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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Is it true that we can't measure position and momentum together?

The uncertainty principle states that there always will be mean variance if we measure position or momentum. It does not state that the measurement is wrong. It only states that there always will be a ...
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Is The Seiberg-Witten Map Unique?

From my understanding the Seiberg-Witten map is a way to convert a non-commutative field theory into a commutative field theory. For example for the commutative relation between positions $[x, y]=i \...
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Commuting operators' eigenfunctions [closed]

So, one of my homework problems reads the following Let $A$ and $B$ be commuting operators and $| \psi_i \rangle$ denote the eigenfunctions of $A$. Show that $\langle \psi_i |B| \psi_j \rangle=\...
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Commuting the time evolution operator [closed]

Given the time evolution operator $U(t, t_0)$, I don't understand why it is true that for a time-independent operator Q, $$[Q, U(t, t_{0})] = 0 \Leftrightarrow [Q, H(t)] = 0 $$ where H is the ...
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Validity of Canonical Quantization

I was studying about what does it mean canonical quantization treatment. But now I have the next question. Why if we establish canonical the commutation relations $$\left[q,p\right]=i\hbar,\quad \left[...
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A question regarding the commutators of operators

Suppose we have got a triple of observables $A,B$ and $C$. Suppose furthermore, that $[A,B]=0$ and $[B,C]=0$ but $[A,C]\neq 0$ . Suppose, also now we do a measurement of $A$ then accordingly we would ...
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Can we measure $x$ and $p_x$ simultaneously by measuring $p_y$ and $y$ as well?

Suppose our plan is to measure experimentally the position $(x,y)$ in the plane and the momentum $(p_{x}, p_{y})$ of a quantum particle. Assuming the canonical commutation relation between $x$ and $p_{...
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68 views

A question involving position and momentum operators

Assume the Hamiltonian of a quantum particle to be independent of time and be of the form $H=\frac{1}{2m}(\hat{p}^{2}_{x}+\hat{p}^{2}_{y})+V(x,y)$. Define a new operator $\hat{p}=\hat{p}_{x}-\hat{p}_{...
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How is the uncertainty principle related to the non-commutativity of the multiplication of operators in quantum mechanics? [duplicate]

The way I understand the uncertainty principle is that it's not even really about quantum mechanics specifically -- it's just a property of waves. e.g. A periodic wave doesn't even have a well ...
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Does following kind of commutator arises anywhere in non-commutative geometry of spacetime?

Pauli matrices satisfy following relation $$[\sigma_i,\sigma_j]=2i\epsilon_{ijk}\sigma_k$$ While looking through models of noncommutative geometry of spacetime I have seen people defining following ...
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Is causality violated in QFT?

I'm studying QFT, and Peskin is his book takes a couple of paragraphs to talk about causality in QFT, using the Klein-Gordon field as an example. The book says on p. 28: To really discuss causality, ...
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Why does $[L_i,\textbf{L}^2]=0$ and $[L_i,H]=0$ imply that $[\textbf{L}^2,H]=0$?

For a particle in a central potential, the orbital angular momentum magnitude operator $\textbf{L}^2$ commutes with the Hamiltonian operator $H$, i.e. $$[\textbf{L}^2,H]=0.$$ I read that one way to ...
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215 views

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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How to show that $[L_i, v_j]=i\hbar\sum_k \epsilon_{ijk}v_k$ for any vector $\textbf{v}$ constructed from $\textbf{x}$ and/or $\nabla$?

In Weinberg's Lectures on Quantum Mechanics (pg 31), he said that the commutator relation $$[L_i, v_j]=i\hbar\sum_k \epsilon_{ijk}v_k$$ is true for any vector $\textbf{v}$ constructed from $\textbf{x}...
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Commutator question: constraint on operator coefficients

I'm trying to understand for which constants $c_{ijkl}$ the two operators $\sum_{n\in\mathbb{Z}}a^\dagger_na_n$ and $\sum_{ijkl\in\mathbb{Z}}c_{ijkl}(a^\dagger_ia_ja_ka_l + a^\dagger_ja^\dagger_ka^\...
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Are equal-time commutators well-defined in QFT?

I'm working through some QFT course notes and I just want to check that my understanding of equal-time commutators is correct. I don't have much to go on -- the notes rather cryptically insert the ...
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Expectation Value of a Commutator [closed]

Lets say we have a quantum system. Its stationary states are described by the energy eigenfunction: $$H|\psi_E\rangle = E|\psi_E\rangle,\qquad H =\frac{1}{2m}P_x^2 + V, \qquad H=H^\dagger$$ I know how ...
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Exponentiated Commutation Relations

A proof about the exponentiated commutation relations is mentioned In this book page 285: The exponentiated momentum operators satisfy: $(e^{itP_j}\psi)(\textbf{x})=\psi(\textbf{x}+t\hbar \textbf{e}...
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Why Locality meant jointly observable?

