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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Placement of indices in canonical commutation relations of coordinates and conjugate momenta as well as fields and conjugate momenta

The canonical commutation relations between generalised coordinates $q_a$ and their conjugate momenta $p^a$ are given by $[q_a,q_b]=[p^a,p^b]=0$ $[q_a,p^b]=i\delta^b_a$. Furthermore, the canonical ...
4
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4answers
388 views

Is commutation relation an equivalence relation?

I'm now learning quantum mechanics with Liboff. In the book it deals with "a compete set of mutually compatible observables" in order to make a state maximally informative. How can one find such set? ...
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2answers
58 views

How to recognize a Complete Set of Commuting Operators (CSCO)

A question about 'completeness'. These two operators are commuting, but I want to know more about their completeness. How do you know if {H}, {B}, {H,B} and/or {$H^2$,B} are forming (a) Complete ...
5
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2answers
277 views

Tricky operator identity: $[L^2,[L^2,\vec{r}]]=2 \hbar ^2 \{ L^2, \vec{r}\}$?

This operator identity showed up in a course I was taking, and it was given without proof. $$[L^2,[L^2,\vec{r}]]=2 \hbar ^2 \{ L^2, \vec{r}\}$$ The curly brackets denote the anticommutator, $AB+BA$. ...
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1answer
29 views

Tensor products of Hilberts spaces: definition of outer products and commutators

Suppose one has two single-particle Hilbert spaces $\mathcal{H}_{A}$ and $\mathcal{H}_{B}$ and consider the tensor product of these such that $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ is a two-particle ...
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0answers
53 views

Elegant method to show $[L^2,[L^2,\vec{r}\,]\,] = 2\hbar^2\{L^2, \vec{r}\}.$ [duplicate]

Show that $[L^2,[L^2,\vec{r}\,]\,] = 2\hbar^2\{L^2, \vec{r}\},$ where $\vec{r} = x\, {\hat x} + y\, {\hat y} + z\, {\hat z}.$ "Edit: $\{A,B\} = AB + BA$ is the anti-commutator." I am able to solve ...
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1answer
46 views

Deriving eigen values of $\hat{N}$

So let's say we have an operator $\hat{a}$ (ladder operator), where $\left[\hat{a},\hat{a}^\dagger\right] = 1$, and $\hat{a}^2 |\phi\rangle = 0$. How do I show that the eigenvalues of ...
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2answers
445 views

Expectation values of commutator and anti-commutator (momentum and position)

What are the expectation values of commutator and anti-commutator for momentum and position operators? In the case of commutator: $$\langle[x,p]\rangle=\langle i\hbar\rangle=~?$$ In the case of ...
4
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3answers
356 views

Schroedinger field operators and their commutation relations

I've got several questions regarding the so called second quantization of the Schroedinger equation. My professor introduced the field operators for the Schroedinger field by simply stating them as ...
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2answers
56 views

Commutation Relations in Second Quantization

I understand that if I have the field operators $\psi(r)$ and $\psi^\dagger(r)$, then I have the canonical commutation relation (in the boson case) $$[ \psi(r) , \psi^\dagger(r')]=\delta(r-r').$$ My ...
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1answer
35 views

The fifth gamma matrix and fermion fields

I am aware of the various relations with Dirac spinors and chirality but how does the fifth gamma matrix $\gamma^5$ behave with fermion fields, $\psi$? Does the fifth gamma matrix have any particular ...
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0answers
27 views

Construct any Hamiltonian that is the linear combination of existing constructable Hamiltonians

In the paper Quantum Computation over Continuous Variables, it states that since $$e^{iAt}e^{iBt}e^{-iAt}e^{-iBt} = e^{-[A,B] t^2} + O(t^3)$$ when $t\rightarrow 0$, if one can apply a set of ...
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4answers
169 views

Help understanding proof in simultaneous diagonalization

The proof is from Principles of Quantum Mechanics by Shankar. The theorem is: If $\Omega$ and $\Lambda$ are two commuting Hermitian operators, there exists (at least) a basis of common eigenvectors ...
4
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1answer
263 views

Virasoro operators commutation relations [closed]

For the commutation relation in quantising the bosonic string $$\left[L_n,L_{m}\right]=(n-m)L_{n+m}+\frac{D}{12}n(n^2-1)\delta_{n+m,0}$$ we can then calculate this for $m=-n$ in between the vacuum ...
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0answers
84 views

Physical significance of non-commutativity of ladder operators in Quantum Harmonic Oscillator

If we apply the raising (creation) operator to $Ψ_n(x)$ and the apply to it the lowering (annihilation) operator, we get $Ψ_n(x)$ times a constant. Does it physically say something? Can we get any ...
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1answer
60 views

Lorentz force derivation in quantum mechanics [closed]

In Sakurai and Napolitano, chapter 2, there's a derivation of the QM Lorentz force. Given $$H=\frac{1}{2m}\left(\mathbf{p}-\frac{e\mathbf{A}}{c}\right)^2+e\phi = \frac{\mathbf{\Pi}^2}{2m}+e\phi$$ ...
2
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1answer
47 views

Why do we use the anticommutation relation for particle-hole and chiral symmetries?

