Commutator Definition and 266 Threads

This isn't really a homework question, it came along in my studying of the chapter, but it is a homework "type" question so I assumed this would be the best place to post this. I am trying to show that [L_x,L_y]=y[p_z,z]p_x+x[z,p_z]p_y=i \hbar L_z This is all the work the book showed. So I...

Homework Statement Prove the following for operators A and B. e^A B e^-A = B + [A,B] + (1/2!) * [A,[A,B]] + (1/3!) * [A,[A,[A,B]]] + ... Homework Equations e^A = 1 + A + (1/2!)A^2 + (1/3!)A^3 + ... The Attempt at a Solution I have no clue how to start. For the highly...

Hello friends! Im trying to get an geometric interpretation of QM and am now confused about the commutation relation between operators. Lets take momentum and position... sure, the fact that they don't commute show that we can not diagonalize them simultanesly. But what is the interpretation of...

The usual answer to this question is that if the commutator between two observables A and B is zero, then there are states that have a definite value for each observable. If [A,B] isn't zero, then this isn't true. Now, in general [A,B] = iC, where C is Hermitian. I'd like to know if there's...

reading that the commutator of rotations on two orthogonal axes is i * the rotation matrix for the third axis but if I commute this \begin{pmatrix}\mathrm{cos}\left( \theta\right) & -\mathrm{sin}\left( \theta\right) & 0\cr \mathrm{sin}\left( \theta\right) & \mathrm{cos}\left(...

Hi! In my textbook the explanation of the expectation value in general covering any observable Q is: \overline{Q} = \int \Psi^\ast (x,t) \hat{Q}\Psi(x,t) dx Then they define the commutator as: [\hat{A},\hat{B}] = \hat{A}\hat{B}-\hat{B}\hat{A} Now for position and momentum...

I now show some derivations regarding quantum commutators,leading to some inconsistencies. Can someone tell what went wrong? What causes the inconsistencies? and what is the correct way of understanding/handling the concepts? Issue # 1 - Hamiltonian and commutation with time (1) The...

assuming that A B C D are all n\times n operators how to evaluate the commutator of direct product? [A\otimes B, C\otimes D]

Homework Statement Calculate [P^m, X^n] Homework Equations [P,X] = PX - XP The Attempt at a Solution P( P^(m-1) * X^(n-1))X - (X^n)(P^n) =(XP +[P, X])(P^(m-1)*X^(n-1)) - (X^n)(P^n) =(P^m)(X^n) + [P, X](P^(m-1))(X^(n-1))- (X^n)(P^n)... I don't think the direction i am...

I found this theorem on Prasolov's Problems and Theorems in Linear Algebra: Let V be a \mathbb{C}-vector space and A,B \in \mathcal{L}(V) such that rank([A,B])\leq 1. Then A and B has a common eigenvector. He gives this proof: The proof will be carried out by induction on n=dim(V). He...

I know how to use a commutator as a mathematical formula but I really don't understand what it means. Can anyone explain it to me. Is a commutator nothing more than a check to see if it commutes or not since I know that if you use a commutator with a wave function and the result equals zero...

Homework Statement If A and B are two operators such that [A,B] = λ , where l is a complex number, and if μis a second complex number, show that: exp[μ(A + B)] = exp(μA)exp(μB)exp(- λμ^2/ 2). Homework Equations The Attempt at a Solution I'm stuck on where to begin. I know...

Homework Statement Hi I want to find the commutator between the momentum operator p and σz, the third Pauli spin matrix. I am not quite sure how to get started on this one. Can I get a push in the right direction? For the record, I would say that it is zero since they act on different...

Homework Statement Calculate [x,px] = (xpx - pxx) Do this for a function f(x). Now calculate [x,py] for f(x,y) Homework Equations px is actually px hat, I'm just not familiar with latex code. px= -i (d/dx) The Attempt at a Solution I believe I got the first part, for...

Homework Statement How do I obtain [H,P_x]? P_x is the polarization operator. Homework Equations H=-\frac{\hbar^2}{2m}\frac{\partial^2 }{\partial x^2}+V(x) P_x=2Re[c_+^*c_-] The Attempt at a Solution I know how to commute H and x. But somehow can't think of a way to...

Homework Statement Show that one can write U=exp(iC), where U is a unitary matrix, and C is a hermitian operator. If U=A+iB, show that A and B commute. Express these matrices in terms of C. Assum exp(M) = 1+M+M^2/2!...Homework Equations U=exp(iC) C=C* U*U=I U=A+iB exp(M) = sum over n...

