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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A simple question on creation and annihilation operators
We know that the KG solution for a Spin-0 particle has the following Hamiltonian
$$\hat{H}=∫ d^{3}p\frac{ω_{p}}{2}(\hat{a}_{p}\hat{a}^{\dagger}_{p}+\hat{a}^{\dagger}_{p}\hat{a}_{p})\hspace{2cm}[\hat{a}...
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Strange definition of the fermion number operator in Polchinski
In Polchinski's exposition of the RNS formalism for the superstring (String Theory: Volume II, chapter 10), in page 8, he mentions the worldsheet fermion number operator, which he calls $F$. He then ...
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Commutator of annichilation and creation operators [closed]
Let $\phi$ be a real scalar field and $\psi$ a complex scalar field. Therefore, we can expand $\phi$ in terms of $a_{\boldsymbol{p}}, a^\dagger_{\boldsymbol{p}}$ and $\psi$ in terms of $b_{\boldsymbol{...
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Operators depending on the same independent variable but commuting between them [closed]
As far as I understood in quantum mechanics two operators can commute even though they are not functionally independent, which means that they can depend on the same independent variable.
On the other ...
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Rotor wave fomulation of quantum mechanics and the associated canonical commutation relations of position and momentum operators
I've been trying to determine whether it would be possible to formulate non-relativistic quantum mechanics entirely in the algebra of physical space (APS) by using rotor waves instead of complex-...
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Commutation relation for Dirac field
In "Quantum Field Theory" by Peskin and Schroeder, I couldn't understand the where they investigate the commutation (as opposed to the anticommutation) relation for the Dirac field (pg. 53):
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Commuting operators and their physical interpretation in QM
I'm studying Quantum Mechanics for the first time at the moment and I have a few questions in mind.
So recently, I saw a proof on that if two operators share the same eigenstates is equivalent to the ...
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Do position and spin commute?
I recently learned that position vectors and spin vectors lie in different spaces, and the complete wave is the tensor product of both. I wanted to know that whether we can talk about commutation of ...
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Product of normally ordered exponentials as a normal ordering of product of exponentials
I want to simplify a product of normally ordered exponentials that are in the following form
$$:e^{x(\hat{a}^\dagger+\alpha_x^*)(\hat{a}+\alpha_x)}:\times :e^{y(\hat{a}^\dagger+\alpha_y^*)(\hat{a}+\...
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Computing the density operator commutation relations (Altland & Simons)
I'm trying to work through Altland and Simons' example of interacting fermions in one dimension. It's in chapter 2, page 70 (you can find it here).
They define fermionic operators
$$
a_{sk}^\dagger
$$...
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Matrix element of two operators, where one does not commute with the operators of the basis kets
If I am considering 2 operators A and B and an initial basis $|F,m_F\rangle$ , where F is the total angular momentum operator of the two systems. While assuming that operator A does commute with $\vec ...
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Variable Dependence of Quantum Operators and Commutator Relationships [duplicate]
EDIT: After doing some digging, I am convinced that the approach taken in this paper was simply an incorrect approach to deriving a quantum version of Hamilton's equations (also related to Ehrenfest's ...
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Why Equal-time commutation relation?
Let $\phi(\mathbf{x},t)$ be a field, and $\pi(\mathbf{x},t)$ be the conjugate momentum field.
A standard practice is to apply the equal time commutation relation:
$$[\phi(\mathbf{x_1},t), \pi(\mathbf{...
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Proving commutator [${L}_i, f(r)]=0$
I am trying to prove that the angular momentum component operator ${L}_i$ commutes with any function of ${r}\equiv \sqrt{{\textbf{x}}\cdot{\textbf{x}}}$, i.e.
$$[L_i, f( r)]=0.$$
I first worked out ...
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Showing that time ordering does not matter for the measurement of commuting observables
Suppose I have two observables $R$ and $S$ who are represented by operator $R$ and $S$ which commute (I will hereafter ignore the distinction between observables and the operators representing them), ...
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Time-ordered exponential operator generated by two commuting Hamiltonians
Define a time-dependent Hamiltonian $$H(t) = H_1(t) + H_2(t),\tag{1}$$ where $$[H_1(t), H_2(t)] = 0 ~ \forall t \in [0,T].\tag{2}$$ Is it true that the unitary operator generated by $H(t)$ is a ...
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Why can't commuting operators allow for full state determination?
For the purposes of this question, suppose that the operator $R$ representing the observable $\mathsf{R}$ has nondegenerate eigenspaces.
In discussing state determination (for some given situation, ...
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Why does $[L_z,L_{\pm}]\neq 0$ imply $[J^2,L_z]\neq 0$?
In a lecture about the angular momentum operator, it is stated that the operator $L_z$ commutes with itself, with $L^2$, with all of spin angular momentum operators, but not with $L_{\pm}$, so;
$$[J^2,...
