Commutator Definition and 274 Threads

The book calculates the commutator $[H,x_i]$ as $$[H,x_i] = \left[ \sum_j \frac{p_j^2}{2m}, x_i \right] = \frac{2}{2m} \sum_j p_j \frac{\hbar}{i} \delta_{ij} = - \frac{i \hbar p_i}{m},$$ where the hamiltonian operator $H$ is $$H = \sum_j \frac{{\mathbf p}_j^2}{2m_j} + V({\mathbf x}).$$ The book...

Homework Statement Given that the function f can be expanded in a power series of a and a^\dagger, show that: [a,f(a,a^{\dagger})]=\frac{\partial f }{\partial a^\dagger} and that [a,e^{-\alpha a^\dagger a}] = (e^{-\alpha}-1)e^{-\alpha a^{\dagger} a}aThe Attempt at a Solution I've tied using...

[A,B] = AB-BA, so the commutator should be a matrix in general, but yet [x,p]=i*hbar...which is just a scalar. Unless by this commutator, we mean i*hbar*(identity matrix) ? I am asking because I see in a paper the following: tr[A,B] Which I interpret to mean the trace of the commutator...

Homework Statement Given that \vec{V} and \vec{W} are vector operators, show that \vec{V}\times \vec{W} is also a vector operator. 2. The attempt at a solution The only way I know how to do this is by showing that the commutator with the angular momentum vector operator ( \vec{J}) is zero...

Homework Statement [T,V]=[TV-VT]ψ Homework Equations T=(-ħ2/2μ)∂2/∂x2 V=(1/2)kx2[/B]The Attempt at a Solution [(-ħ2/2μ)∂2/∂x2((1/2)kx2ψ)]-[(1/2)kx2(-ħ2/2μ)∂2/∂x2(ψ)] I think my problem is with executing the chain rule on the first term: (-ħ2/2μ)[x2ψ''+2xψ'+2xψ'+2ψ-x2ψ''][/B] The x2ψ''...

Homework Statement Hey guys, So I have to show the following: [P^{\mu} , \phi(x)]=-i\partial^{\mu}\phi(x), where \phi(x)=\int \frac{d^{3}k}{(2\pi)^{3}2\omega(\vec{k})} \left[ a(\vec{k})e^{-ik\cdot x}+b^{\dagger}(\vec{k})e^{ik\cdot x} \right] and P^{\mu}=\int...

Homework Statement I am practicing problems from the textbook, but have no idea how to get to some of the solutions available in the back of the textbook... 6-16. Evaluate the commutator [C,D] where C and D are given below: (e) C = d2/dx2, D = x (g) C = integral (x = 0 to infinite) dx, D =...

I'm having some difficulties with a certain commutator producing inconsistent results. Specifically I'm referring to [p_x,x^3] Depending on how i expand this it seems i get different coefficients, i.e [p_x,x^3]=[p_x,x]x^2+x^2[p_x,x]=-i\hbar x^2 -x^2i\hbar=-2i\hbar x^2 However...

Homework Statement Calculate the commutator ##[\hat{L}_i, (\mathbf{rp})^2]## Homework Equations ##\hat{\vec{L}} = \sum\limits_{a=1}^N \vec{r}_a \times \hat{\vec{p}}## ##[r_i,p_k] = i\hbar\delta_{ik}## The Attempt at a Solution Okay so here is what I have so far: $$ \begin{eqnarray}...

Homework Statement Given that U upon acting on the creation operator gives a creation operator for the transformed momentum $$U(\Lambda) a_p^\dagger U(\Lambda)^\dagger = a_{\boldsymbol{\Lambda} \mathbf{p}}^\dagger $$ and ##\Lambda ## is a pure boost, that is ## U(\Lambda) = e^{i...

According to my teacher, for any two operators A and B, the commutators [f(A),B]=[A,B]df(A)/dA and [A,f(B)]=[A,B]df(B)/dB He did not give any proof. I can easily prove this for the particular cases [f(x),p]=[x,p]df(x)/dx and [x,pn]=[x,p]npn-1 But I don't see how the general formula is true. I...

When performing matrix multiplication with 2 matrices A and B ;AB might exist but BA might not even exist. Hermitian operators can be thought of as matrices but in everything I have seen so far AB and BA always exist even though they can be different depending on the value of the commutator. Do...

I am trying to derive the algebra and I get a factor of 2 wrong... Consider the Lorentz group elements near the identity \Lambda_1^\mu\,_\nu = \delta^\mu\,_\nu + \omega_1^\mu\,_\nu, \quad \Lambda_2^\mu\,_\nu = \delta^\mu\,_\nu + \omega_2^\mu\,_\nu and write a representation as...

Definition/Summary The commutator subgroup of a group is the subgroup generated by commutators of all the elements. For group G, it is [G,G]. Its quotient group is the maximal abelian quotient group of G. Equations The commutator of group elements g, h: [g,h] = g h g^{-1} h^{-1} The...

Hi all, I haven't been able to find an answer online but this seems like a pretty basic question to me. What are the commutator relations between the position/momentum operators and the field operator? I'm not even certain what the commutation relations between X/P and a single ladder operator...