1. Prove that [A,B^n] = nB^{n-1}[A,B] Given that: [[A,B],B] = 0 My Atempt to resolution We can write that: [[A,B],B] = [A,B]B-B[A,B] = 0 So we get that: [A,B]B = B[A,B] After some working several expansions, and considering that [X,YZ] = Y[X,Z] + [X,Y]Z I arrived...

Let me just head off the first waves of posts this thread will likely get. I am very fluent in quantum mechanics. I am completely aware of the behaviour of a commutator structure: simultaneous eigenbasis, etc. I understand how commutators model the structure that quantum mechanics has. My...

Hi, If we have two commuting operators A and B, is true that any function of A will commute with any function of B? I have a result which takes [L_{z},r^{2}]=0 and claims that [L_{z},r]=0. How can this be proved? Thank you

I have this review question: If operators A and B are hermitian, prove that their commutator is "anti-hermitian", ie) [A,B]†=-[A,B] What has me confused is the placement of the dagger on the commutator. Why [A,B]† and not [A†,B†]? Also, I am using Griffith's Intro to QM as a text. I have...

Hey guys, maybe you can help me with the following problem. I have to calculate the commutator relations in position representation: a) [V,ρ] b) [p,ρ] c) [p^2,ρ] Note that <q'|ρ|q>=ρ(q',q) is a matrix element of the density operator I already solved the first one. You just have to...

Homework Statement Is the commutator of x and any function of x zero? Homework Equations Taylor's theorem allows such a function to be expanded into polynomials, so that [f(x),x] may be expanded into terms of [x^n,x], which are all zero. Hence, f(x) and x commute. The Attempt at a...

I'm curious if there's a chain rule for the commutator (I'll explain what I mean) just like there's a product rule ([AB,C]). So, say you have an operator, which can be expressed in terms of another operator, and we know the commutation relationship between x and another operator, y. I'll call...

Homework Statement Let U and V be the complementary unitary operators for a system of N eingenstates as discussed in lecture. Recall that they both have eigenvalues x_n=e^{2\pi in/N} where n is an integer satisfying 0\leq n\leq N. The operators have forms U=\sum_{n}|n_u\rangle\langle n_u...

Homework Statement Show that: [p,x] = -iħ, Show that: [p,x^n] = -niħ x^(n-1), n>1 Show that: [p, A] = -iħ dA/dx Where p = -iħ d/dx, and A = A(x) is a differentiable function of x. Homework Equations [p,x] = px - xp; The Attempt at a Solution So far I understand part of each...

Hello, I am looking to find a closed-form formula for the following commutator [J_{-}^{n},J_{+}^{k}] where the operators are raising and lowering operators of the \mathfrak{su}(2) algebra for which [J_{+},J_{-}]=2J_0 and [J_{0},J_{\pm}]=\pm J_{\pm} I've already made some progress and I...

To prove: Commutator of the Hamiltonian with Position: i have been trying to solve, but i am getting a factor of 2 in the denominator carried from p2/2m Commutator of the Hamiltonian with Momentum: i am not able to proceed at all... Kindly help.. :(

Hi, I don't understand a particular coordinate expansion of the commutator of 2 vector fields: [X, Y ]f = X(Y f) − Y (Xf) = X_be_b(Y _ae_af) − Y _be_b(X_ae_af) = (X_b(e_bY_ a) − Y _b(e_bX_a))e_af + X_aY _b[e_a, e_b]f X,Y = Vector fields f = function X_i = Components of X and...

According to Dyson, Feynman in 1948 related to him a derivation, which, from 1) Newton's: m\ddot{x}_i=F_i(x,\dot{x},t) 2) the commutator relations: [x_i,x_j]=0m[x_i,\dot{x}_j]=i\hbar\delta_{ij} deduces: 1) the 'Lorentz force': F_i(x,\dot{x},t)=E_i(x,t)+\epsilon_{ijk}\dot{x}_j B_k(x,t) 2)...

Homework Statement Could someone please explain what is meant by the term: \partial_{[ \mu}F_{\nu \rho ]} Homework Equations I have come across this in the context of Maxwells equations where F^{\mu \nu} is the field strength tensor and apparently: \partial_{[ \mu}F_{\nu \rho...

Homework Statement show that [L_z,L_x]=i(\hbar)L_y The Attempt at a Solution [A,B]=AB-BA L_z=xP_y-yP_x L_x=yP_z-ZP_x So do i just use the fact that [A,B]= AB-BA and then use the momentum operators and substitute everything in an churn out the algebra to reduce it...

