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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Canonical transformations in Quantum Mechanics
Heisenberg's famous commutator of a pair of conjugated canonical variables is formulated for position and conjugated momentum
$$[q,p] = i\hbar.$$
Intuitively I would guess that it would also work ...
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How does Dirac prove that the canonical coordinates $q_r$ form a complete commuting set of observables?
I'm reading "The principles of quantum mechanics" by Paul Dirac, and I've reached the point where he introduces the momentum operator.
After showing that the $p_r$'s commute with each other ...
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If an operator $A$ commutes with the Hamiltonian $H$ do they have common eigenstates?
In this book Intermediate quantum mechanics by Hans A. Bethe and Roman Jackiw they write at page 103:
In the case of spin, the Hamiltonian commutes with the spin operator of each electron $\...
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Question about commutator involving fermions and Pauli matrices [closed]
Suppose $\lambda_A$ and $\bar{\lambda}_A$ are fermions (A goes from 1 to N) and $\{ \lambda_{A \alpha}, \bar{\lambda}_B^{\beta}\} = \delta_{AB}\delta_{\alpha}^{\beta}$.
Let $\sigma^i$ denote the Pauli ...
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Exponential operator approximation: Suzuki-Trotter Expansion
One cannot solve the transition amplitude $\langle{x}\vert e^{-iHt}\vert{y}\rangle{}$ with $H=H_0+V$ by just applying the operators one after another on the bra/ket, because the free hamiltonian $H_0$ ...
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Are partition functions invariant under Bogoliubov transformations?
Consider a Hamiltonian $H(a_i, a^{\dagger}_i)$ as a function of some ladder operators $a_i, a^{\dagger}_i$. Now, consider a partition function $H(a'_i, a'^{\dagger}_i)$ where $a', a'^{\dagger}$ are ...
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Uncertainty on the sum of two non-commuting operators
Suppose that I have an observable
$$
\hat{E} = \sin(\alpha) \hat{Q} + \cos({\alpha}) \hat{P}
$$
with $\hat{Q}, \hat{P}$ being non-commuting operators satisfying
$$
[\hat{Q}, \hat{P}] = i \hbar
$$
It ...
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From Klein-Gordon equation to Dirac equation: a wrong "derivation" [closed]
So let us start with the Klein-Gordon equation
$$\tag{KG}
(-p^\mu p_\mu + m^2)\phi = 0
$$
The idea is to "factorize" the operator $-p^\mu p_\mu + m^2$.
\begin{equation}\tag{1}
-p^\mu p_\mu + ...
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Non-vanishing amplitude outside light cone doesn't violate causality? [duplicate]
I am following Peskin & Schroeder's QFT book. And on equation 2.51, we get an expression for the free Klein-Gordon propagator for timelike intervals $x^0-y^0=t$, $x-y=0$:
$$D(x-y) \sim e^{-imt}\...
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Commutation relation between pairs of conjugated variables
Suppose I have four operators $\hat{q}_1$, $\hat{p}_1$, $\hat{q}_2$ and $\hat{p}_2$ such that
$[\hat{q}_1, \hat{p}_1] = [\hat{q}_2, \hat{p}_2] = i$
$[\hat{q}_1, \hat{q}_2] = [\hat{p}_1, \hat{p}_2] = [...
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Deriving $\hat{p}_ x =-iℏ∂/∂x$ [duplicate]
(Taken from "The meaning of Quantum Theory, from Jim Baggott).
Starting from [x,^Px]= iℏ, if we chose the position operator to be simply 'multiplication by x', this forces the linear momentum ...
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Commutation relation of position and Hamiltonian operator [duplicate]
In the book Quantum Mechanics volume 2 by Cohen-Tannoudji, in Electric dipole approximation, it was written that $\left[\boldsymbol{Z}, H_0\right]=i \hbar \frac{\partial H_0}{\partial P_{\boldsymbol{z}...
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Commutable operators and eigenfunctions
My lecture notes says Since $[\hat{L^2} ,\hat{L_{z}}] = 0$
Then $Y(\theta,\phi)$ solves the below equations simultaneously:
$$\hat{L^2}Y(\theta,\phi) = A Y(\theta,\phi)$$
$$\hat{L_{z}}Y(\theta,\phi) = ...
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Momentum operator and Space operator
This may be a silly question, but given that the momentum operator (say in the $x$-direction) can be written as $$p_x = -i \hbar \frac{\partial}{\partial x},$$ would it be correct to say that $$p_x^2 ...
