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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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 ...
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2answers
559 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 ...
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63 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
134 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
73 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 ...
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53 views

Hamiltonian in commutator contradiction [duplicate]

Consider the following: $$[ \hat H, \hat x]=\left[-\frac{\hbar^2 \hat p^2}{2m}+V,\hat x\right]\ne0 \text{ in general}$$ But $$[ \hat H, \hat x]=\left[i\hbar \frac{\partial }{\partial t},\hat x\right]...
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1answer
41 views

Quantum Field Theory: commutator of covariant derivatives

I just started studying qtf and I dont understand the last lecture. In the lecture script a shortcut is defined. $P^j = \frac{1}{i}\partial_j - qA^j$ With this: $[P^j,P^k] = -\frac{q}{i}(\partial_j A^...
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1answer
308 views

Commutation of vector operators

I'm supposed to show that $\left[\mathbf A,\mathbf B\right]=0$ (for two vector operators $\mathbf A$ and $\mathbf B$) if and only if all components of $\mathbf A$ commute with all components of $\...
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14 views
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1answer
84 views

Relativistic Commutation relation for momentum and position

We all know that the canonical commutation relation give you $$[x_i,p_j]=i\hbar\delta_ij,$$ is there a relativistic version such as $$[x^a,p_b]=i\hbar\delta_a^b?$$ If so what is the time ...
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1answer
123 views

Canonical Commutation Relations in arbitrary Canonical Coordinates

If one were to formulate quantum mechanics in an arbitrary canonical coordinate system, does he impose canonical commutation relations using Dirac's recipe? $$[\hat{Q}_i,\hat{P}_j]~=~i\hbar~\{q_i,p_j\...
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166 views

QM angular momentum commutator solution using index notation

there are a few answered questions regarding the commutator of any two 3D angular momentum operator components $[L_i, L_j]$ , however, I am trying to go through fully using index notation so that I ...
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180 views

Is there a simple expression for $[x,e^{ixp}]$?

I'm sure this exists somewhere, but somewhat surprisingly it is not that easy to google.* The commutators $$ \left[x,e^{i(ax^2+b(xp+px)+cp^2)}\right] $$ of position and the exponential of a quadratic ...
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2answers
67 views

Quantum field operators in HEP and CMT

For a real scalar field (which is a bosonic field) we have these commutation relations : $$ \left[\phi(x,t),\phi(y,t)\right]=0 \qquad \qquad \left[\phi(x,t),\pi(y,t)\right]=\delta(x-y).\tag{1}$$ But ...
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117 views

Commutator of position and momentum

I'm reading Sakurai's Quantum Mechanics. One of the problem in the book asks to use the relation $$ \langle{x}|p\rangle=\frac{1}{\sqrt{2\pi\hbar}}e^{\frac{ipx}{\hbar}} $$ to evaluate $\langle{x}|[X,P]|...
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58 views

Deriving the form of generators of transformations

I'm struggling to understand a bit of quantum mechanics relating to the transformation generators. This specific bit contains quite a few guesses and assumtions which probably do make sense in ...
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40 views

Regarding properties of matrices involved in Dirac equation

In this document, after equation 62 on page 9, the author says that we can rewrite $\alpha^i \alpha^j \partial_i \partial_j$ as $\frac{1}{2} (\alpha^i \alpha^j + \alpha^j \alpha^i)\partial_i\...
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319 views

How do you prove that the number operator commutes with a general Hamiltonian?

If you have a hamiltonian, in the case of a bosonic system $$ H=\sum_{ij}H_{ij}a_i^\dagger a_j, $$ and the number operator $$ N=\sum_{i}a_i^{\dagger}a_i. $$ How do you show that they commute? I have ...
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285 views

Meaning of expectation value of product of non-commuting operators

Let $\hat{A}$ and $\hat{B}$ be Hermitian observables with spectra labeled by $a$ and $b$. Then we can write \begin{equation} \hat{A} = \sum_a a\, \hat{P}_a \end{equation} \begin{equation} \hat{B} = \...
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5answers
427 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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332 views

Time-ordering and Dyson series

In Dyson series we use a time-ordered exponential by arguing that a Hamiltonian at two different instants of time does not commute. Why is it that so? Can anyone explain with example why should the ...
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2answers
99 views

$[A_1, H] =[A_2, H] = 0$ but $[A_1, A_2] \neq 0$?