Homework Statement by considering the action of [p-hat (subscript x), H-hat] on a general state, show that [p-hat (subscript x), H-hat] =-ihbar dV/dx Homework Equations H-hat = (((p-hat)^2)/2m) +V(x) p-hat (subscript x)= -i*h d/dx (partial derivative) The Attempt at a...

Homework Statement [\hat{H},\vec{r}]= ? The Attempt at a Solution The answer is given, and I KNOW that factor of 6 shouldn't be there. The answer should be -\frac{\hbar^2}{m} \nabla Anyway I've always been lurking these forums and I enjoy the discussions here, but this factor is...

The commutator for group theory is [X,Y]=X^{-1}Y^{-1}XY whereas the quantum commutator is [X,Y]=XY-YX . At first glance, the two commutators seem to be totally unrelated because the quantum commutator speaks of two binary operations whereas group theory has one binary operation. However...

can someone please help me with this. it's killing me. Homework Statement to show \left[\vec{L}^{2}\left[\vec{L}^{2},\vec{r}\right]\right]=2\hbar^{2}(\vec{r}\vec{L}^{2}+\vec{L}^{2}\vec{r})Homework Equations I have already established a result (from the hint of the question) that...

I have been told that L and P^2 do not commute, but I don't see why. It seems like the commutator should be zero. \left[ \vec{L} , P^2 \right] = \left[ L^k , P_i P_i \right] = \left[ L_k , P_i \right] P_i - P_i \left[ L_k , P_i \right] = \left( - i \hbar \epsilon_{i}^{km} P_m \right)...

So I am reading a book on ferromagnetism, the author writes \left[S_J,S_{J-1} \cdot S_J\right] = S_J cross S_{J-1} (I couldn't get the cross product x in latex code for some reason) Where [tex]S_J[\tex] and [tex]S_{J-1}[\tex] are the spin operators for atoms j and j-1. I was...

Homework Statement Let H be the hamiltonian H = p²/2m+V(r) Let r.p be the scalar product between the vector r and p. Calculate the Commutator [r.p , H] (Commutator of [A,B]=AB-BA ) Homework Equations The equations citated we should be using are: [x_i, p_i]=i \hbar And...

I have a potential of -1/r and I need to compute \left[H , \ \mathbf{p} \right] . I got the result of i \hbar \left( \frac{1}{r^{2}}, \ 0 , \ 0 \right) . Am I wrong about this?

Hi, I haven't posted this in the homework section, as I don't really see it as homework as such. I'm trying to derive the Heisenberg equations of motion for the Klein Gordon field (exercise 2.2 of Mandl and Shaw). I'm trying to derive the commutator of the Hamiltonian and canonical momentum...

hi I found this in textbook: [A,B] = [\DeltaA, \DeltaB] Experimenting witht he expressions of \DeltaA and \DeltaB, I find [\DeltaA, \DeltaB] = [A,B] - [A, <B>] - [<A>, B] + [<A>,<B>] A, and B are two hermitean operators, and \DeltaA = A - <A> etc, so <A> and <B> do not commute in general...

Homework Statement Calculate [\hat X, \hat P^2]. Homework Equations [\hat A, \hat B] \Psi =[\hat A \hat B - \hat B \hat A ] \Psi.The Attempt at a Solution I am confused by P^2. P is worth -i \hbar \frac{\partial}{\partial x}. So I believe P^2= \hbar ^2 \left ( \frac{\partial}{\partial x}...

Homework Statement I must calculate [X,P].Homework Equations Not sure. What I've researched through the Internet suggests that [\hat A, \hat B]=\hat A \hat B - \hat B \hat A and that [\hat A, \hat B]=-[\hat B, \hat A]. Furthermore if the operators commute, then [\hat A, \hat B]=0 obviously...

Hi guys I'm quit confuse here I want to know what's the difference between lie product and commutation relation? I've been told that every commutator is lie product but not the way around, but I can't see the difference?

I'm working on a question where the operator B has a commutator [A,B] which is not equal to AB-BA, and instead has a squared term in it. Maybe I just don't understand commutators but what does this mean/how do I use the commutator (to find eigenvectors of A)?

Homework Statement Verify the following commutation relations using \vec J = \vec Q \times \vec p and [Q_{\alpha},p_{\beta}]=i \delta_{\alpha \beta} I 1. [J_{\alpha}, J_{\beta}]=i \epsilon_{\alpha \beta \gamma} J_{\gamma} 2. [J_{\alpha}, p_{\beta}]=i \epsilon_{\alpha \beta \gamma}...

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...