Episode #125 of the Stack Overflow podcast is here. We talk Tilde Club and mechanical keyboards. Listen now

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.

Filter by
Sorted by
Tagged with
3
votes
1answer
207 views

What is the commutator of a Poisson bracket and the covariant derivative?

Consider a classical vector field $V^\mu$ on a curved background. We make a 3+1 split of coordinates into $t,x^i$, where $x^i$ are coordinates on spatial hypersurfaces $\Sigma$ and $t$ the parameter ...
3
votes
0answers
393 views

Expected value of commutator using path-integral

Consider a real scalar field theory in finite temeperature. According to the book by Kapusta and Gale, Finite-Temperature Field Theory, its retarded Green's function is given by $$iD^R(x,x') = Tr\{\...
3
votes
1answer
3k 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 ...
3
votes
0answers
265 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 $\langle x | p\rangle$...
3
votes
0answers
292 views

Commutators with function [duplicate]

I have following exercise: If $[C,D]$ is a c-number and $f(x)$ is a well-behaved function (i.e. all derivatives exist and are finite), show that: $$[C, f(D)]=[C,D]f'(D)$$ where $f'(D) = \frac{d}{dx}f(...
3
votes
0answers
593 views

The correspondence between Poisson bracket and Commutators in Quantum Mechanics

I don't understand canonical quantization. In passing from classical to quantum, one replaces the Poisson brackets with the commutators. I don't really understand this. How can we generally show that ...
2
votes
3answers
427 views

Why should not $\hat X$ and $\hat P$ commute in quantum mechanics?

I have heard if we want to obtain classical results from quantum mechanics, we have to choose the commutator of $\hat X$ and $\hat P$ to be $\left[\hat X,\hat P\right]=i\hbar$. Is there any reason ...
2
votes
2answers
2k views

Symmetry and degeneracy in quantum mechanics

If an operator commutes with the Hamiltonian of a problem, must it always admit degeneracy? It appears that not necessarily. For example, the parity operator commutes with the Hamiltonian of a free ...
2
votes
2answers
1k views

Operator relation involving the logarithm of an operator?

Dirac gives the relation: $\exp(iaq)f(q,p) = f(q, p - a\hbar)\exp(iaq)$ where $\hbar$ is Planck's constant. Can anybody give me the corresponding relation when the $\exp$ function is a $\ln$?
2
votes
3answers
267 views

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?
2
votes
2answers
827 views

Yang-Mills field strength tensor

In basically every QFT book the Yang-Mills strength tensor $F_{\mu\nu}$ is defined as $$F_{\mu\nu}=[D_\mu,D_\nu]$$ where $D_\mu$ is the covariant derivative $$D_\mu=\partial_\mu-A_\mu$$ and $A_\mu$ is ...
2
votes
3answers
927 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 ...
2
votes
1answer
14k views

Proof of weaker Baker-Campbell-Hausdorff Formula [duplicate]

Prove the weaker form of the BCH Formula: $$e^A e^B = e^{A + B + \frac{1}{2}[A,B]} $$ with the assumption $[A, [B, A]] = 0; [B, [B,A]] = 0$ Start with $f(\lambda) = e^{\lambda A} e^{\lambda B} e^{-\...
2
votes
2answers
42 views

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 ...
2
votes
2answers
198 views

Commutator $[\hat{A},\exp(\hat{A})]= 0$

In Equation (4) of this Physics.SE post, Qmechanic wrote, $$\tag{4}\frac{d}{dt}e^{t\hat{A}}~=~\hat{A}e^{t\hat{A}}~=~e^{t\hat{A}}\hat{A}.$$ How does one get this equation?
2
votes
2answers
916 views

Commutator with exponential $[\exp(A),\exp(B)]$

$A,B$ are quantum mechanical operators. $[A,B]\neq 0$ that is given. $e^{A}=\sum_{n=1}^{\infty} \frac{A^n}{n!} $ Is the following correct? $$[e^{A},e^{B}]=e^{A}e^{B}-e^{B}e^{A}=e^{A+B}-e^{B+A}=0 $$...
2
votes
2answers
526 views

Commutation relation under time ordering

Consider a quantum system with the following Hamiltonian: $$H(t)=H_0+H_1(t),\tag{1}$$ where $H_0$ is a noninteracting Hamiltonian and $H_1(t)$ a time-dependent perturbation. To formulate the linear ...
2
votes
1answer
986 views

Why does $[xp_{y},x]$ commute?

