Stack Exchange Network

Stack Exchange network consists of 174 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
0answers
43 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
74 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
60 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
45 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
30 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
54 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
85 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
47 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
42 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
82 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
82 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
46 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
18 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
44 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
50 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
106 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
80 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
64 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
116 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
134 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}...
0
votes
1answer
41 views

A question about the commutator

I'm self-studying quantum mechanics and have a question regarding the commutator. Since the commutator of two operators is defined as $[A,B]$ = $AB$ - $BA$ Assuming that these operators do not ...
0
votes
1answer
42 views

Commutator of $\hat {L}_x$ and $\hat{V}(\hat{r})$ [duplicate]

Consider the angular momentum operator $\hat{L_x}=\hat y\hat{p}_z-\hat{z}\hat{p}_y$ and the potential operator $\hat{V}$ where the potential $\hat{V}=\hat{V}(\hat{r})$ is spherically symmetric. It ...
1
vote
0answers
61 views

What is radial ordering?

In my String theory lecture radial ordering was introduced and I don't understand what it is. My first guess was $$R(A(z)B(w)) = A(z)B(w)\Theta(|z|-|w|) + B(w)A(z)\Theta(|w|-|z|).$$ But then we have ...
2
votes
1answer
31 views

Quantizing first class constraints

Let $\gamma$ denote a first class constraint. Then if there exists a function on phase space $f(q,p)$ for which the Poisson bracket with the constraint does not vanish $\lbrace f, \gamma\rbrace \neq 0$...
1
vote
1answer
92 views

Commutator of a quark current

In Quantum Chromodynamics, when we take the limit in which the u, d and s quarks have no mass, there exists a global symmetry $G \equiv SU(3)_L \otimes SU(3)_R$ in flavour space. The corresponding ...
0
votes
1answer
42 views

Finding the uncertainty from a probability distribution?

If you have two properties, $A$ and $B$, that do not commute, and thus have a commutator $C$, and the uncertainties $\Delta A$ and $\Delta B$ obey the relation $$(\Delta A)(\Delta B)\geq \frac{1}{2}|C|...
4
votes
2answers
492 views

Does the Heisenberg equation for fields and canonical momentums hold as well for the hamiltonian density operator instead of the Hamiltonian operator?

In quantum field theory, with the field $\phi$ and the momentum $\pi$ being operators, their time evolution is governed (in the Heisenberg-picture) by the Heisenberg equation: \begin{align} \dot{\phi}...
2
votes
1answer
50 views

Quantum statistics from the (anti)commutation relations of the operators?

From a QFT point of view, the difference between bosons and fermions is that their creation/annihilation operators ($a^{\dagger}$, $a$ and $c^{\dagger}$, $c^{\dagger}$ respectively) obey the following ...
2
votes
1answer
66 views

Joint Spectral Measure theorem

I want to gain an intuition to understand the joint spectral measure theorem. In the case that operators involved in this theorem have purely discrete spectrum, the theorem should be reduced to the ...
0
votes
1answer
94 views

Plausibility of the Weyl-Relations of position and Momentum - Physical meaning of the Heisenberg group

In this question I asked about the uniqueness of the momentum operator $\hat{p}$ for a given position operator $\hat{x}$, and wether the uniqueness was fixed by the commutation relations that position ...
1
vote
1answer
72 views

Ladder operators vs creation/annihilation operators

I am trying to figure out the difference between the ladder operators (for harmonic oscillator) $a^\dagger$, $a$ and the creating/annihilation operators $c^\dagger$, $c$. Are they the same? I have ...
2
votes
1answer
83 views

What does it mean for 2 observables to be compatible?

If I have 2 observable operators $A$ and $B$, if $A$ and $B$ commute: $[A, B] = 0$, then they must necessarily form a complete set of commuting observables (CSCO). Essentially, if 2 observables are ...
1
vote
1answer
82 views

Evaluating an Equation Using Einstein Summation Notation [closed]

Problem As part of a proof for the commutation relation of a vector operator and the angular momentum operator, I need the evaluate the expression $$(R_\omega)_{ij}V_j = \big([1-\cos(\omega)]\hat{\...
1
vote
1answer
84 views

Commutation relation in quantized electromagnetic field theory

I have a question regarding a proposed problem (Problem 4.8) in Rodney Loudon's book "The Quantum Theory of Light". Let $U(t)$ be an operator defined by $$ U(t)=\exp\left\lbrace\frac{i}{\hbar}\int\...
0
votes
1answer
71 views

What is the state of particle at time $t$ if at $t=0$ it is in an eigenstate of $\hat{A}$, and $\hat{A}$ commutes with $\hat{H}$?