When using Fourier's trick for determining the allowable energies for stationary states, Griffiths introduces the a+- operators. When factoring the Hamiltonian, the imaginary part is assigned to the momentum operator versus the position operator. Is there a reason for this? If : a-+ = k(ip +...

Homework Statement Find ##[\hat{x},\hat{T}]##. Homework Equations ##[\hat{x},\hat{T}]=\hat{x}\hat{T}-\hat{T}\hat{x}## The Attempt at a Solution I wind up with ##\frac{i\hbar}{m}\hat{p}##. Did I do good, boss? Chris

Hi, everybody: I encountered a problem when I am reading a book. It's about the atom-photon interaction. Let the Hamiltonian for the free photons be H_0=\hbar \omega(a^{\dagger}a+\frac{1}{2}). so the commutator of the annihilation operator and the Hamiltonian is [a,H_0]=\hbar\omega a and I...

If the group G/[G,G] is abelian then how do we show that xyx^{-1}y^{-1}=1? Thanx

Hi there! I have tried for hours to calculate the commutator of angular momentum in the differential form, but I cannot get the correct answer. This is my first experience with actually checking if two operators commutes, so there may be some beginner's misunderstandings that causes the...

This is not a homework question, I just can't find a good resource on this topic. I am working in quantum mechanics on commutator relations. My book (Griffiths) lacks information on how to manipulate the commutator relations. For instance, when I have [AB,C], when can I make it A[B,C]? Or...

Hi, suppose that the operators $$\hat{A}$$ and $$\hat{B}$$ are Hermitean operators which do not commute corresponding to observables. Suppose further that $$\left|A\right>$$ is an Eigenstate of $$A$$ with eigenvalue a. Therefore, isn't the expectation value of the commutator in the eigenstate...

Homework Statement Find ##[L_i,L_j]##. Homework Equations [x_i,p_j] = \delta_{ij}i\hbar The Attempt at a Solution [L_i,L_j] = \epsilon_{ijk}\epsilon_{jlm} [x_jp_k,x_lp_m] = \left( \delta_{jl}\delta_{km} - \delta_{jm}\delta_{kl}\right)[x_jp_k,x_lp_m] = [x_jp_k,x_jp_k] - [x_jp_k,p_kx_j] =...

In infinitely potential well problem ##V(x)## is zero inside the box and ##\infty## outside the box. Is it possible to calculate commutator ##[V(x),p]##? If case that this commutator is zero ##\sqrt{\frac{2}{a}}\sin \frac{n\pi x}{a}## will also be eigenstate of ##p##. I am confused with this.

Homework Statement |phi (n)> being eigen states of hermitian operator H ( H could be for example the hamiltonian of anyone physical system ). The states |phi (n)> form an orthonormal discrete basis. The operator U(m,n) is defined by: U(m,n)= |phi(m)><phi(n)| Calculate the commutator...

If we have two fermion operators with a known anti-commutator AB+BA, what do we do if we find ourselves with AB-BA in an equation? Does this automatically vanish for fermions? if not, is there anything we can say about in general?

I'm doing something horribly wrong in something that should be very easy. I want to show that: [L^2, x^2] = 0 So: [L^2, x x] = [L^2, x] x + x [L^2, x] L^2 = L_x^2 + L_y^2 + L_z^2 Therefore: [L^2, x] = [L_x^2 + L_y^2 + L_z^2, x] = [L_x^2, x] + [L_y^2, x] + [L_z^2, x] = L_y [L_y...

How do I compute the commutator [a,e^{-iHt}], knowing that [H,a]=-Ea? I tried by Taylor expanding the exponential, but I get -iEta to first order, which seems wrong.

Homework Statement This is more about checking a solution I've been given is correct, because either my professor is consistently getting this wrong or I am.The Attempt at a Solution My professor's answers say [ \left( \begin{array}{ccc} 1 & 0 \\ 0 & 4 \end{array} \right), \left(...

I know that the commutator of the position and momentum operators is ihbar. Can any other combination of two different operators produce this same result, or is it unique to position and momentum only?

Homework Statement Hello! I'm having troubles with this proof. given two operators A &B , such that [A,B] = λ where λ is complex,and μ is also complex, show that exp{μ(A+B)} = exp{μA}exp{μB}exp{(-μ^2λ)/2} Homework Equations [A,B] = λ. [A,B] = AB-BA = λ The Attempt at a...

Calculate the commutator [px, xn] Homework Statement There are 3 tasks. 1) No other information is given. only that I have to calculate the commutator [px, xn]. For task 2 and 3 a relevant equation is given below. 2) calculate the commutator [x, Kx] 3) calculate the commutator [px, Kx]...

Homework Statement For the SHO, find these commutators to their simplest form: [a_{-}, a_{-}a_{+}] [a_{+},a_{-}a_{+}] [x,H] [p,H] Homework Equations The Attempt at a Solution I though this would be an easy problem but I am stuck on the first two parts. Here's what I did at first...