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Angular momentum commutator in QFT/the roles of coordinates
this is my first question on this site, so please correct me if I break any convention.
It arose as part of homework, but I suppose that there is some deeper irritation behind it.
We are supposed to ...
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Proving a commutator identity $[\hat{x}^n,\hat{p}]= i \hbar n \hat{x}^{n-1}$ in quantum mechanics [duplicate]
I was trying to show that:
$$[\hat{x}^n,\hat{p}]= i \hbar n \hat{x}^{n-1}.$$
I know from the definition of a pair of a commutator in QM they act on a wave function like this:
$$[\hat A, \hat B] = \...
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For hydrogen atom wave function, why eigenstate of Hamiltonian isn't only spherical harmonics if $H$ commutes with square of angular momentum? [closed]
I was reading quantum mechanics and I read about CSCO. So, commuting operators should share common eigen states, does that mean same eigen states? because in hydrogen atom wave function, Hamiltonian ...
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What is the Heisenberg-Weyl Algebra?
I did all the courses on Quantum Mechanics and QFT which my faculty offers and up to now no one defined to me what a Heisenberg-Weyl algebra actually is.
This appears in my studies when studying the ...
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What is the value of the commutator $[L_x, S_y]$? [closed]
I would like to know how to calculate the commutator $[L_x, S_y]$ in quantum mechanics, if $L_x$ is the x-component of the angular momentum operator and $S_y$ the y-component of the spin operator.
I ...
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Integrals and Laplace Operator with a Commutator
If I have a Hamiltonian which looks like:
$H=\int_{}^{} \psi(x)^\dagger \nabla^2_{x} \psi(x)d^3x $
and a total number operator like:
$N=\int_{}^{} \psi(x)^\dagger \psi(x)d^3x $
and I want to ...
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Canonical Commutation relations in gravity
The canonical commutation relations in gravity are sometimes written
$$
[\gamma_{ij}(x),\pi^{kl}(y)]=\frac{i\hbar}{2}(\delta_i^k\delta_j^l+\delta_i^l\delta_j^k)\delta^3(x-y),\tag{0}
$$
where $\gamma_{...
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Uniform losses commute with linear optics: how does it work?
In many papers about quantum optics and interferometry, it's assumed or said that "it's well known" that linear optics commutes with uniform losses. In particular if we have a beam splitter ...
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What is $\left<x\right|\hat x\hat p-\hat p\hat x\left|x\right>$? [duplicate]
This is simple question, but I don't know how to do it.
$\left<x\right|\hat x\hat p-\hat p\hat x\left|x\right>=?$,
I can solve it in two ways.
One of them is
$$\left<x\right|\hat x\hat p-\hat ...
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Understanding exception to: Two non-commuting Hermitian operators commute with the hamiltonian implies degenerate energy eigenvalues
For context, I am working through the exercises in Modern Quantum Mechanics by Sakurai and Napolitano Second Ed. I have previously completed (years ago in undergrad) the Griffiths 3rd ed. Introduction ...
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Help with Commutators [closed]
I'm trying to self study quantum mechanics and am having a little trouble manipulating commutators. I get two different answers below, depending on the method I'm using. The second method gives me the ...
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Why does the interaction hamiltonian not commute with itself at different times?
If you have a poincare invariant Hamiltonian $H$, then the Hamiltonian must commute with itself at different times and not explicitly depend on time. If the Hamiltonian $H$ can be written as $H$ = $H_{...
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Uncertainty of an observable when on energy eigenstate
Suppose we are on an energy eigenstate
$$ \varphi_n $$
We can calculate the expected value
$$ <[\hat H , \hat A ] >_{φ_n} $$ and show that it will be equal to zero.
Now, according to Robertson ...
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Uncertainty principle when in position (or momentum) eigenstate [duplicate]
Suppose we want to calculate the expectation value
$$\langle x|[\hat{x},\hat{p}]|x\rangle,$$
where $|x\rangle$ is an position eigenstate, so that $\hat{x}|x\rangle=x|x\rangle$ and $\langle x|\hat{x} = ...
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Commutation relations interacting fields
I am reading Schwartz's "Quantum field theory and the standard model". I have a question on how he derives the Feynman rules for an interacting scalar field from a Lagrangian formalism.
In ...
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Why do the Canonical Commutation Relations hold in Interacting Theories? [duplicate]
The canonical commutation relations for a scalar field theory stating
$$
[\phi(\vec x, t), \partial_t\phi(\vec x', t)] = i \hbar \delta^3(\vec{x} -\vec{x}').\tag{7.4}
$$
Schwartz in Section 7.1 in his ...