Tobias Osborne's lecture around 20:00, he mentioned that the ideal of "Locality" could be expressed as such If $x-y$ were space-like, then for all observable $A_{j,x}$ and $A_{k,y}$ were ...
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Relation between $\vec{L}^2$, $\vec{r}$ and $\vec{p}$ [closed]

I'm trying to prove equation (1.35) $$\begin{align} (\mathbf{a}\times\mathbf{b})^2 &= \mathbf{a}^2\mathbf{b}^2 - (\mathbf{a}\cdot\mathbf{b})^2 \\ &− a_j[a_j,b_k]b_k + a_j[a_k,b_k]b_j − a_j[...
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Expanding commutator in terms of an eigenbasis

I'm trying to get some extra intuition into commutators by considering an eigenbasis expansion (where this is possible). However, when I try to expand the operators into their own eigenbasis, and do a ...
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1answer
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How to solve differential equation involving commutator and anti-commutator?

In one of my exercise, I got following differential equation for density matrix $\rho$, $$ \frac{d\rho}{dt}=-i[H_1,\rho]+\{H_2,\rho\} $$ where $H_1$ and $H_2$ are the Hermitian Hamiltonian, and $[.,.]$...
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How to verify the Klein-Gordon field commutation relations, Peskin and Schroeder Equation (2.30)

I am trying to verify the commutation relation given in Peskin and Schroeder. In particular, I don't know how to go between these two lines: $$[\phi(\textbf{x}), \pi(\textbf{x}')] = \int \frac{d^3p ...
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45 views

Effect of commuting observables on the probability of measuring a certain value [closed]

Say you can measure $3$ observables $(A, B, C)$ and you do the measurements in two different ways. $\newcommand{\ket}[1]{|#1\rangle} \newcommand{\bra}[1]{\langle#1|} \newcommand{\braket}[2]{\langle#1|#...
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Does $[A , B] \not = 0 $ necessarily mean $\Delta A \Delta B \ge k$ where $k \not = 0$?

For example : $[\hat{x},\hat{p}] = i \hbar \hat{I}$ and $\Delta x \Delta p \ge \hbar/2$ but in case of number states $|n \rangle $ $$[\hat{C},\hat{n}] = i \hat{S}\\ \Delta C \Delta n \ge 0 $$ where ...
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Explaining Kronecker delta from operator [closed]

I'm trying to prove the relation for angular momentum operator. $[\hat{L}_i,\hat{r}_j] = i\hbar \sum_{k} \epsilon_{ijk} \hat{r}_k $ $ [\hat{L}_i,\hat{r}_l]$ = $ \sum_{jk} \epsilon_{ijk} [\hat{r}_j\...
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Is Planck's Constant Really a Constant?

I am going through Groenewold's theorem and in his book: On The Principles of Elementary Quantum Mechanics, page 8, eq. 1.30: $$[\mathbf{p}, \mathbf{q}]=1\left(\text { i.e. } \mathbf{p q}-\mathbf{...
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1answer
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Unexpected divergence in expectation value

I'm currently trying to calculate the expectation value \begin{equation} \langle\psi(p,s)|\bar{\psi}(x)\Gamma_\rho \psi(x)|\psi(p,s)\rangle, \end{equation} where $\Gamma_\rho$ is understood to be ...
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219 views

Is symmetrization $xp-px$ required for commutation $[H,x]=0$?

Given a Quantum Hamiltonian: $$\hat{H}=ax^2+bp^2$$ It does not commute with either $x$ or $p$. Suppose we have a Hamiltonian :$$H = k \hat{p}\hat{x}$$ why do we need it to be: $$H = k (\hat{p}\hat{x} -...
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What does the comma mean in this commutation rule between quantum operators?

The Theorem about quantum operators commutation relation says: Consider pairs $(U, V )$ of unitary representations on a Hilbert space $H$, satisfying the commutation rule: $$U(x) V(y)=\exp (i \...
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Is the commutator relation $[\hat{x}, \hat{p}_x]=i\hbar 1\!\!1$ an *assumption* in the quantum theory?

This question is somewhat related to (but not by any means the same as) the question I asked recently. In his Lectures on Quantum Theory, Isham essentially says (reference given below) that if an ...
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275 views

How do you know if a operator commutes with the hamiltonian?

In the question there is a central potential within a Hamiltonian, and I have to find the appropriate quantum numbers. They say that $j, m, s$, $\ell$ are the appropriate quantum numbers to describe ...
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Why does Peskin and Schroeder move normal ordering move outside a commutator?