In physics we say that a quantity is conserved if its operator commutes with Hamiltonian. For example, in condensed matter systems, when the momentum $k$ commutes with the Hamiltonian $H$ as ...
4
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3answers
360 views

Relation between representations of boson operators?

I have a simple (I think !) question about the representations of boson operators and how they are related. First of all let's define two conjugate observables $Q$ and $P$ (i.e. $\left[Q,P\right]=i$ ...
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1answer
32 views

Structure constant of the commutators of generators in broken symmetry

When I read a paper related to spontaneously global symmetry breaking, I cannot understand a statement: If we use the notation $T^i$ for the unbroken group generators in $H$ and $X^a$ the broken ...
3
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3answers
112 views

Does the canonical commutation relation relate to the fact that momentum is the generator of spatial translations?

In classical mechanics momentum is the generator of spatial translations. This remains true in quantum mechanics. The way we define the momentum operator in one-dimension, for example, already shows ...
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0answers
94 views

Commutation relations in quantum mechanics

As we know, simple harmonic oscillator can be solved only by commutation relations between creation and annihilation operators, and the Hamiltonian expression. The spin energy is either solved only ...
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6answers
2k views

Is there something behind non-commuting observables?

Consider a quantum system described by the Hilbert space $\mathcal{H}$ and consider $A,B\in \mathcal{L}(\mathcal{H},\mathcal{H})$ to be observables. If those observables do not commute there's no ...
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1answer
78 views

Commutation relations in Quantum Field Theory [closed]

\begin{align} [a, a^\dagger] =& \left[\int d^3 x e^{-ikx} (\omega \phi(x) + i \Pi^\dagger(x)), \int d^3 x' e^{ikx'} (\omega \phi^\dagger(x') - i \Pi(x')) \right] \\ =& \int d^3x \, d^3x' \, ...
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2answers
71 views

Commutator relationships and the exponential

I am currently trying to prove that the two following commutator relationships are equivalent (for an operator $\hat{A}(s)$ that depends on a continuous parameter $s$), so if one holds the other one ...
4
votes
3answers
100 views

Eigenstates of Ladder Operators

According ot Griffith's Intro to Quantum Mechanics (page 147), if some function $f$ is an eigenfunction of $L^{2}$, then $L_{-}f$ is also an eigenfunction of $L^{2}$. Is $f$ also an eigenfunction ...
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1answer
48 views

Sum and Different and angular momentum operators

Why is $\overrightarrow{L_{1}}+\overrightarrow{L_{2}}$ an angular momentum operator, but not $\overrightarrow{L_{1}}-\overrightarrow{L_{2}}$? What does this show about the applicability of the vector ...
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2answers
100 views

Mathematical Proof the angular momentum and Hamiltonian commute?

I'm in a quantum mechanics class, and it is given in the book that the operators $\hat{L^{2}}$ and $\hat{H}$ commute for the 3D Harmonic Oscillator, but no definite mathematical proof is given, and ...
0
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1answer
43 views

Spherical Polar Proof for Non-Commutativity of Indivdual Quantum Angular Momentum Operators

How can the following commutation relation be solved through spherical polar coordinates $[\hat{L}_{z}$,$\hat{L}_{x}]$ = $\imath\hbar\hat{L}_{y}$ I understand the derivation through partial ...
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1answer
85 views

Ladder operators - commutation relations and their properties

At the beginning of Fetter, Walecka "Many body quantum mechanics" there is a statement, that every property of creation and annihilation operators comes from their commutation relation (I'm ...
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1answer
58 views

How does this quantisation relation come about?

I'm currently doing a course in string theory and in the lecture notes it is stated: $$ [x^-, p^+]~=~-i \tag{1}$$ I am fine with this. However, after trying (and failing) a question, I looked at ...
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1answer
227 views

What is the physical importance of the commutation relations of angular momentum?

What is the physical meaning of these commutation relations: $$[L_{z},L_{\pm}]=\pm\hbar L_{\pm}\tag{1}$$ and $$[L_{+},L_{-}]=2\hbar L_{z} ~?\tag{2}$$
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2answers
80 views

Fermions, different species and (anti-)commutation rules

My question is straightforward: Do fermionic operators associated to different species commute or anticommute? Even if these operators have different quantum numbers? How can one prove this fact in a ...
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5answers
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Trace of a commutator is zero - but what about the commutator of $x$ and $p$?

