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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35 views

Quantum fields as differential operators

As I understand it, there was initially two formalism for QM, before Dirac reunites them both with his famous braket notation: Schrödinger's formalism that involved differential operators acting on ...
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Show $[\gamma^\mu,\eta^{\nu\lambda}I]=0$ for Dirac matrices

I am trying to show the following commutation relation for the Dirac matrices $\gamma^\mu$ and the metric $\eta^{\nu\lambda}$: $$2[\gamma^\mu,\eta^{\nu\lambda}I]=0$$ where $I$ is the 4x4 identity ...
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Mathematical problem in 1D bosonization

I am reading the following article on bosonization : https://arxiv.org/abs/cond-mat/9805275 and I encountered the following set of equalities. $$\begin{align} [\phi_\eta (x),\partial_{x'}\phi_{\eta'}(...
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What's the commutator of $|\mathbf{\hat{x}}|$ and $|\mathbf{\hat{p}}|$?

Straight to the point: what's the result of the commutator of the magnitude of the position and the momentum operators and how can I approach it, i.e., $[|\mathbf{\hat{x}}|,|\mathbf{\hat{p}}|]=$ ? My ...
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Solving an equation that contains commutators

I am reading Luttinger's theory of thermal transport coefficients (1964). It has the following equation $$[H,f] -isf = [\rho,F].\tag{1.7}$$ $H$ and $\rho$ is Hamiltonian and the density matrix of the ...
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Proof of commutation relation for lattice QFT

How do you prove the following commutation relation for the lattice QFT \begin{equation} [\phi(X),\Pi(y)]=\text{i}a^{-d}\delta_{x,y}\mathbb{I}? \end{equation}
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Quantization and Commutation Relations

Why do we use commutation relations when quantizing any system? In the case of developing quantum mechanics from classical mechanics, we write the hamiltonian and then quantize it by having the ...
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Derivation of Poisson bracket and commutator of position and conserve charge [closed]

How can I prove these two relations? Assuming $\mathbf{D}$ is defined as $$\mathbf{D}=\sum\frac{\partial\mathcal{L}}{\partial\dot{x}_i}\delta x_i-\mathcal{L}=\mathcal{H}t-\frac{1}{2}\mathbf{p}\cdot\...
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Momentum commutation for boson field

Given a boson field described by $\psi(\vec{x})$, conserved momentum from the Lagrangian (which isn't relevant here) is $\vec{P} = \frac{\hbar}{2i} \int d^3 x \left( \psi^\dagger \nabla \psi - \nabla \...
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Can spin operator expectation value be time-independent while commutator with Hamiltonian is non-zero?

Considering the following (magnetic field) hamiltonian: $\hat{H}=-\gamma B_z \hat{S}_z$ ($\gamma$ is a constant). Suppose an electron is in an eigenstate of $S_x$, and we ask ourselves the question ...
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Complicated bosonic commutations

Given that $a$ and $a^\dagger$ are bosonic annihilation and creation operators (in the language of second quantisation); are there any simpler ways to calculate the commutators of arbitrary products ...
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What is commutation of $J^2$ with $L_z$?

What is $[J^2,L_z]$ and $[J^2,S_z]$ such that $[J^2,J_z]=0$?
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Combination of 'transposition operators': do they commute?

Suppose I have the Hamiltonian defined as $H =\hat A\hat B+\hat C\hat D$, where the operator $A,B,C and D$ are square matrices. If I label the positions of $A,B,C,D$ as $1,2,3,4$. Now I want to apply ...
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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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Topological Descent Equation

Assume that we have a cohomological field theory, with an odd symmetry generated by an odd operator $Q$ and an exact energy momentum tensor $T_{\mu\nu}=[Q,G_{\mu\nu}]$. Then by integrating over an ...
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Shared eigenbasis of commuting Operators

Suppose I have two Hamiltonian pieces $H_1$ and $H_2$ such that $[H_1,H_2]=0$. Then we know that the two pieces have shared eigenbasis. Assume both $H_1$ and $H_2$ have eigenvalues 2 and -2. Let $|\...
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Velocity operators in quantum mechanics

According to the Heisenberg equation of motion, the velocity operator is given by $$\hat{v}=\frac{d \hat{r}}{dt} = \frac{1}{i\hbar}[\hat{r},\hat{H}].$$ Question 1: How can I find the velocity operator ...
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Commutation relation(s) for integrated operators

Suppose I have two operators $\hat{A}$ and $\hat{B}$, where $\hat{A}=\hat{a_1}+\hat{a_2}+...+\hat{a_m}$, and $\hat{B}=\hat{b_1}+\hat{b_2}+...+\hat{b_n}$. Is there a necessary and sufficient condition ...
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Proof of the canonical commutator relationship from $\hat{p}=-i\hbar \nabla$

