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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How to remove the $\exp(-i(p^0+q^0)x^0)$ term in the canonical commutation?

Using the convention A Modern Introduction to Quantum Field Theory by Michele Maggiore Eq. 4.2 or equivalently the quantum theory of fields by Steven Weinberg Eq.1.2.63. $\phi(x)=\int \frac{d^3p}{(2\...
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Simultaneous observables

Hi I understand that when two observables are simultaneous we can measure them both at the same time without affecting each other however is there a condition that the commutator between the two ...
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How $[x,p]=i \hbar$ implies that $x$ and $p$ do not have simultaneous eigenstates?

I am reading Quantum Comuputing Explained by David McMohan. Here is a portion which I am not able to understand. Let $x$ be the position and $p$ be the momentum of the particle, then we know that $$[...
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How Can I Find The Commutator of this of these $2$ operators (quantum)?

I am a little confused about this. So take this operator here: $$\hat a = \frac{m\omega \hat x + i\hat p_x}{\sqrt{m\omega \hbar}}$$ Where $\hat x$ is the position operator which is just $x$. And $\hat ...
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Does a quantum commutator exist for energy and time? [duplicate]

In quantum mechanics the position operator $\hat{x}$ and the momentum operator $\hat{p}$ have a commutator $$ [\hat{x}, \hat{p}] = i\hbar $$ Does a similar commutator also exist for the uncertainty ...
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What does “commuting with the Hamiltonian” mean?

In quantum mechanics an observable or an attribute to a particle (like spin) is conserved if and only if it commutes with the Hamiltonian. What does this mean? What observables do not commute with the ...
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Two operators commute with the Hamiltonian, but do not commute with each other

I was reading Griffiths, and he made a statement that if two operators commute with the Hamiltonian, but do not commute with each other, then the energy spectrum has to be degenerate. He gave the ...
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Commutation relation of four vectors

I was trying to prove that: $$[P_\mu, J_{\rho \sigma}] = i(\eta_{\mu \sigma} P_\rho - \eta_{\mu \rho} P_\sigma) $$ $\textbf{Attempt}$ $$\begin{align} [P_\mu, J_{\rho \sigma}] = [P_\mu, x_\rho P_\sigma ...
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Calculating commutators in quantum mechanics symbolically with the help of Mathematica [migrated]

Say I have a hamiltonian $H=p_x^2+p_y^2+x^2+y^2+x^4+y^4$, and I want to calculate the commutator $\left[A,B\right] \equiv AB-BA$ of arbitrary operators $\mathcal{O}=x^ap_x^by^cp_y^d$, where $a,b,c,d\...
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What is the meaning of $i\hbar S_{z}$? Does the commutation relation between $S_{x}$ and $S_{y}$ mean anything?

OK for a system with spin $1/2$, if one measures $S_{x}$ the information on $S_{y}$ is lost and measuring $S_{y}$ after an $S_{x}$ measurement one gets a $50\%$ probability for $S_{y}$ up. My question ...
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An attempt at deriving canonical commutation relations

I am reviewing basic quantum mechanics since I feel like I am struggling with the fundamentals. I am not sure exactly how much I am "taking for granted" or whether or not my logic is clear. ...
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Operator integral [closed]

Consider the following integral: \begin{equation} L =2 \int_{0}^{\infty} d t \exp\left\{-\hat{\rho}_{\lambda} t\right\} \partial_{\lambda} \hat{\rho}_{\lambda} \exp\left\{-\hat{\rho}_{\lambda} t\right\...
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Issue with a sign in the commutator calculation of field operators for a real scalar field

In the derivation for the commutator (real scalar field, Klein-Gordon equation) $$[\phi(x),\phi(y)]=0$$ I have solved up to $$[\phi(x),\phi(y)]=\frac{1}{2(2\pi)^3}\int \frac{d^3 p}{\omega_p}[e^{ip(x-y)...
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If two operators say $D$ and $B$ commutes then why a non-degenerate eigenfunction of operator $D$ is also an eigenfunction of operator $B$?

I have following derivation which is not understandable for me and I am unable to understand it.Consider a eigenfunction-value equation $$D{\Psi}=d{\Psi} $$ Now $B$ operates on above equation and ...
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Quantum commutator simplification

I have a given hamiltonian and want to calculate the commutator of various operators with the commutator. This results in long and tedious computations. Is there a software designed simplify these ...
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Question regarding commutator of $\hat{p}$ and $V(\hat{r})$

I recently did an exercise for which I had to calculate $[\hat{H}, \hat{p}] = [V(\hat{r}), \hat{p}]$ with $- \nabla V(r) = F(r)$ and calculated it the following way: $[\hat{H}, \hat{p}]\psi(r) = [V(\...
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Origins of the canonical commutation relation

I have recently been reading Gunter Ludwig's book wave mechanics to get a better understanding of quantum mechanics and in reading through the book I came across the relation $$m\sum_s \{|q_{rs}|^2\...
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Why can't derivative notation of Hamiltonian and momentum prove $[H, p] = 0$?