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Incompatible observables
Suppose that $A$ and $B$ are observables satisfying $[A, B] \neq 0$, and $|\phi\rangle$ is an eigenstate of $A$. Then is it necessarily the case that $|\phi\rangle$ can be expressed as a superposition ...
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Why is the anticommutator of the uncertainty principle omitted if it serves to increase the accuracy of our "knowledge" of a quantum state?
The generalized uncertainty principle can be derived and shown to be this which is fine and rigorous.
$\langle ( \Delta A )^{2} \rangle \langle ( \Delta B )^{2} \rangle \geq \dfrac{1}{4} \vert \langle ...
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Directly get the rotation meaning of $e^{-i\frac{\theta}{2}\sigma _{\vec{n}}}$ from the commutation relation? [duplicate]
Suppose I have three hermitian operators $\sigma _x,\sigma _y,\sigma _z$ that I don't explicitly say they are Pauli matrices but still use the similar notation $\sigma_j$. They only need to satisfy ...
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Commutator on product space
Let's say we have two Hilbertspaces $H_1$ and $H_2$, with angular momentum operators $j_1$ and $j_2$. On their product space $H_1 \otimes H_2$ we define the total angular momentum operator as:
$J = ...
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Momentum commutation relations in QFT [closed]
Given the following relations for scalar fields at equal time $t$:
$$ [\phi_{\alpha}(x) , \pi_{\beta} (x')] = i \delta_{\alpha \beta} \delta( x - x') $$
$$ [\phi_{\alpha}(x) , \phi_{\beta} (x')] = [\...
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If a Hermitian and Unitary operator commute, do they have a simultanous eigenbasis?
I was learning about Hamiltonian with discrete translational symmetry. We showed that there is a simultaneous eigenbasis of the Hamiltonian $H$ and the discrete translation symmetry being the bloch ...
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Proof of commutation relation in Lattice Vertex Operator Algebra
In DGM [1] on page 548 below Equation 5.4, it is claimed that the operators $\frac{dX^j(z)}{dz}$ and $\frac{dX^k(\zeta)}{d\zeta}$ commute, where
\begin{equation}
X^j(z)=q^j-i p^j \log z+i \sum_{n \neq ...
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Power Series Expansion of Unitary Operators in Weinberg
For a Lie group $T(\theta)$ depending on a finite set of real parameters $\theta^a$,Weinberg (in his QFT book - Equations 2.2.17, 2.2.18, 2.2.19, p54) expands the unitary representations of $T$ in the ...
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Where is the Yennie gauge useful in Gupta-Bleuer formalism (or QED in general)?
Consider the Lagrangian of the Gupta-Bleuer formalism given by:
$$\mathcal{L}
=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}
-\frac{1}{2\xi}(\partial A)^2.$$
I understand the necessity of the gauge fixing term: ...
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Well-definedness of commutation relation in commuting local Hamiltonians
I'm reading the famous paper by Haah: Local stabilizer codes in three dimensions without string logical operators. In the last sentence of the introduction, he wrote:
A logical operator is a Pauli ...
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Do physical quantities that appear in dot product in quantum mechanics always commute?
I am a sophomore learning quantum mechanics, and I got a confusion in my study, I tried to search for answers online but failed to find a satisfying answer. Hence, I wanted to post my question here ...
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Very simple translational invariant Hamiltonian and adaptation for physically sensible hamiltonian
If we consider the Hamiltonian of a system as given by
$$H_0 = \hat{p}.\tag{1}$$
Then the unitary evolution operator will be given by
$$U(t) = \exp\left({-\frac{iH_0t}{\hbar}}\right)= \exp\left({-\...
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Virasoro Algebra - Why does normal ordering change the value of the commutator?
In a classical theory, the functions
$$l_n = \frac{\eta_{\mu\nu}}{2} \sum_{m} a_m^\mu a_{n-m}^\nu$$
form a Virasoro Algebra, i.e. they satisfy
$$\{l_n, l_m\} = i (n-m) \delta_{n+m,0}.$$
When we ...
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Commuting material time derivative and material space derivative
Let's note $x$ the coordinates in the current configuration and $\nabla$ the associated gradient; similarly, let's note $X$ and $\nabla_0$ for the reference configuration. I will also note the ...
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Non-local commutator $[\phi(x), \phi(y)] = i\pi \textrm{sgn}(x-y)$ of 1D chiral boson: why?
It is well-known from bosonization that 1D chiral boson has the commutation relation
$$[\phi(x), \phi(y)] = i\pi \mathrm{sgn}(x-y). \tag{1}$$
This gives the correct anticommutation relation for the ...
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Commuting observables and an eigenstate [closed]
Consider the observables $A$ and $B$ with $[A, B]=0$. Suppose that the state of the system satisfies the equation $A|a_1\rangle=a_1 |a_1\rangle$. After a measurement of the observable $A$, in what ...