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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Covariant derivative with respect to commutator

I have some confusion with the notion of $\nabla_{[A, B]}\bf{v}$, that expression, with a commutator of vector fields as the subindex of the connection appears for instance in the definition of the ...
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107 views

Quantum mechanics, Fourier transformation

Why do we use $p=-i\hbar\frac{\partial}{\partial x}$ in quantum physics? (I know the reason for $i\hbar$, quantization). Is this right to say we can't measure velocity and position of electrons at the ...
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Explicit form of Klein factors in Giamarchi

In Giamarchi, Quantum Physics in One Dimension, Appendix B, I don't understand how he did his last step in equation B.8, as shown below. If anyone has gone over the derivation, I would really ...
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Proof of Heisenberg's uncertainty principle for operators [duplicate]

Why is Heisenberg's uncertainty principle for the position and momentum operators in the Cartesian coordinates system defined as $$\sigma_{x}^{2} \sigma_{p_{x}}^{2} \geqslant\left(\frac{1}{2 i}\...
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How does one obtain $\hbar$ as $\frac{h}{2\pi}$?

I'm reading Dirac's Principles of Quantum Mechanics. He defines $\hbar$ to be the real number satisfying the following relation $$ uv - vu = i\hbar[u,v]$$ where $u$ and $v$ are dynamical variables, ...
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Why does equal commutator relation imply equal operator?

In 1d bosonization, Giamarchi (Quantum Physics in One Dimension) Chap 2, shows that fermionic Hamiltonian $$H_f=\sum_k k(R_k^* R_k -L_k L_k)$$ is equal to the bosonic representation $$H_b = \sum_k |...
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38 views

Commutation relations in Gupta-Bleuler quantization

Quantization of the free electro-magnetic field has essential differences in comparison to quantization of say scalar or massive vector fields. In fact there are different approches to it. One of ...
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Why is the Schrödinger field an annihilation operator?

The relativistic scalar field operator is not a ladder operator. Its commutation relations are $$\begin{align} \left[\hat{\phi}\left(\vec{x}\right), \hat{\phi}\left(\vec{y}\right)\right] = \...
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Baker-Hausdorff for normal ordering exponential

Let $A=A^+ +A^-$ where $A^+,A^-$ denote the creation and annihilation portion of the field. Then in Eduardo Fradkin, Field Theories of Condensed Matter Physics, equation (5.284), it states that $$ :e^...
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Show that when angular momentum $L_x$ and $L_y$ commute with operator $G$, then $L_z$ also commutes with $G$

I want to prove that if Angular momentum $L_x$ and $L_y$ commute with an operator $G$, angular momentum $L_z$ also commutes with $G$. if $[L_x , G] = [L_y, G] = 0$ then $[L_z , G] = 0$ I know that $...
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Calculating the Commutation Relation of the Generators of $SO(n)$ [duplicate]

I'm working through problems in the book Einstein Gravity in a Nutshell by Zee, and I'm stuck on one of the harder problems. The problem is Calculate $[J_{(mn)}, J_{(pq)}]$. We are given that $[J_{(...
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Ehrenfest theorem and correlation among observables at the quantum scale

I am studying quantum mechanics and I encountered the famous Ehrenfest Theorem, which states that given an observable $A$, its expectation value time evolution is governed by $\partial_t\langle A\...
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How to differentiate exponentials of operators?

Suppose we have $$e^{At}e^{Bt}=F(t),$$ where $$A, B$$ - operators that do not commute. Now I need to take the derivative $$dF(t)/dt.$$ In which order do I write the operators? $$dF(t)/dt = Ae^{At}e^{...
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Virasoro algebra commutation (part 2)

This was a sub-question in my previous post that I ask separately now. In Introduction to Conformal Field Theory by Blumenhagen and Plauschinn (springer link) the Virasoro algebra is introduced the ...
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Virasoro Algebra commutation

In Introduction to Conformal Field Theory by Blumenhagen and Plauschinn (springer link) the Virasoro algebra is introduced the central extension of the Witt algebra. They give the central extension $$\...
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Uncertainity relation of Kinetic energy with position