I'm undergraduate student in physics and have question about first quantization. We already know that in quantum mechanics, hamiltonian and momentum don't commute with each other in general sense. ...
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Background on the Stone-von Neumann theorem

I'm actually a mathematician. I'm required to give a lecture on the Stone-von Neumann theorem. I already have all the mathematical details figured out, but I wish to make the lecture more interesting ...
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Alternative representations of the momentum operator in position space

The fundamental relation between the position and momentum basis in quantum mechanics is summed up in the canonical commutation relation: $[x,p]=i\hbar.I$ From here, one can get to the matrix elements ...
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On the normal ordering of Fermi fields

From my understanding, the normal ordering of Klein-Gordon fields in QFT is valid because of the ambiguity that comes with quantizing a classical theory, in the sense that the conmutator of fields is ...
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How to prove with and without using Einstein summation method? [closed]

To proof : [A.L,B.L] = i(AxB).L Where two vectors A & B commute with each other and with L also. L is angular momentum operator
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Rotation matrices in Schwinger's oscillator model of angular momentum

I am Section 3.9 in Sakurai's Modern QM, 3rd ed (which is Section 3.8 in 2nd ed.) I am trying to obtain the given form for $\hat D(R)|jm\rangle$: I employ $\hat D^{-1}\hat D=1$ and ignore the ...
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Matrix elements for the particle on a ring

I was learning about the particle on a ring and was attempting to calculate the matrix elements $\langle m\mid p \mid n\rangle$ and $\langle m\mid x \mid n\rangle$ where $$\mid m\rangle = \frac{1}{\...
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Show that if $\psi$ is an eigenfunction for the operator $A$ and $[A, B]\psi =0$ then $\psi$ is an eigenfunction for the operator B also [closed]

It is only possible for a state to have definite values for both $A$ and $B$ if the wave function $\psi$ satisfies $[A, B]\psi =0$. This is a statement from Lectures on Quantum Mechanics by Weinberg ...
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How do you show that a state is a simultaneous eigenstate of $\hat L_z$ and $\hat L^2$?

what is the general process for showing that a given state is a simultaneous eigenstate of the angular momentum operators $\hat L_z$ and $\hat L^2$? I've searched for a while but I'm not really ...
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Is $[E_i(x), B_j(y)]=i \hbar \varepsilon_{ijk}\partial_k \delta^3(x-y) $ ? (and how to derive it?)

In a comment, user @Andrew says that $$[E_i(x), B_j(y)]=i \hbar \varepsilon_{ijk}\partial_k \delta^3(x-y). $$ Is this the case? Considering one of Maxwell's equations can be written: $$\nabla \times \...
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Four-momentum and potential cross product identity

In my script about the dirac equation, i come across the following equation identity: $$ \vec p \times \vec A = -e\vec A\times\vec p + \hbar/i e (\nabla\times\vec A) .$$ $p$ momentum vector, $A$ ...
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Disentangling exponential of number operator and creation and annihilation operators

Is there a way to disentangle the exponential of the sum of the number, the annihilation and the creation operator? For example, $$e^{\alpha N + \beta a + \gamma a^\dagger } = e^{G a^\dagger}e^{A N}e^{...
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Commutation relations of generators of $SO(N)$

That is, $J_{i j}$ is a tensor. We can take this a step further, and let $R^{\prime}$ itself be an infinitesimal rotation, of the form $R^{\prime} \rightarrow 1+\omega^{\prime},$ with $\omega_{i j}^{\...
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What does it “physically” mean, in terms of uncertainty and measurements, for a commutator to be different than zero in quantum mechanics?

Let's consider the commutator $[L_i,L_j] = i \hbar L_k$ of the angular momentum. The consequence of this equation is that two components of the angular momentum cannot be simultaneously measured. I ...
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How to evaluate Commutator Bracket $\left[x,\frac{\partial}{\partial x}\right]$ indirectly using Poisson Bracket?

I want to evaluate $\left[x,\frac{\partial}{\partial x}\right]$ using a Poisson bracket. Can this be done? I have heard that the commutator bracket is $i\hbar$ times the Poisson bracket. I tried to do ...
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Integral eigenvalues in compact rank-2 symmetric $U(1)$ gauge theory

I am reading a paper related to rank-2 symmetric $U(1)$ gauge theory: Fracton topological order from the Higgs and partial-confinement mechanisms of rank-two gauge theory (or arXiv:1802.10108). My ...
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How to prove the non-communtative relation of magnetic translation operator?

