Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [commutator]

A mathematical construct quantifying the difference in effect of applying two operators in two alternate successions. It is the defining product of a Lie algebra, the efficient underlying description of Lie groups, of use in several areas of physics, most notably quantum field theory.

0
votes
1answer
59 views

How to efficiently compute the commutator $[\hat{r},\nabla^2]$?

Given a system with Hamiltonian $ \hat{H} = \frac {\hat{p} ^2}{2m} + \hat{V}(r)$ in a certain state $|\psi \rangle$, I want to find if $\langle r \rangle$ varies with time. From $$ i \hbar\frac {d ...
-1
votes
1answer
61 views

Proof for $\langle i[A,B]\rangle$ [on hold]

I have to prove the following equation: $$ \langle i[A,B]\rangle = 2\mathfrak{Im}\left[\int dV(\overline{B\psi)}(A\psi)\right]\,,$$ where A,B are hermitian operators. Here is my calculation, but I don'...
0
votes
0answers
19 views

Source for mathematical methods [duplicate]

I am just curious about any question and example sources for linear vector spaces, bra-ket notation, operators, commutators and hilbert spaces.
1
vote
1answer
58 views

Field operator commutation: If two operators commute, then their fourier transforms also commute?

Im doing this in the context of field operators $$\psi(x)=\sum_k a_k e^{ikx},$$ $$\psi^T(y)=\sum_k a_k^T e^{-iky},$$ and their being defined as the fourier transform of the creation/annihilation ...
0
votes
2answers
50 views

Lie algebra vs. position and momentum commutators

Most theoretical texts on high energy physics make statements like below: $$[A_i , A_j] = i C^k_{i,j} A_k $$ (I suppose $\hbar$ may or may not be needed) and of course they describe this as being ...
1
vote
1answer
61 views

How does Sakurai reduce a product to a commutator?

The following section is from Modern Quantum Mechanics by Sakurai; can any one help me finding out how this is done? In contrast, if we follow approach 2, we obtain \begin{align} \vert\alpha\...
1
vote
1answer
35 views

Commutator of spacetime translation

In Srednicki's textbook Quantum Field Theory, eq. (95.7) reads: \begin{equation} [\Phi (x, \theta, \theta^{*}), P^{\mu}] = -i\partial^{\mu}\Phi (x, \theta, \theta^{*}). \end{equation} where $\Phi (x,...
0
votes
1answer
41 views

Commutation relations in Gupta-Bleuler formalism

When quantising the EM field thanks to the Gupta-Bleuler formalism, Itzykson and Zuber assume that the canonical commutation rules are $$ [\hat{A}_\rho (t,\vec{x}), \hat{\pi}^\nu(t,\vec{y})]= i \, ...
10
votes
2answers
991 views

Symmetry in quantum mechanics

My professor told us that in quantum mechanics a transformation is a symmetry transformation if $$ UH(\psi) = HU(\psi) $$ Can you give me an easy explanation for this definition?
0
votes
0answers
30 views

Canonical commutation relations for a real scalar field

I am taking my first course in QFT and have come across this problem From the canonical commutation relations for a real scalar field $\hat{\phi}$ show that $$[\partial_i \hat{\phi} , \hat{\phi}...
0
votes
0answers
25 views

Derivation for the commutation relation of $\alpha_m^\mu$ and $\alpha_n^\nu$

I have been trying to derive the commutation relation of $\alpha_m^\mu$ and $\alpha_n^\nu$ in a closed-string mode expansion, but I found an extra factor of $2$ that ruins things out: Given $\dot X = ...
1
vote
0answers
39 views

The products of powers of Hermitian operators

Let's say I have two operators, $\hat{x}^k$ and $\hat{p}_x^l$, where $\hat{x}$ and $\hat{p}_x$ are the ordinary position and momentum operators. It seems fairly straight forward to show that $\hat{x}^...
1
vote
1answer
71 views

Lie algebra: Proof that the commutator of infinitesimal motions is an infinitesimal motion

I am following Classical and Quantum Mechanics via Lie Algebras by Neumaier and Westra. Setup I am stuck at part of Thm 2.3.1. Consider the matrix group $\mathbb{G}$. The set of $\mathbb{G}$-motions ...
2
votes
3answers
237 views

If infinitesimal transformations commute why don't the generators of the Lorentz group commute?