EDIT: added (assuming $\lambda$ to be non-degenerate). Based on the specifics of the question, we don't in fact know whether this is the case, so it may be that $\left|\lambda\right>$ is not an ...
1
vote
1answer
58 views

Commutators in Gupta-Bleuler formalism for quantization of the electromagnetic field

In the Gupta-Bleuler formalism we have for the canonical momenta $$\pi_\mu=F_{\mu0}-g_{\mu0}\partial_\alpha A^\alpha. $$ Every resource I find online say that the equal time canonical commutation ...
-1
votes
1answer
103 views

Are there any ways to exclude uncertainty in the values of any non-commuting operators? [closed]

If two similar systems are created and In the first system the position is measured with accuracy and in the second one the momentum is measured with accuracy can this avoid the uncertainty in the ...
2
votes
1answer
103 views

Proving that an operator is hermitian [closed]

Let $A$ be an operator that is the product of two hermitian operators. Am I at liberty to say that if those two hermitian operators commute and their commutator is zero, then $A$ is hermitian?
0
votes
1answer
56 views

Definitions of operators and commutativity in quantum mechanics

If $[\hat A,\hat B] = 0$, where $\hat A$ and $\hat B$ are operators, then the operators commute. This also means that, when applied to a wavefunction, that one can measure observables $A$ and $B$ in ...
1
vote
1answer
99 views

Do all well-measured observables effectively commute?

Do all well-measured observables effectively commute with each other? The rest of this long post clarifies what I mean by that simple-looking question. Consider quantum field theory in Minkowski ...
1
vote
1answer
66 views

Canonical commutation relation from Pauli-Jordan function

In Steinmann's book, Perturbative Quantum Electrodynamics and Axiomatic Field Theory is stated that the commutator of two fields $\varphi$, which satisfy the Klein-Gordon equation $$ (\square-m^2)\...
0
votes
0answers
45 views

Dirac field local observables

This is actually a continuation of calculation I've been working on. It is well known that, in the case of Dirac fields $\psi(x)$, they satisfy anticommutatation relationships since they're fermionic ...
6
votes
2answers
273 views

Commutator expectation value in quantum mechanics

Suppose $A$ and $B$ are operators, $A$ is Hermitian, $B$ anti-hermitian, and their commutator is the identity, i.e. $$[A, B] = I \, .$$ Denoting the eigenvectors of $A$ as $\lvert a \rangle$, so that $...
0
votes
2answers
73 views

If the commutator $[x,p]=i$, why does $[x,p^2]=2ip$?

According to Arfken et al. Mathematical Methods p.277 $$[x,p^2]=xp^2 - pxp + pxp -p^2x =[x,p]p + p[x,p]= 2ip \, .$$ According to the text this follows solely from $[x,p]=i$. I'm not understanding ...
0
votes
1answer
35 views

Commutator of normal ordered squared scalar field

The teacher left us prove the following statement. Let $\phi(x)$ be a scalar field $$ \phi(x) = \int \frac{d^3p}{2\pi \sqrt{2\omega_{\boldsymbol{p}}}} \left[ a_{\boldsymbol p} e^{-ip \cdot x} + a^+_{...
0
votes
1answer
67 views

If $\psi$ acting on $a_+$ and $a_-$ operator just moves up and down the ladder, why is $[a_-, a_+] = 1$ and not 0? [duplicate]

If $\psi$ is acted upon by both the operators one by one, it should return the same wave function. Thus order in which you increase or decrease the energy shouldn't matter. Then why is it so that the ...
2
votes
3answers
187 views

Is the Heisenberg-picture commutator $[x(t),p(t)]$ between position and momentum always equal to $i\hbar$?

Some misconceptions over here, For $x=$position and $p=$ momentum, I know $[x,p]=i\hbar$ but does $[x(t),p(t)]$ still have the same relation where $t$ here represents time.
0
votes
1answer
103 views

Propagator Causality with commutators all the way

We know that two fields commute - by locality and causality - iff there is spacelike separation $\left[\phi_l^k(x) , \phi_m^{k'}(y)\right] = 0$ for $(x-y)^2<0$ In the canonical quantization of ...
1
vote
1answer
56 views

A problem in three sequential selective measurements in case of incompatible observables

In case (a) the B filter takes care of B's eigenvalues and we sum all of their probabilities to calculate the probability of obtaining $|c'\rangle$. In case (b) B filter is not used and this creates ...
0
votes
1answer
70 views

The creation and annihilation operators in quantum mechanics

What is the result of the commutation relation between the creation operator and a power of the annihilation operators in simple harmonic oscillator problem?