h}{2\pi}\frac{\partial}{\partial t}, then it would appear to commute with the position operator x. Then, if the energy and position operators commute, I ought to be...

software to calculate simple commutator relation ?? Dear All: I have hundred terms of commutators needs to be calculate. Each one looks like [{\epsilon_{i m}}^n\eta^m\frac{\partial}{\partial\eta^n},C\eta_j\eta^l\frac{\partial}{\partial\theta^l}] ,where C is function of \theta^i and...

Consider the SUSY charge Q= \int d^3y~ \sigma^\mu \chi~ ~\partial_\mu \phi^\dagger~ The SUSY transformation of fields, let's say of the scalar field, can be found using the commutator i [ \epsilon \cdot Q, \phi(x)] = \delta \phi(x) using the equal time commutator...

When simplifying this \int d^3x' [\pi(x), \frac{1}{2}\pi^2(x') + \frac{1}{2} \phi(x')( -\nabla^2 + m^2)\phi(x')] we know that [\pi(x), \pi(x')] = 0 [\phi(x), \pi(x')] = -i\delta(x-x') how does that simplify to \int d^3x' \delta(x-x')( -\nabla^2 + m^2)\phi(x') I know that...

Homework Statement When calculating this commutator, [ \pi(x), \int d^3x' { \frac{1}{2} \pi^2(x') + \frac{1}{2} \phi(x')(-\nabla^2 + m^2) \phi(x') }] I almost get the right answer, but not sure if this is valid, or if there is an identity The Attempt at a Solution when I get to this...

Perhaps someone will help me in this. I need to prove that the group [G,G] of elements of the form gh g^{-1}h^{-1} where g,h in G, is normal in G, i.e if k is in G, then kghg^{-1}h^{-1}k^{-1}=aba^{-1}b^{-1} for some a,b in G. I tried writing it as kghkk^{-1}g^{-1}h^{-1}k^{-1}, but here is...

Homework Statement Verify that (2.16) follows from (2.14). Here \Lambda is a Lorentz transformation matrix, U is a unitary operator, M is a generator of the Lorentz group. Homework Equations 2.8: \delta\omega_{\rho\sigma}=-\delta\omega_{\sigma\rho} M^{\mu\nu}=-M^{\nu\mu} 2.14...

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Homework Statement Show that, [a, \hat H] = \hbar\omega, [a^+, \hat H] = -\hbar\omega Homework EquationsFor the SHO Hamiltonian \hat H = \hbar\omega(a^+a - \frac{\ 1 }{2}) with [a^+, a] = 1 [a, b] = -[b, a] The Attempt at a Solution I have tried the following: [a, \hat H] = a\hat...

Hi all, i cannot find where's the trick in this little problem: Homework Statement We have an hermitian operator A and another operator B, and let's say they don't commute, i.e. [A,B] = cI (I is identity). So, if we take a normalized wavefunction |a> that is eigenfunction of the operator A...

Can somone remind me how to see that the Lie derivative of a vector field, defined as (L_XY)_p=\lim_{t\rightarrow 0}\frac{\phi_{-t}_*Y_{\phi_t(p)}-Y_p}{t} is actually equal to [X,Y]_p?

Homework Statement If A is a Hermitian operator, and [A,B]=0, must B necessarily be Hermitian as well? Homework Equations The Attempt at a Solution

Is there a commutation relation between x^{\mu} and \partial^{\nu} if you treat them as operators? I think I will need that to prove this [$J 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 \sigma} J^{\nu...

so, is the commutator relation between two observables just a Lie bracket? And if so, I have two questions: I know from differential geometry that the Lie bracket of two vector fields gives me a third vector field. So, what do we mean when we say that [x,p] = i*hbar? In fact, is there at all...

I saw the following video: Lecture Series on Electronics For Analog Signal Processing I by Prof.K.Radhakrishna Rao, Department of Electrical Engineering,IIT Madras. For more details on NPTEL visit http://nptel.iitm.ac.in Category: Education Tags: Voltage Multiplier So I was wondering...

I want to prove this formula e^Ae^B = e^Be^Ae^{[A,B]} The only method I can come up with is expand the LHS, and try to move all the B's to the left of all the A's, but it is so complicated in this way. i.e. e^Ae^B=\frac{A^n}{n!}\frac{B^m}{m!} = \frac{1}{n!m!}\Big(A^{n-1}BAB^{m-1} +...