Consider the boson with charge +1 hopping in a square lattice in the xy plane with an uniform magnetic field B perpendicular to the xy plane. The magnetic field effect in hopping process can be ...
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Commutation of 2 linear operators [closed]

I have to show that $$\left[\hat A,\hat B^n\right]=\sum_{s=0}^{n-1}\hat B^s\left[\hat A,\hat B\right]\hat B^{n-s-1}$$ Is it enough to prove this for $n=0,1,2,..$ and generalize it to the above ...
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What are the best book to learn commutator algebra in quantum mechanics?

I am looking for a pretty handy and well described quantum mechanics book where I can find the concept behind commutators in general.
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A unitary transformation induces a change $\delta\alpha$ on an operator that commutes with a complete set. Why must $\delta\alpha\propto 1$?

I'm reading Schwinger's 1951 paper "The Theory of Quantized Fields I". He described $\alpha$ as a complete set of commuting Hermitian operators, and considers a unitary transformation of ...
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Two questions on commutators [closed]

So, in quantum mechanics we know that $[x,p] = i\hbar$, but how do we know that it is true no matter what coordinates you use? Also, is it true or not that $e^{xp/\hbar} e^{-px/\hbar} = e^{i}$? Can we ...
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54 views

What is the relationship between expectation values of two commuting operators?

Given a state $|\psi(t)\rangle$, if the operators A and B commute, and the expectation value of A is constant in time, (i.e., $\langle\psi(t)|A|\psi(t)\rangle$ = constant) what can we infer about the ...
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Doesn't the non-commutivity of $x$ and $p$ complicate the measurement of a quantum system's (e.g.) Energy?

A classical system is parameterized by $x$ and $p$ coordinates, and so any other observable -- such as energy -- is some function $E(x,p)$ of them. I assume, then, that to measure the energy one must ...
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Is there a Hermitian operator on $L^2(\mathbb{R})$ which is outside the C*-algebra generated by $\hat{x}$ and $\hat{p}$?

This question seems to have been popping up here in a variety of forms that I feel don't seem to really get at (and get criticized for being vague or less than perfectly defined) what I believe the ...
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Generalisation of Seiberg-Witten Map?

Given the following algebra, $$[\hat{x}_i,\hat{p}_j] = i\hbar\delta_{ij};~[\hat{x}_i,\hat{x}_j] = i\theta_{ij};~[\hat{p}_i,\hat{p}_j] = i\eta_{ij}$$ in a space, where $\theta_{ij},\eta_{ij}$ are ...
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58 views

(Anti)commutation of creation and annhilation operators for different fermion fields

The Fourier expansion of the fermion field operator is such that $$ \hat\psi=\int\!d^3p\,\left[ f_b(p)\hat b(p) +f_d(p)\hat d^\dagger\!(p) \right] ~~, $$ for some sufficiently complicated $f_b$ and $...
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66 views

Measurements, QFT and Wightman's axiom 3

I think I might have misconceptions about the conceptual core of QFT. Let me explain where I am puzzled. In QM, the measurement process is accounted by the postulate of collapse of the wave function: ...
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Can there be an infinite set of commuting quantum mechanical operators, and how are their common eigenfunctions to be determined?

The quantum mechanical atomic kinetic energy and momentum operators commute, and essentially are powers of the gradient operator: it seems that any power of the gradient commutes with any other, and ...
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Does microcausality plus the time-slice property imply local primitive causality?

In quantum field theory, observables are associated with regions of spacetime. One of the basic principles of relativistic quantum field theory is microcausality, which says that observables ...
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Quantum unitary transformation

In quantum mechanics, we know $\dot{\psi}=-\frac{i}{\hbar}H\psi$, but why is $U\dot{\psi}=-\frac{i}{\hbar} \left(UHU^\dagger \right) U\psi$? Does it mean $UHU^\dagger = H$ ? I think $UU^\dagger H = H$,...
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Why commutator of positive and negative parts of scalar field is equal to the Feynman propagator?

Peskin & Schroeder state that the contraction of two fields, defined as the commutator: $$ [\phi^+(x),\phi^-(y)]\qquad \text{assuming}\ x^0>y^0$$ is equal to the Feynman propagator $D_F(x-y)$. ...
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Does $L\cdot S$ commute with $J^2$?

I'm trying to calculate the commutator $[L\cdot S,J^2]$ The only way to proceed seems to write $L\cdot S=\frac{1}{2}(J^2-L^2-S^2)$ Now I'm left with $[L^2+S^2,J^2]$ and I'm stuck here. Any hint on how ...
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58 views

Why does $[\hat{n},\hat{\phi}] = i$ imply $e^{i \hat{\phi}} |n\rangle = |n+1\rangle$?

When reading in quantum mechnics books and papers, I often come across a statement about two 'conjugate' variables. E.g. Let be $\hat{n}$ and $\hat{\phi}$ two variables satisfying $[\hat{n}, \hat{\phi}...

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