The equation trying to prove that Wick's theorem by induction in P&S on page 90 implies that normal ordering can be moved outside a commutator (at least with a positive frequency field), which I ...
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Three operators commuting with each other

It is well known that if two operators commute, the it is possible to find common eigenfucntions for them. What if we have 3 operators that commute with each other? Will we find common eigenfunctions ...
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Is every pair of conjugate variables associated with a Fourier transform?

For example, in quantum mechanics, the commutator of the position and momentum is $$[\hat{P_i} ;\hat{Q_j} ] =i\hbar\delta_{ij}\neq 0, i\neq j$$ I know that the position space representation of the ...
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What can be said about the commutator of an operator with itself at different times?

In general for some smooth and bounded $\hat{V}$ $$ \left[\hat{V}(t_1), \hat{V}(t_2) \right] \neq 0 \text{ if } t_1 \neq t_2 $$ But what more can be said about commutators of this type? I am ...
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Can current and voltage be linked by an uncertainty relation when electrons tunnel through a barrier?

Quantum tunneling has been shown to be linked to uncertainty relations for some observables involved in the system. For instance, if we consider electrons tunneling through a potential barrier it can ...
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Degeneracy and Complete Sets of Commuting Observables

I want to understand how the degeneracy of an operator is related to the existence of a complete set of commuting operators that includes it. I know that if a set of operators commute, they possess a ...
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1answer
93 views

Does the canonical commutation relation give a unique solution for the momentum operator? [duplicate]

So lets say we are in a 1d system and in the position basis just for simplicity. The CCR is: $$ [x,p]=i $$ and the momentum operator is $-i\partial_x$. Is this solution unique or are there other ...
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The commutation relation between the square components of angular momentum

So my question is as follows. I was reading about Angular Momentum from Griffiths, Introduction to Quantum Mechanics and it is a well known fact that the components of angular momentum do not commute- ...
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Commutation relations/sturcture constants for Lorentz algebra

I am trying to compute the curvature for a gauge theory based on the pure (local) Lorentz group. The final hurdle is working with monstrous structure constants. My objective is to show that \begin{...
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1answer
54 views

Commutation relation for deviation of two hermitian operators

On page 35, right after equation 1.4.60, Sakurai says that the commutator $$[\bigtriangleup A, \bigtriangleup B] = [A,B]$$ where $\bigtriangleup A = A - \langle A \rangle$, and $A$ is a hermitian ...
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1answer
238 views

Eigenfunctions of compatible observables that are not shared

I'm using D.J. Griffiths's Introduction to Quantum Mechanics (3rd. ed), reading about the angular momentum operators $\mathbf L=(L_x,L_y,L_z)$ and $L^2$ in chapter 4. The author discusses ...
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1answer
87 views

Virasoro modes commutation calculation

Given the commutation relations: $$ [\alpha_m,\alpha_n]=m\delta_{m+n,0} $$ and $$ L_m=\frac{1}{2}\sum_\rho\alpha_{m+\rho}\alpha_{-\rho} $$ I am trying to calculate the commutator between $L_m$ and $...
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4answers
883 views

Are all operators really associative?

Operators are associative as seen here. But when we try to calculate $[\hat{x}, \hat{p}]$ for example, we use a test function and apply $\hat{p}$ to both $\hat{x}$ and the function, instead of ...
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86 views

Product of non-commuting operators

I want to expand the product: $$\left(\hat{A}_{1}+\hat{A}_{2}\right)\left(\hat{B}_{1}+\hat{B}_{2}\right)$$ $\hat{A}_1$ and $\hat{B}_1$ are operators both working on the same particle, and do not ...
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64 views

Commutator of net position and net momentum

It is well known that $$[\hat{x},\hat{p_x}] = [\hat{y}, \hat{p_y}] = [\hat{z}, \hat{p_z}] = i\hbar$$ But what if instead we wanted to know the commutator of the net displacement $\hat{r} = \sqrt{\...
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1answer
168 views

How does preservation of the Lorentz algebra demonstrate Lorentz invariance of a QFT?

In his book "Quantum Field Theory of Point Particles and Strings", Brian Hatfield makes the following claim (on p. 46) after canonically quantizing the free scalar field theory: We started with a ...
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641 views

Why is this sum the delta function?

I am reading the first chapter from Fetter and Walecka, which is on second quantization, and I have understood everything up to this section. It seems to me that these field operators essentially ...
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145 views

For $[A,B]=0$, if an eigenfunction of $A$ not an eigenfunction of $B$, does that imply degeneracy of one operator?

When two operators $A$ and $B$ commute, there can be functions which are eigenfunctions of $A$ but not that of $B$. For example, in case of the one-dimensional harmonic oscillator, any linear ...

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