I am having a difficult time understanding this problem. Suppose $[A_1, A_2] \ne 0,$ $[A_1, H] = 0,$ $[A_2, H] = 0.$ Show that the energy eigenstates of $H$ are in general ...
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1answer
128 views

Commutation relations in second quantization

I know that for operators $a(\chi_1), a(\chi_2)$ of the same type (fermionic or bosonic) $$ [a(\chi_1), a(\chi_2)]_{-\xi} = [a^\dagger (\chi_1), a^\dagger (\chi_2)]_{-\xi} = 0 \tag{1}$$ where $$\xi ...
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1answer
157 views

Prove: $A$ and $B$ commute, therefore functions $f(A)$ and $g(B)$ will always commute with one another [closed]

How do I / can I actually prove the relationship $[a,b]=0 \Rightarrow [f(a),g(b)]=0$ for all functions $f,g$. I'm asking because the following sentence in the solution to my quantum mechanics ...
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1answer
94 views

Minus sign in the time ordering operator

The time ordering operator is usually defined as $$\mathcal{T} \left\{A(\tau) B(\tau')\right\} := \begin{cases} A(\tau) B(\tau') & \text{if } \tau > \tau', \\ \pm B(\tau')A(\tau) & \text{if ...
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1answer
182 views

Angular Momentum Operators - Commutation Relations

I was going over past PGRE exam questions, and came across this one. The components for the angular momentum operator $\mathbf{L}=(L_x,L_y,L_z)$ satisfy the following commutation relations. \...
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2answers
325 views

Commutation relations in QFT and the principle of locality

My question is, given two space-time points $x^{\mu}$ and $y^{\mu}$, if the events that occur at these points are simultaneous, i.e. $x^{0}=y^{0}$, are the two events necessarily space-like separated? ...
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60 views

The relation between commutation and quanta

This question discusses discretization in some sense, and this question talks about how quantization and Hilbert Spaces are related (the answer seems to to be not at all), but what I'm curious about ...
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1answer
230 views

How is quantization related to commutation? [duplicate]

How are commutation (of observables) and quantization related? Reading about the Stone-Von Neumann Theorem, it seems that commutativity is the classical limit of quantum mechanics, and hence non-...
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1answer
139 views

What is this nested bracket notation?

The following is an excerpt from K. Varga's paper, Precise solution of few-body problems with stochastic variational method on correlated Gaussian basis: ...The function $θ_{LM_L}(\mathbf{x})$ in ...
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1answer
55 views

Commutators and Operators [closed]

Is commutator of two operators an operator? I searched google but still got no success! I'm very curious to know the answer to this!
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2answers
289 views

Commuting observables and CSCO's

I've been looking at some basic quantum mechanics all day in an attempt to better my understanding of the subject. While going over the proof that commuting operators are compatible, I started getting ...
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2answers
114 views

Would $[\hat{Q},\hat{H}]$ correspond to an observable? [closed]

Would $[\hat{Q},\hat{H}]$ correspond to an observable? Where $\hat{Q}$ is an observable and $\hat{H}$ is the Hamiltonian. Surely that would just mean that $[\hat{Q},\hat{H}]$ would commute i.e. = 0?: ...
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306 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}} =...
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1answer
965 views

How to derive Uncertainty Principle relation?

How to derive Heisenberg Uncertainty Principle relation?
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108 views

Derivatives of Fields in E&M

In QED the field strength tensor $F_{\mu\nu}$ is given by the commutator of the covariant derivatives $$D_\mu=\partial_\mu-ieA_\mu$$ where $A_\mu$ is the gauge field. Explicitly we have $$[D_\mu,D_\...
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When trying to see what symmetries an operator generates, how do you “decide” what coordinate to apply it to?