If infinitesimal transformations commute as proved e.g. on this mathworld.wolfram page, why are the commutators for the generators of the Lorentz group nonzero?
1
vote
3answers
95 views

Why doesn't the anticommutator $\{x,p_x\}$ have an unique value?

The commutator of position and momentum, $[x,p_x]$, has a unique value given by $i\hbar$. Why doesn't the anticommutator $\{x,p_x\}$ also have a definite value?
0
votes
0answers
50 views

Uncertainty Principle and Commutators

In preparation for an exam I stumbled upon a quantum mechanics task I can´t really solve right know. So i hope someone can maybe give me a hint or two how to understand this. Here is the task: Let $...
1
vote
1answer
43 views

Canonical commutation relation for spherical coordinates?

What coordinates systems can the canonical commutation relation be generalised to? I also ask specifically for spherical coordinates. This is because I want to prove $\hat{\bf p}\cdot{\bf e}_r-{\bf e}...
1
vote
1answer
63 views

Does $[L_z,H] = 0$ imply the state is also an Eigenstate of $H$ is also an eigenstate of $L_z$?

Given that the Hamiltonian $\mathcal{H}$ is rotationally invariant then we know $[L_z,\mathcal{H}] = 0$. Does that imply that an eigenstate of H is also an eigenstate of $\mathcal{H}$? More ...
1
vote
1answer
47 views

Does total $\hat{S}^2$ always commute with total $\hat{S}_z$ even for interacting spins?

I was given the following operator $\hat{f}$ describing the interaction of two spin-$\frac12$ particles: $$\hat{f}=a+b{\hat{\bf S}_1}\cdot{\hat{\bf S}_2}.$$ I was told that I can prove that $\hat{f}$...
0
votes
0answers
40 views

Commutation in coupled Harmonic Oscillators

Starting with a coupled Harmonic Oscillator problem $$ H = \frac{p_1^2 + p_2^2}{2m} + \frac{K}{2}\left[x_1^2 + x_2^2 + \left(x_1 - x_2\right)^2\right] = \left(\frac{p_1^2}{2m} + \frac{2K}{2}x_1^2\...
0
votes
1answer
40 views

Hausdorff expansion

Could someone explain me, what the Hausdorff expansion is? In my quantum mechanics homework I should use something called the Hausdorff expansion which looks like the following: $$e^ABe^{-A}=B+[A,B]+\...
0
votes
1answer
32 views

Square bracket notation of the basis of 16 independent gamma matrices

The question is very simple and I couldn't find an answer. What the notation $\gamma^{ [ \mu} \gamma^{\nu} \gamma^{\rho ]}$ and $\gamma^{ [ \mu} \gamma^{\nu} \gamma^{\rho} \gamma^{\sigma ]}$ means? ...
1
vote
0answers
24 views

Geometrical way to view discretization of energy in quantum mechanics. How commutation relation implies discreteness?

The relation from which discreteness in eigenvalue of the energy of bound state arises is $[x, p]=i\hbar$ followed by the rule that wavefunction should be normalizable. So my question is there a ...
1
vote
0answers
41 views

Finding a closed formula using Baker-Hausdorff formula for a unitary transformation; An endless commutator

Consider the Baker-Hausdorff formula for two operators $a_1$ and $iHt$: $$e^{iHt}a_1 e^{-iHt} =a_1+[iHt,a_1]+\frac{1}{2!}[iHt,[iHt,a_1]]+\frac{1}{3!}[iHt,[iHt,[iHt,a_1]]]+....,$$ where $[A,B]=AB-BA$. ...
0
votes
1answer
41 views

The charge given by a commutator

I saw in the text that $[Q,X]=cX$ and says the operator $X$ has charge $c$ under the generator $Q$. I tried to understand why the coefficient $c$ means the charge. So I used this relation to get the ...
0
votes
1answer
22 views

Symmetry in Fock-space 2-body interaction

The simplest two body interaction term for fermions is $$H = \sum_{ijkl} U_{ijkl} a_i^\dagger a_j^\dagger a_k a_l$$ and I'm trying to determine the symmetries on $U$. Unfortunately I keep getting ...
6
votes
2answers
129 views

Geometric interpretation of the second Bianchi identity?