Operators can be cyclically interchanged inside a trace: $${\rm Tr} (AB)~=~{\rm Tr} (BA).$$ This means the trace of a commutator of any two operators is zero: $${\rm Tr} ([A,B])~=~0.$$ But what about ...
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2answers
943 views

Weyl exponential form of the Canonical Commutation Relations

What is the physical meaning of the $c$-numbers $Q, P\in \mathbb{R}$ in the exponent of the Weyl system $\exp\left[\frac{i}{\hbar} Q \hat{p}\right]$ and $\exp\left[\frac{i}{\hbar}P\hat{q}\right]$? ...
3
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1answer
148 views

How I can prove the Commutation between hamiltonian and Runge-Lenz vector? [closed]

I am a undergraduate student in physics. I found this page that shows a way to prove the commutator between Runge-Lenz vector and Hamiltonian .$\left [\hat{A}_{i},\hat{H}\right]=0$ I believe he did a ...
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0answers
55 views

Commutation of two vector operator [closed]

Consider vectors $\overrightarrow { A } $ and $\overrightarrow { B } $ as operators or vector of operators. If this commutation holds$$[\overrightarrow { A },\overrightarrow { B }]=0$$ Then, is that ...
3
votes
2answers
303 views

$\hat{L}_{x}$ and $\hat{L}_{y}$ do not commute… or do they?

So $\hat{L}_{x}$ and $\hat{L}_{y}$ do not commute: $$ [ \hat{L}_{x}, \hat{L}_{y}] = i\hbar \hat{L}_{z}$$ But, what if we perform this operation on a state such that: $$\hat{L}_{z} \phi_{l, m_{l}} ...
3
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5answers
383 views

Eigenspaces of angular momentum operator and its square (Casimir operator)

The casimir operator $\textbf{L}^2$ commutates with the elements $L_i$ of the angular momentum operator $\textbf{L}$: $$ [\textbf{L}^2, L_i] = 0. $$ However, the $L_i$ do not commute among ...
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1answer
60 views

Limit of the position and momentum commutator [closed]

The commutator of position and momentum operator, $\hat{p}$ and $\hat{x}$, respectively is derived as $[\hat{x},\hat{p}]=i\hbar$. Let $\lim_{x\rightarrow x_{o}} [\hat{x},\hat{p}]=\lim_{x\rightarrow ...
2
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3answers
2k views

Commutators involving functions

I am looking for the commutator: $$[e^{aq},p]$$ My approach is to Taylor expand the function: $$[\sum_n \frac{1}{n!}(aq)^n,p]$$ I know that $[q^n,p]=ni\hbar q^{n-1}$ So how do I account for $n$ ...
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43 views

The commutator between observable and unit radius vector

As I encounter the commutator relating to unit radius vector, I am quite confused. I have just started the learning of quantum mechanics and all I know about the commutator is based on two identities: ...
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47 views

commutation relations in terms of eigenstates scalar product

This question has caught my attention because I was unaware of the fact that the position-momentum canonical commutation relations could be derived out of the only assumption for $\langle x | ...
3
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2answers
163 views

Derivation of canonical position-momentum commutator relation

We know that the position-momentum commutator is fundamental in quantum mechanics, but would it be possible to derive it starting from a different set of first principles, more specifically starting ...
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1answer
62 views

How to impose canonical commutation relations when quantising a field

I believe this is a simple question, however I cannot find it explained anywhere what the term: "Impose canonical commutation relations" means. If I have a classical equation, and I wish to quantise ...
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1answer
48 views

Second Quantization: The Identity Operator does not Commute?

Let me take the simplest possible example. Consider the fermonic Fock-space $\Lambda^*(\mathbb{C}^n)$ built out of a finite-dimensional, oriented single-particle Hilbert space $\mathbb{C}^n$ with ...
5
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2answers
384 views

Mutual or same set of eigenfunctions if two operators commute

If two operators commute, do they have "a mutual set of eigenfunctions", or "the same set of eigenfunctions"? My quantum chemistry book uses these as if they are interchangeable, but they do not seem ...
12
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2answers
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What does the Canonical Commutation Relation (CCR) tell me about the overlap between Position and Momentum bases?

I'm curious whether I can find the overlap $\langle q | p \rangle$ knowing only the following: $|q\rangle$ is an eigenvector of an operator $Q$ with eigenvalue $q$. $|p\rangle$ is an eigenvector of ...
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1answer
62 views

Is it right to write $\varepsilon_{ijk} \delta_{jl}=\varepsilon_{ilk}$? (indices notation)

Consider the $l$ component of vector position $\vec{r}$, $r_l$, and the $i$ component of angular momentum $\vec{L}$, $L_i$. We have that $$L_i=[r\times p]_{i}=\varepsilon_{ijk}r_jp_k$$ ...
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1answer
109 views

Expectation value of an operator and commutator relation

I have a quantum operator $A.$ It's expectation value is constant respect to time. I mean $$\langle \psi(t)|A|\psi(t)\rangle$$ is a constant values. If I know $|\psi(t)\rangle$ is not an eigenstate ...
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0answers
68 views

Commutation between angular momentum and Hamiltonian

Consider the following Hamiltonian of a 3-dimensional system: $$H=\frac{p^2}{2m}+V(r)$$ If the components of the angular momentum, $L_i$, commute with $H$, then: $$[H,L_i]=0$$ This condition can ...