Given that $\hat{r} \psi = r\psi$ where $r$ is the position of a quantum particle, and where $\hat{p}=-i\hbar \nabla$, the notes I have simply state that $$[\hat{r}_i, \hat{p}_j] = i\hbar \delta_{ij}$$...
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Transformation operator in Sequence of linear QND measurements

I am following the book Braginsky and Khalili. Consider a measurement scheme where we connect a object to be measured to another quantum system which is then measured by classical devices.(Example: ...
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Diagonal and block diagonal matrix

What's the significance of diagonal and block-diagonal matrices in quantum mechanics? For instance, let $S$ be the symmetry operator, since $[S,H]=0$, they have a shared eigenbasis. If I use a basis ...
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Commutator of $p$ and $x^n$ [closed]

I found this calculation about the commutator $[p, x^n]$: In lines 4 to 5, it seems like they take an $x$ from $px$ and move it to the left to make $x \cdot x = x^2$. Is that legal? Wouldn't that ...
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Commutators with Hamiltonian of the form $H=\frac{p^2}{2m} + V(x)$ [closed]

Consider a one-dimensional problem with a Hamiltonian \begin{equation*} H=\frac{p^2}{2m} + V(x) \end{equation*} where $x$ and $p$ are the position and momentum operators, $m$ is the mass and $V(x)$ ...
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Time ordering operator if commutator is $c$-number function

I have a question concerning the time ordering operator. Let's suppose we have a time evolution generated by some Hamiltonian $H(t)$ given by $$ U(t)=T_\leftarrow\exp\left(-\mathrm{i}\int_0^t\mathrm{d}...
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Units of $\hat{a}$ and $\hat{a}^\dagger$ in discrete vs continuous $k$ and normalization

Consider the quantization of the electromagnetic field. In the discrete case, given in Wikipedia, the operators $\hat{a}$ and $\hat{a}^\dagger$ are dimensionless $$[\hat{a}]=[\hat{a}^\dagger] = 1,$$ ...
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Time-evolution operator written through a commutator [duplicate]

I found this expression for the time-evolution operator: $$\begin{split} U(t) & = T_{\leftarrow}\exp\left[-i\int_0^t ds H(s)\right] \\ &= \exp\left[-\frac{1}{2}\int_0^t ds\int_0^t ds' [H(s),H(...
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Use the commutation relation to show that the conjugate momentum acts on eigenstates of $\hat{\Phi}$ as $ - i \delta / \delta\phi_a(\mathbf{x})$

This is part (b) of Schwartz's Problem 14.3 in his Quantum Field Theory and the Standard Model textbook. Suppose that we have a real scalar field operator $\hat{\Phi}(x^0,\mathbf{x})$ with conjugate ...
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How do fermionic operators transform?

In quantum mechanics, if we have an operator $\Omega$, then under the transformation $T$, with infinitesimal generator $G$ (i.e. $T(\epsilon)=1-i\epsilon G + \ldots$), then operator transforms as $$\...
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Order of measurement of two observables

Let $\hat A$ and $\hat B$ be two operators representing the observables $A$ and $B$, and let $\Psi(t)$ be the state of a quantum system. Let's suppose that we measure $A$ at $t_0$ and just after that ...
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Common eigenfunctions of commuting operators: case of degeneracy

As proved in the answer to this post, if the operators $\hat A$ and $\hat B$ commute, then they have the same eigenstates. Let $$\hat A\psi_{A_i}=A_i\psi_{A_i}\qquad \Rightarrow\qquad \hat B\hat A\...
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Operators as complex numbers

I recently came across a paper where the following manipulation had been done after writing considering Heisenberg Operators as complex numbers $$\delta a^\dagger*a_s=a_s*\delta a^\dagger$$ (where $a$...
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Why is radial ordering necessary?

Suppose I have some conserved charge in a 2 dimensional CFT $$Q(|z|)=\int_{w=|z|}\text{d}w\,T(w).\tag{1}$$ The infinitesimal transformation induced on a field $\phi$ at $z$ is then $$[Q(|z|),\phi(z)]=\...
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How do disentangling and reordering of exponential operators work?

I have seen in several sources that by invoking Lie groups, $$e^{\alpha_1 g_1+\alpha_2 g_2 + \dots} = e^{\beta_1 g_1}e^{\beta_2 g_2}\dots $$ where $g_i$ are elements of a Lie algbera. For example, ...
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What is the “secret ” behind canonical quantization?