Assuming a torsion free Christoffel symbol, the covariant derivative can be shown to satisfy the second (differential) Bianchi identity: \begin{equation} [[\nabla_a,\nabla_b],\nabla_c]+[[\nabla_c,\...
1
vote
1answer
44 views

What is the commutator of a reasonably-behaved function of an operator and the derivative of that function? What is $[f(\hat x) ,f'(\hat x) ]$?

I tried to do the usual procedure and expand the commutator, but couldn't proceed after I Taylor-expanded $f(\hat x)$. $$\Big[f(\hat x),\frac{d}{dx}f(\hat x)\Big]=$$ $$f(\hat x)f'(\hat x)-f'(\hat x)...
2
votes
2answers
74 views

Angular momentum coupling

I read about angular momentum coupling on wikipedia and there are a few things i dont understand. What does this mean "spin and orbital angular momentum of a single object belong to different Hilbert ...
0
votes
0answers
55 views

What is the QFT state with two distinguishable fermions present?

I want to describe a system with two non-interacting and definitely different fermions, say an electron neutrino, $\nu_e$, and an electron, $e^-$. The state describing a single electron is given ...
0
votes
0answers
45 views

Commutator of Lie Group Generators

This is from Maggiore's "A Modern Introduction to Field Theory", Page 15. I have a Lie group with matrix generators $$ T^{a}_{R}$$ Where $a$ takes values from 1 to the dimension of the Lie group. ...
0
votes
2answers
79 views

Simple question on Angular Momentum

Need to know why $L^2$ and ONLY ONE of $L_x$, $L_y$, $L_z$ are constants of motion. Main problem arrives when $V = f(r, \theta, \phi)$ causing none of the $L_x$, $L_y$, $L_z$ to commute with ...
2
votes
1answer
65 views

How does one deal with derivative operator in quantum field theory properly?

Given creation and annihilation operators, ${a^{\dagger}(x,t)}$ and $a(x,t)$ in non-relativistic quantum field theory, respectively, which satisfy the following properties: Now, I want to prove $$[...
0
votes
1answer
50 views

Does $x$-component $\hat{L}_x$ of angular momentum commute with $\hat{x}$? [closed]

Question from lecture notes: What of the following operators does not commute with $\hat x$? A. $ \hat L_x $ B. $ \hat L_y $ C. $ \hat L_z $ D. None of the above. The answer ...
0
votes
0answers
32 views

Commutator $\vec{L}$ with $\vec{X}\cdot\vec{P}$

Let $\vec{X}=(X_1,X_2,X_3)^T$ and $\vec{P}=(P_1,P_2,P_3)^T$. Define $\vec{L}=\vec{X}\times\vec{P}$. Then, I can calculate $\vec{L}=(X_2P_3-X_3P_2,\,X_3P_2-X_2P_3,\,X_1P_2-P_1X_2)^t$. For all ...
1
vote
1answer
88 views

“Commuting observables share common eigenstates”

I am struggling to find a precise definition of this line from my quantum mechanics textbook: If $[A,B] = 0$, then the operators commute, and "commuting operators share common eigenstates". This ...
2
votes
2answers
89 views

Does $i\hbar \frac{d \hat A }{d t} = [\hat A (t_0), \hat H]$ hold when $H$ is time-dependent, but $[H(t_0), H(t'_0)] = 0$?