Consider the following commutator for the product of the creation/annihilation operators; [A*,A] = (2m(h/2∏)ω)^1 [mωx - ip, mωx + ip] = (2m(h/2∏)ω)^1 {m^2ω^2 [x,x] + imω ([x,p] - [p,x]) + [p,p]} Since we have the identity; [x,p] = -[p,x] can one assume that.. [x,p] - [p,x] =...

Two hermitian commutator anticommute: {A,B}=AB+BA=0.Is it possible to have a simultaneous eigenket of A and B?illustrate... Thank you in advance

Hi. I'm trying to understand a derivation of the Bianchi idenity which starts from the torsion tensor in a torsion free space; $$ 0 = T(X,Y) = \nabla_X Y - \nabla_Y X - [X,Y]$$ according to the author, covariant differentiation of this identity with respect to a vector Z yields $$$ 0 =...

Hello, i try to prove that ∂μFμ\nu + ig[Aμ, Fμ\nu] = [Dμ,Fμ\nu] with the Dμ = ∂μ + igAμ but i have a problem with the term Fμ\nu∂μ ... i try to demonstrate that is nil, but i don't know if it's right... Fμ\nu∂μ \Psi = \int (∂\nuFμ\nu) (∂μ\Psi) + \int Fμ\nu∂μ∂\nu \Psi = (∂\nuFμ\nu) [\Psi ]∞∞...

I think most of them are replaced by brushless types or Induction Motor with VFD (variable frequency drive). Is there some industries where they are still preferred today?

Homework Statement [S2 total, Sz ∅1] Homework Equations S2 total = S2∅1+ 1∅S2+2(Sx∅Sx+Sy∅Sy+ Sz∅Sz) The Attempt at a Solution I calculated it in steps: (1∅Sx 2 +Sx 2∅1) * Sz ∅1 =[S2x, Sz] ∅1 + Sz∅Sx 2 =-h_cut i (SxSy+SySx)∅1 + Sz∅Sx 2 Is it correct way of doing it? I...

I know how to derive below equations found on wikipedia and have done it myselt: \begin{align} \hat{H} &= \hbar \omega \left(\hat{a}^\dagger\hat{a} + \frac{1}{2}\right)\\ \hat{H} &= \hbar \omega \left(\hat{a}\hat{a}^\dagger - \frac{1}{2}\right)\\ \end{align} where...

I am new to group theory, and read about a "universal property of abelianization" as follows: let G be a group and let's denote the abelianization of G as Gab (note, recall the abelianization of G is the quotient G/[G,G] where [G,G] denotes the commutator subgroup). Now, suppose we have a...

Homework Statement So we know that for two operators \hat{A} and \hat{B} + \hat{C} we have the following rule for the commutator of the two: [\hat{A},\hat{B} + \hat{C}] = [\hat{A},\hat{B}] + [\hat{A},\hat{C}] However, if I'm commuting [\hat{p_{x}}, \hat{H}] where \hat{H} is the...

1) Find the commutator subgroup of the Frobenius group of order 20. 2) I have the Cayley table. 3) What is the definition of a commutator subgroup? I am absolutely sure we haven't heard this term all semester.

Homework Statement Homework Equations i think the most relevant equations would be some commutator algebra theorems i do not know of ! The Attempt at a Solution

Homework Statement Evaluate [x, SinPx] given [x, Px]=ih Homework Equations Px = h/I( d/dx) The Attempt at a Solution Let f (x) be a function of x.[ x, Sin Px ] f( x) ⇒ [ x sin{( h / i )d / dx} f ( x ) -Sin { (h/i) d / dx } x f(x)] . Does anybody concur .

Greetings, I would like to find the commutator \left[Lx^2,Ly^2\right] and prove that \left[Lx^2,Ly^2\right]=\left[Ly^2,Lz^2\right]=\left[Lz^2,Lx^2\right] I infer from the cyclic appearance of the indices that using the index notation would be much more compact and insightful to solve the...

Homework Statement Hello. I am supposed to find the commutator between to operators, but I can't seem to make it add up. The operators are given by: \hat{A}=\alpha \left( {{{\hat{a}}}_{+}}+{{{\hat{a}}}_{-}} \right) and \hat{B}=i\beta \left( \hat{a}_{+}^{2}-\hat{a}_{-}^{2} \right), where alpha...

For every operator ##A##, ##[A,A^n]=0##. And if operators commute they have complete eigen- spectrum the same. But if I look for ##p## and ##p^2## in one dimension ##sin kx## is eigen- function of ##p^2##, but it isn't eigen-function of ##p##. p^2 \sin kx=number \sin kx p\sin kx \neq number...

Homework Statement Using (x,p) = i (where x and p are operators and the parentheses around these operators signal a commutator), show that: a)(x^2,p)=2ix AND (x,p^2)=2ip b) (x,p^n)= ixp^(n-1), using your previous result c)evaluate (e^ix,p) Homework Equations For operators, in...

If ##\hat{n}=\hat{a}^+\hat{a}## is number operator and \hat{a}^+,\hat{a} are Bose operators. Is there then some formula for [f(\hat{n}),\hat{a}] [f(\hat{n}),\hat{a}^+]