I'm looking at a solution in my book that says $[xp_{y},x]$ commutes. Does bracket notation imply: $[A,B]=AB-BA$ so that $[xp_{y},x]=xp_{y}x-xxp_{y}$ Taking the comment from Max Graves and ...
2
votes
3answers
484 views

Is the Heisenberg-picture commutator $[x(t),p(t)]$ between position and momentum always equal to $i\hbar$?

Some misconceptions over here, For $x=$position and $p=$ momentum, I know $[x,p]=i\hbar$ but does $[x(t),p(t)]$ still have the same relation where $t$ here represents time.
2
votes
2answers
351 views

Does Schroedinger equation depend on the sign of Poisson bracket?

Let's consider Poisson bracket $$\left\{ A,B\right\} =\alpha_{p} \left( \frac{\partial A}{\partial p_{k}}\frac{\partial B}{\partial q^{k}}-\frac{% \partial A}{\partial q^{k}}\frac{\partial B}{\...
2
votes
2answers
106 views

Can the necessity of using anti-commutators for Dirac fields and commutator for Klein-Gorden be deduced from the field equations?

We all learned to use the commutator for quantizing the KG field and the anti-commutator for the Dirac field. We are told (which is correct) so that KG-excitations are bosons and Dirac-excitations ...
2
votes
1answer
3k 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 ...
2
votes
1answer
254 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 ...
2
votes
1answer
288 views

Proving that an operator is hermitian [closed]

Let $A$ be an operator that is the product of two hermitian operators. Am I at liberty to say that if those two hermitian operators commute and their commutator is zero, then $A$ is hermitian?
2
votes
1answer
81 views

Second quantization canonical commutation relation : $\{c_\alpha,c_\beta^\dagger\}=\delta_{\alpha,\beta}$ a counter example?

Suppose two different states $\alpha$ and $\beta$ of some system of fermions such that each state only allows zero or one particle. The canonical commutation relation $\{c_\alpha,c_\beta^\dagger\}=\...
2
votes
1answer
354 views

Why must fermion fields anticommute and bosons commute?

Fermion fields must satisfy anticommutation relation. But why? I know that unless they anti-commute the Pauli exclusion principle cannot be satisfied. But is there some other deeper/fundamental ...
2
votes
2answers
473 views

Interpreting the commutators of the Poincare generators

Suppose we have the usual commutators ($J$=Angular Momentum, $P$=Momentum, $K$=Boosts, $H$=Hamiltonian.) $$ [J_i,J_j]=i\epsilon_{ijk}J_k\quad[J_i,K_j]=i\epsilon_{ijk}K_k\quad[J_i,P_j]=i\epsilon_{ijk}...
2
votes
1answer
121 views

Restriction of an operator

I am reading Cohen-Tannoudji's Quantum Mechanics Vol. 1. In problem 11 from Chapter II there are two given operators defined in a space generated by $\{\left|u_1\right>,\left|u_2\right>,\left|...
2
votes
2answers
335 views

Second Quantization: Do fermion operators on different sites HAVE to anticommute?

In second quantization, we assume we have fermion operators $a_i$ which satisfy $\{a_i,a_j\}=0$, $\{a_i,a_j^\dagger\}=\delta_{ij}$, $\{a_i^\dagger,a_j^\dagger\}=0$. Another way to say this is that $$ ...
2
votes
2answers
11k views

Commutation of Hamiltonian with momentum

In which case does the Hamiltonian $H$ commutes with the momentum $P$? Can anybody help me? With an example? (No particular or strange Hamiltonians and no particular momenta are involved). How can I ...
2
votes
3answers
1k views

Measuring position and momentum at the same time?

