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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.

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

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 ...
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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 ...
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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 $\lbrace f, \gamma\rbrace \neq 0$...
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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 ...
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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|...
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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}...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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{\...
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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\...
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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 ...
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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 ...
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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 ...
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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?
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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 ...
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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 ...
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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)\...
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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 ...
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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 $...
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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 ...
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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^+_{...
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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 ...
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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.
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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 ...
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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 ...
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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?
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The commutator of position and momentum operators in three-dimensional Cartesian coordinates

I'm to calculate the commutator of the following operators : $\mathbf{\widehat{r}}=\mathbf{e}_{x}x+\mathbf{e}_{y}y+\mathbf{e}_{z}z$ and $\mathbf{\widehat{p}}=-i\hbar\left ( \mathbf{e}_{x}\frac{\...
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States created by translation operator

Quantum Mechanics Volume One page 188 by Claude Cohen Tannoudji. In $q$ and $p$ state vectov formalism. $QS(\lambda) |q\rangle=(q+\lambda)S(\lambda)|q\rangle$, where $S(\lambda)=e^{-i\lambda P/\hbar}...
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Operating a Commutator as an observable

I came across the Baker-Hausdorff theorem in my quantum mechanics course and wanted to know how an observable commutes with a commutator it was a part of. Well if I have $[p,x]$, it should give me $-...
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How does $[A,B]=0$ imply the possibility of measuring the corresponding eigenvalues simultaneously?

I came across a webpage where they showed $[A,B]=0$ implies that we can measure it's corresponding eigenvalues simultaneously. I don't understand which step of the mathematical proof points to this ...
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Relation between groups and commutators

I am getting introduced into supersymmetry following chapter 11 of Ryder's Quantum Field Theory. I have a question about why $[\delta_1,\delta_2]=\Delta$ being this result independent of the field ...
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Second quantization canonical commutation relation : $\{c_\alpha,c_\beta^\dagger\}=\delta_{\alpha,\beta}$ a counter example?

Suppose two different states $\alpha$ and $\beta$ of some system of fermions such that each state only allows zero or one particle. The canonical commutation relation $\{c_\alpha,c_\beta^\dagger\}=\...
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How to prove commutation relation between charge and current in current algebra?

I am reading Gauge Theory of Elementary Particle Physics by Tapei Cheng and Lingfong Li. Proceeding equation 5.54, there is a statement which says Then from Lorentz covariance, we can include the ...
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Commutation relation, field and conserved charges

One more question on basic commutation relation for fields. Let $\phi(x)$ be a scalar field and $$ P^\alpha = \int d^3x T^{0\alpha}, $$ where $T^{\alpha\beta}$ is the energy momentum tensor. Now, ...
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Physical meaning of commutation

I was reading the solution to quantum harmonic oscillator by J.J. Sakurai. He uses the annihilation and creation operators and there's a key step (I think) which is $$[a,a^{\dagger}]=1$$ I know we can ...
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Can non-central hamiltonians commute with $\vec{L}^2$?

Central potentials $V(r)$ trivially commute with the operator $\vec{L}^2$ in quantum mechanics because the latter is a function of the angular coordinates $(\theta,\phi)$ only. Non-central potentials, ...
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Find the commutator $[AX+BY,Z]$

The problem I'm asked to solve is on quantum mechanics: Find the commutator $[xH+pH, p^2]$, where $H$, $x$ and $p$ are the Hamiltonian, space and momentum operator respectively A the moment, ...
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Show that $\vec L$ and $\vec S$ commute with each other

It is stated in Griffiths in a hint to a question that $\vec L$ and $\vec S$ commute with each other but no proof is given. $\vec L$ is given in the differential form and $\vec S$ is given in matrix ...
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Momentum operator ambiguous?

In nonrelativistic quantum mechanics, are different operators possible as a candidate for the momentum operator, given that one has fixed one position operator and a hilbert space that this position ...
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How to prove the translation generator commutes with the spinors in SUSY algebra?

I was reading Modern Supersymmetry by John Terning, the book starts with SUSY algebra and says $$ \left[ P_{\mu} , Q_{\alpha} \right] = \left[ P_{\mu} , Q_{\alpha}^{\dagger} \right] = 0 $$ I am ...
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Solving the von-Neumann equation explicitly [closed]

The von-Neumann equation reads: $$\frac{d\rho}{dt} = -\frac{i}{\hbar}[H,\rho]$$ The solution: $$\rho(t)=U\rho U^{\dagger}$$ with $U=e^{-\frac{iHt}{\hbar}}$ is easily obtained when starting from the ...
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Strange question involving finding a relation between a commutator and the time derivative of an operator

In order to get to the parts I am stuck at, I will add the examiners' solutions to each subquestion, which is needed to get to the subquestion that I am querying. The following is a bizarre question ...
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Energy $E_n$ for eigenstates $n$ and position probabilistic distribution

Energy and position operator does not commute. Uncertainty relation and Energy-Position interference So how come "For a particle in a box with given states $n$," and we obtained the $E_n$ for exact ...
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Simmons-Duffin CFT Lecture Notes: Exercise 3.3

Here is the exercise 3.3 in the Simmons-Duffin CFT lecture notes: Show that in $d \ge 3$, $$[Q_\epsilon,T^{\mu\nu}] = \epsilon^\rho \partial_\rho T^{\mu\nu} + (\partial^\rho \epsilon_\rho)T^{\...
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How to change a commutator of SUSY super-charges into an anti-commutator?

I would like to understand an apparently rather simple calculation which checks the closure of the Supersymmetry algebra via the commutator of 2 supersymmetric variations of the type: $$\delta \phi = ...
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Misner, Thorne and Wheeler, Box 9.2 Commutator … doesn't make sense to me

I apologize for the goofy commutator $\left[\left[\_,\_\right]\right]$ notation. MathJax doesn't like my \llbracket \rrbracket notation. And I religiously use $\left[\dots\right]$ for function ...
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What is the mistake in calculating such a commutator? [duplicate]

$B$ is an Hermitian operator in Hilbert space, and $|b\rangle$ is the eigenstate of $B$. We can have $[A, B] = 1$ where A is arbitary operator. Then we can calculate as below: \begin{align} &\...