It is known that - given in Sakurai, ch2.2, p83 - in Heisenberg's picture, for a Hamiltonian, $H$, independent of time, the time evolution of any operator $\hat A$ is given by $$i\hbar \frac{d \hat ...
0
votes
0answers
51 views

Commutation of differential operators with boundary conditions

First post ever. Let's see how this goes... My question concerns the commutation of differential operators in the presence of boundary conditions. If it is of any help, this is relevant to me in the ...
2
votes
0answers
44 views

Commutator of $f(\hat{x})$ and $\hat{p}$ [closed]

I have a homework problem that asks us to determine $\big[\,f(\hat{x}),\hat{p}\big]$ only by use of $$\big[\hat{x}^n,\hat{p}\big] = ni\hbar \hat{x}^{n-1}$$ as well as assuming $f(\hat{x})$ can be ...
0
votes
2answers
94 views

Notation and concepts of Yang Mills Theory

I am studying loop quantum gravity using the book by Pullin and Gambini. I am having some trouble understanding and getting past the chapter on Yang Mills theory, mainly because I am confused about ...
3
votes
1answer
83 views

Is is possible to have a pair commuting observables only in a single direction?

In quantum mechanics, for two observables to be compatible, successive measurements of the observables, say $A$ and $B$, should yield the same result as earlier, i.e if we do the measurements with the ...
1
vote
0answers
75 views

How to find a CSCO (Complete Set of Commuting Operators/Observables) of an eigenfunction?

So, say you have an eigenfunction defined as: $\psi \left(x_1,x_2\right)=\int _{-\infty }^{+\infty }\:e^{p\left(\frac{2\pi i}{h}\right)\left(x_1-x_2+x_o\right)}dp$ (The spinless 2 particle system ...
0
votes
0answers
23 views

Conjugate of total spin operator

I got a lattice, and the total spin operator for x and for y, for that lattice. I know that the x component conmutes with an operator called staggered spin operator in y. I also know that the ...
-1
votes
1answer
65 views

What is the meaning of “representation of the canonical commutation relation in the form of Heisenberg for symplectic locally convex space”?

What is the meaning of "representation of the canonical commutation relation in the form of Heisenberg for symplectic locally convex space"?
-1
votes
1answer
55 views

Commutation of $J^2$ and $J_i$ [closed]

For the Hermitian operators $\hat{J_1},\hat{J_2},\hat{J_3}$ that satisfies the commutation relations $$[\hat{J_1},\hat{J_2}]=i\hbar\hat{J_3},$$ $$[\hat{J_2},\hat{J_3}]=i\hbar\hat{J_1},$$ $$[\hat{J_3},\...
4
votes
2answers
116 views

Significance when expectation of a commutator is zero

It is clear to me what it means when the commutator of two operator $[A, B]$ is zero and what it implies. However, is there any significance when the expectation of the commutator $\langle[A, B]\...
2
votes
2answers
92 views

Commutation relations

Given that the Hamiltonian for Muonium spin in zero magnetic field is $$\hat{H} = a \vec I \cdot \vec J$$ where $\vec I$ is the spin of a muon, and $\vec J$ is the spin of the electron, what is the ...
1
vote
2answers
71 views

Commutator generating transformations

Lately I am encountering the commutator of variations of the variables and I'm not quite sure about its physical meaning. Some examples. 1) "The composition of two supersymmetries generates a time ...
2
votes
3answers
122 views

Operators commutation and relation between eigenvalues

If $H$ and $L_i$ are commuting ( $[H, L_i] = 0$ ) could we deduce that the eigenvalues of $H$ depend/ do not depend on $m$ and $\ell$ ( eigenvalue of $L_z, L^2$ )? I don't think so since it does not ...
0
votes
1answer
154 views

What is the time derivative $\frac{d}{dt}(\exp(\hat A))$ of operator exponential $\exp(\hat{A})$? [duplicate]

Assuming that the operator $\hat{A}$ does not necessarily commute with $d \hat A/dt$, what is $$\frac{d}{dt}(\exp(\hat A))~?\tag{1}$$ Is it $\exp(\hat A(t)) \frac{d \hat A}{dt}$ or $\frac {d\hat A}{dt}...