In a non-relativistic quantum mechanical system in an infinite potential well. I try to measure the energy and the position of the system simultaneously. Since, the respective operators do commute ...
2
votes
1answer
441 views

Help Simplifying a Commutator Equation

For the SHO, our teacher told us to scale $$p\rightarrow \sqrt{m\omega\hbar} ~p$$ $$x\rightarrow \sqrt{\frac{\hbar}{m\omega}}~x$$ And then define the following $$K_1=\frac 14 (p^2-q^2)$$ $$K_2=\frac ...
2
votes
1answer
966 views

Can the quantum angular momentum operator be derived from its commutation relations with position and momentum?

Exercise 12.2.2 in Shankar's Principles of Quantum Mechanics asks to derive the expression for the angular momentum operator $L_z$ \begin{equation} L_z = XP_y-YP_x \end{equation} using its ...
2
votes
2answers
2k views

Constructing the exponential form of a unitary operator

I think I've got this figured out but wanted to make sure I'm doing this right. Working with operators that satisfy bosonic commutation relations $$[b,b^\dagger] = 1,$$ I define a very general ...
2
votes
2answers
2k views

Non-commuting operators can't share any eigenvector

In an introductory Quantum Mechanics textbook, I found the following statement: For two Hamiltonians $H$ and $H'$, non commuting with each other, but commuting with the same group of translations ${...
2
votes
1answer
2k views

Expectation of a commutation relation

Is there any significance to: $\langle[H,\hat{O}]\rangle =0$ (which can easily be shown) where $H$ is the Hamiltonian, $\hat{O}$ is an arbitrary operator? Thanks.
2
votes
1answer
41 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 ...
2
votes
1answer
56 views

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}{...
2
votes
2answers
97 views

Angular momentum coupling

I read about angular momentum coupling on wikipedia and there are a few things i dont understand. What does this mean "spin and orbital angular momentum of a single object belong to different Hilbert ...
2
votes
1answer
90 views

Quantum statistics from the (anti)commutation relations of the operators?

From a QFT point of view, the difference between bosons and fermions is that their creation/annihilation operators ($a^{\dagger}$, $a$ and $c^{\dagger}$, $c^{\dagger}$ respectively) obey the following ...
2
votes
2answers
236 views

What does it mean for 2 observables to be compatible?

If I have 2 observable operators $A$ and $B$, if $A$ and $B$ commute: $[A, B] = 0$, then they must necessarily form a complete set of commuting observables (CSCO). Essentially, if 2 observables are ...
2
votes
1answer
299 views

Why do position operators in orthogonal directions commute?

In three dimensions, we have $\hat x$, $\hat y$, $\hat z$ as the position operators in the three orthogonal directions. If the components of angular momentum don't commute, why must these all commute? ...
2
votes
1answer
91 views

Show eigenvalue does not depend on magnetic quantum number $m$

We have a scalar operator $A$, being invariant under rotations which commutes with the angular momentum, i.e. $$[A,J_i]=0 \text{ where } i=x,y,z$$ $$[A,J^2]=0 $$ So eigenfunctions of $A$ can be ...
2
votes
1answer
709 views

Commutation between energy and momentum

The composition property of Lorentz transformations forces the commutation relation: $$[P^{\mu},P^{\rho}]=0$$ where $P$ is the four-momenta. The above seems to imply that Energy and 3-momentum ...
2
votes
3answers
323 views

Are there any theorems that support the commutation relations in QFT?

I am learning Quantum Field Theory. I am confused about the commutation relations, which says that the field and its conjugate momentum don't commute. Are there any theorems that support the ...
2
votes
1answer
115 views

According to the Galileo Algebra, space translations commute with time translations. Does this mean that $[\vec P,H]=0$?

The Galileo Algebra is discussed in, for example, the wikipedia article Representation theory of the Galilean group. In that article, we can see that, for example, $$ [E,P^i]=0 $$ which means that ...
2
votes
1answer
666 views

Under what condition is angular momentum conserved in both classical and quantum physics?

Classically, angular momentum is only conserved in a central potential by considering the torque (correct me if I am wrong). In quantum mechanics, it is also true, isn't it? If this is the case, ...
2
votes
1answer
961 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
votes
2answers
4k 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 I'...
2
votes
1answer
582 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 ...