In R. Shankar's Principle of quantum mechanics book in the problem Now $$\Delta T = \frac{-\hbar^2}{2m} \Delta( p^2)$$ And I don't arrive anywhere using this, but I also know that $\Delta A \...
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Commutation relation Lorentz Algebra

Related question, which I don't understand either. I think is easier to get the Lorentz group algebra as defined by Maggiore, $$ [J^{\mu\nu},J^{\rho\sigma}] = i(\eta^{\nu\rho}J^{\mu\sigma} - \eta^{\...
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Vanishing correlation function

Mirror Symmetry p. 206, Eq. 10.192. I have an operator $\mathcal{O}$ that commutes with my supercharge $\overline{Q}_+ $, $\left[\overline{Q}_+, \mathcal{O} \right]=0$. Why does the correlation ...
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Commutation of alpha dirac matrix

I want to calculate the commutation of $[\hat{x},\vec{\alpha}\;\vec{p}]$. This boils down to $$[\hat{x},\vec{\alpha}\;\vec{p}] = i\hbar\hat{\alpha_x}+\left[\hat{x},\hat{\alpha_x}\right]\hat{p_x} +\...
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Generators of conformal transformations change of basis

I recently started going through Introduction to Conformal Field Theory by Blumenhagen and Plauschinn ( springer link ). On page 11, they glue together the generators of conformal transformations as ...
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Normal ordering of operators: Commutator or no commutator [duplicate]

My exercise for a quantum optics course tells me to "find a normal ordering" for an Operator $\hat{O}(\hat{a},\hat{a}^\dagger)$, which is given as a (rather complicated) string of $\hat{a}$'s and $\...
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Heisenberg Picture from $\left[f(A), B\right] = \left[A, B \right]\frac{\partial f}{\partial A}$

I have a question about the equation below: $$\left[f(A), B\right] = \left[A, B \right]\frac{\partial f}{\partial A}$$ Is this equation valid in the Schrödinger picture, Heisenberg picture, or in ...
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Commutation relations, complex scalar field

Why for the complex scalar field $$ \hat\phi = \int \frac{d^3p}{(2\pi)^{3/2}(2E_{\vec{p}})^{1/2}}\left(\hat{a}_{\vec{p}}e^{-p \cdot x} + \hat{b}_{\vec{p}}^\dagger e^{p \cdot x}\right), $$ the ...
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Question about commutators acting on wavefunctions

Consider a commutator acting on a 1D wavefunction: $$[\frac{\hbar}{i} \frac{d}{dx},x]\psi(x)=(\frac{\hbar}{i} \frac{d}{dx}x-x\frac{\hbar}{i} \frac{d}{dx})\psi(x).$$ Now does this mean $\frac{\hbar}{...
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Why symmetry transformations have to commute with Hamiltonian?

Let us consider a unitary or antiunitary operator $\hat{U}$, that associates with each quantum state $| \psi \rangle$ another state $\hat{U} | \psi \rangle$. I have read that to $\hat{U}$ be a ...
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General commutation question

If I have three general observables, $\hat{C}$, $\hat{H}$, and $\hat{L}$, and the commutation relation between $\hat{C}$ and $\hat{H}$ is given by, $$ [\hat{C}, \hat{H}] = \hbar \hat{L} $$ At the ...
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QFT Complex scalar field and commutators

The conserved charge is $$Q=i\int\ d^3x(\phi\pi-\phi^\dagger\pi^\dagger)$$ Expressing this in terms of creation annihilation operators gives $$Q=i\int d^3 x \frac{d^3p d^3k}{(2\pi)^3(2\pi)^3}\frac{i\...
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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 ...
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Proof for $\langle i[A,B]\rangle$ [closed]

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'...
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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.
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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 ...
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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 ...
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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\...
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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,...
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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 \, ...
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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?
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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}...
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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 = ...
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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}^...
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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 ...
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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?
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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?
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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 $...
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86 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}...
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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 ...
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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 $\hat{f}$...
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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\...
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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]+\...
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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? ...
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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 ...