Suppose I have $\hat{O}_{1}=-i\hbar\partial_{x}$ then \begin{eqnarray} e^{-i\gamma\hat{O}_{1}/\hbar}x\,e^{i\gamma\hat{O}_{1}/\hbar}=x+\gamma \end{eqnarray} and \begin{eqnarray} e^{-i\gamma\hat{O}_{...
2
votes
2answers
73 views

Transition from 4-potential to E and B [closed]

In my lecture notes there is a step that i cannot follow: $$\frac{i}{2}[\gamma^{\mu},\gamma^{\nu}] (\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu})=: \sigma^{\mu\nu}F_{\mu\nu}=i\vec{\alpha} \vec{E}-\...
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1answer
140 views

Derivation of Pauli Hamiltonian

In my lecture notes there is a step that i cannot follow: $$\frac{i}{2}\epsilon_{ijk}\sigma_k [\pi_i,\pi_j] = -e\epsilon_{ijk}\partial_iA_j\sigma_k$$ with $\vec{\pi}=\vec{p}-e\vec{A}(x)$ When I ...
4
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2answers
357 views

What is the meaning of commuting Hamiltonians?

I have two quantum mechanical Hamiltonians such that \begin{equation} [\hat{H}_1,\hat{H}_2] = 0, \end{equation} where $\hat{H}_1$ and $\hat{H}_2$ act on the same set of states. What is there to ...
2
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2answers
93 views

What does a left-right arrow in a tensor formula mean?

I need help with some some notation I've not seen before. Is using the left-right arrow in this formula $$[P^μ,M^{ρσ}]=i\hbar(g^{\mu\sigma}P^\rho-(\rho\leftrightarrow\sigma))$$ equivalent to writing ...
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1answer
150 views

Commutator and Hamiltonian [closed]

Assume that $[\hat{A},\hat{H}]_-=0$ and $[\hat{B},\hat{H}]_-=0$ but we know that $[\hat{A},\hat{B}]_-\neq 0$. Then there exists degenerate stationary states of $H$. How to prove it?
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Why is it “disconcerting” if the components of an operator do not commute?

A symmetrized operator is given by $$\hat{R}=\frac{1}{2\hat{H}}\hat{N}+\hat{N}\frac{1}{2\hat{H}}.$$ With $\hat{H}$ the Hamiltonian and $\hat{N}$ the first moment of energy. The defined $\hat{R}$ is ...
3
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1answer
164 views

Commutator of fermionic operators

The fermionic creation/annihilation operators are defined by the anti-commutation relations: $$ \{a_k^{\dagger},a_q^{\dagger}\} = 0 = \{a_k,a_q \} $$ $$ \{a_k^{\dagger},a_q\} = \delta_{kq} \, .$$ I ...
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3answers
754 views

Does it mean anything if the commutator of an operator with the Hamiltonian is equal to the Hamiltonian?

Question says it all, really. I have $[\hat{H},\hat{O}]=-2i\hbar\hat{H}$. Does this mean that the operator $\hat{O}$ (an observable) is special in some way?
3
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1answer
189 views

Spin operator: tricky proof using gamma matrices

I have not dealt with the gamma matrices extensively so I am having a bit of trouble here. Basically I want to show that the spin operator defined by $$ \mathbf{\hat{S}} = \frac{1}{2}\gamma^5 \gamma^...
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1answer
197 views

Translation Operators

Show that if the wave function $\langle x|\psi\rangle$ is modified by a position-dependent phase $\langle x|\psi\rangle \to e^{\frac{ip_ox}{\hbar}}\langle x|\psi\rangle$ then $\langle x\rangle \to \...
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3answers
276 views

How do I prove that the del squared operator commutes with the angular momentum operator? [closed]

I need to prove in Cartesian coordinates that $[\nabla^{2},\hat{L_{z}}]= 0$ I know that the angular momentum operator is defined as: $\hat{L_{z}}=x\hat{p_{y}}-y\hat{p_{x}}$ And the del squared is ...
2
votes
1answer
72 views

Anticommutator difference [closed]

What is the value of this difference of anticommutators $$\{x^2,p^2\}-(\{x,p\}^2)/2$$ if the commutator $$[x,p]=i\hbar ~?$$ I have tried and obtained a value $$-3\hbar^2/2 - 2i\hbar px.$$ But the ...
2
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0answers
96 views

Prove that $e^{\frac{i\lambda}{\hbar} S_x}S_ze^{-\frac{i\lambda}{\hbar}S_x}=S_z\cos(\lambda)+S_y\sin(\lambda)$ [closed]

Prove using Hadamard's lemma that $$e^{\frac{i\lambda}{\hbar} S_x}S_ze^{-\frac{i\lambda}{\hbar}S_x}=S_z\cos(\lambda)+S_y\sin(\lambda) $$ where $\lambda$ is a complex number. I get: $$\begin{align} e^...