The way (and perhaps most students around the world) I was taught QM is very weird. There is no intuitive explanations or understanding. Instead we were given a recipe on how to quantize a classical ...
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Why is it OK to commute a quantum operator with the cross product?

I am going through lecture notes relating to the parity operator $\mathcal{P}$ My confusion relates to the derivation of the symmetry transformation of the orbital angular momentum $$\mathcal{P} \vec{...
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What is the meaning of the commutation relations (spin $1/2$ particles)?

I've often seen spin 1/2 commutation rules as a principle valid for every angular momentum. In some text books there is a derivation from symmetries principles. My question is, if I have a spin $1/2$ ...
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Why do we have $\hbar$ in the commutation relation?

Let's think of the Planck constant as of the slope of the electromagnetic field dispersion relation, $E=\hbar \omega$. Planck constant is not independent of the electron charge, both can be rescaled ...
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Question on eigenstates of hermitian operators that fulfill canonical commutation relation [duplicate]

Two operators, in a suitable basis with matrix representations $A$ and $B$ have the following commutator $[A, B] = AB − BA = kI$, where $k$ is a non-zero complex number and $I$ is the $n × n$ identity ...
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Commutator of field operator with function of number operator [closed]

Consider the commutator $$\left[\hat{a},\sqrt{1-\hat{a}^\dagger\hat{a}}\right].$$ Because $\hat{a}^\dagger\hat{a}=\hat{n}$ is the familiar number operator, and $[\hat{a},\hat{n}]=\hat{a}$, I would ...
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What's wrong with this “proof” that QFT violates causality?

In An Introduction to Quantum Field Theory, by Peskin and Schroeder, when discussing the quantized real Klein-Gordon field ($\phi=\phi^\dagger$), they show the commutator $[\phi(x),\phi(y)]$ vanishes ...
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Second quantization and bosons in Tomonaga's “The story of spin”

TL; DR: According to a book by Tomonaga in a chapter introducing second quantization, the formula $$e^{\pm i \Theta/\hbar} \,\psi(N) = \psi(N\pm 1)$$ is supposed to prove that the second quantization ...
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(Anti)commutators at different times

Why does the commutator of two operators evaluated at different times vanish? Take for instance a fermonic field $\psi_\sigma (\vec{x},t)$, which satisfies the well known anti-commutation relations ...
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When studying the hydrogen atom, why do we seek simultaneous eigenfunctions of $\hat{L}^2$, $\hat{L}_z$, and $\hat{H}$?

When solving the Schrödinger equation for the hydrogen atom, textbooks invariably work in a more constraint situation, whereby not only an eigenfunction for the Hamiltonian operator $\hat{H}$ is ...
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Commutator of tetrad and Lorentz generators

While reading "Four lectures on Poincaré gauge field theory" (available at RG) the authors present a relationship between a tetrad $e^i_{\;\gamma}$ (with Latin indices coordinates, Greek ...
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Physics that calls for deeply nested Lie/Poisson brackets

I've been scouring physics for non-associative situations, particularly where study of quasigroups and loops might come in handy (they always seem to be left out). The poisson and lie brackets form a ...
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1answer
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Creation and annihilation from basic position/momentum operators [closed]

I was reading these lecture notes and they show how if you start with two operators $p$ and $q$ such that $[q,p]=i$, you can define $a:=\frac{1}{\sqrt{2}}(q+ip)$ and $a^\dagger:=\frac{1}{\sqrt{2}}(q-...
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Baker–Campbell–Hausdorff and Gaussian density operators for fermions

I'm trying to understand a passage from the paper "Gaussian operator bases for correlated fermions", J. F. Corney and P. D. Drummond (https://arxiv.org/abs/quant-ph/0511007), specifically ...
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Quantization of an $\mathcal{c}$-algebra

I don't know if this is a valid question to ask, but I am wondering about the following: We are given a set of $\mathcal{c}$-number Lie-Brackets $$ [q_i,q_j] = 0= [p_i,p_j] \\ [q_i,p_j] = c_{ij}, $$ ...
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1answer
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Addition of a constant to the operator due to quantization

Groenewold in his book On the Principles of Elementary Quantum Mechanics (1946, Springer Netherlands) page 45, maps the canonical momentum $p^2$ in the classical phase space to a general canonical ...
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For two commuting operators $A$ and $B$ and in absence of any degeneracy, is every eigenstate of $A$ is also an eigenstate of $B$ and vice-versa?

Two commuting operators $\hat{A}$ and $\hat{B}$ always share a complete set of common eigenfunctions. However, in the presence of degeneracy, every eigenstate of $\hat{A}$ need not an eigenstate of $\...

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