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Canonical Commutator Relation from Translation operator
In the book of Sakurai & Napolitano, the Canonical Commutation relation is derived using the unitary translation operator:
I agree with the derivation of the book until i reach this step, and ...
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Quantization of radiation with canonical conjugate variables
I am reading the Introduction to quantum optics book and I am a bit stuck at the quantization of electromagnetic field (around page 317).
The problem in this case is that we can't just express the ...
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Why is $[ \hat{A},\hat{B} ] \rightarrow i \hbar \text{{A, B}}$?
If we have two classical quantities $A$, $B$, and their corresponding quantum operators $\hat{A}$, $\hat{B}$, then their commutators and Poisson brackets are linked by
$$ [ \hat{A},\hat{B} ] \...
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Why do we only consider commutators and anticommutators in QFT?
While studying canonical quantization in QFT, I observed that we quantize fields either by a commutation or an anticommutation relation
\begin{equation}
[\phi(x), \phi(y)]_\pm := \phi(x) \phi(y) \pm \...
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Dressing an operator by Wilson line in Quantum Electrodynamic
I am reading a paper arXiv:1507.07921 which introduce gravitational dressing. The paper compare it to dressing in QED.
Consider the scalar QED lagrangian
$$\mathcal{L}=-\frac{1}{4}(F^{\mu\nu})^2-|D_\...
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How to correctly use the Mean-Field Approximation to simplify a Commutation Relation in Excitonic Physics?
I am following the work in 'Theoretical Methods for Excitonic Physics in 2D Materials'1.
They are aiming to derive the BSE equation for exciton physics. I am however stuck on the use of the Mean-field ...
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Proving Non-degenerateness for Compatible observables [closed]
Hi everybody I was trying to verify the following fact but I've been having some trouble.
I need to prove that if $A,B$ are two compatible observables and their eigenvalues are non-degenerate, then ...
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What's the inverse of an adjoint endomorphism?
$\def\ada{\text{ad}_A}$Suppose $A,B$ are operators in Quantum Mechanics, the adjoint endomorphism $\text{ad}_A$ can be expressed by
$$
\text{ad}_A B=[A,B]=AB-BA
$$
and
$$\text{ad}_A^0B=B, \quad \ada^n ...
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Commutation relations for angular momentum operators [closed]
I was reading Sakurai's Quantum Mechanics and I'm having some trouble with the following computation:
I need to verify that:
$$[ S_i, S_j] = i \hbar \epsilon_{ijk}S_k , i \in \{1,2,3\}$$
For $i = 1,y ...
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Set of Compatible Observables and Simultaneous Eigenkets Basis
I have managed to prove that a finite set of compatible observables [i.e. pairwise commuting (possibly unbounded) selfadjoint operators] on an Hilbert space $\{H\}$ admits an orthonormal basis of ...
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Maximality and Completeness for Set of Compatible Observables
Let $\mathcal{A} = \{A_j\}_{j=1}^{N}$ be a finite family of compatible observables (they commute pairwise) on an Hilbert space $H$. Consider the following definitions:
$\mathcal{A}$ is Maximal if and ...
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If something commutes with $x$, it commutes with any $f(x)$
I often read that if some operator commutes with (say) the position operator $x$, it commutes with any function $f(x)$ of it. For analytical functions, this is obvious, but I was wondering if there ...
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Do operators commute if they depend on the same operator? [duplicate]
Consider two arbitrary operators $\hat{A}(\hat{R})$ and $\hat{B}(\hat{R})$ that are both functions of the position operator $\hat{R}$ (and do not depend on any other operators). Does it follow that $$[...
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Can I upper-bound uncertainties of $B$ in an eigenstate of $A$ using an appropriate norm of $[A,B]$?
There are times when I would really like a "reverse" uncertainty principle to hold, allowing me to upper-bound certain uncertainties in terms of properties of commutators. I'll try to ...
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Commutator of gamma matrices with scalar product of 4-vectors
I know about the commutator relationship given in this question:
$$\left[\gamma^{\mu},\gamma^{\nu}\right]=2\gamma^{\mu}\gamma^{\nu}-2\eta^{\mu\nu}$$
then, my question is if there exist a similar ...
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What went wrong in the following calculation of $\langle p'|[x,p]|p'\rangle$? [duplicate]
We know that $$[x,p]=i\hbar. $$
Consider now the diagonal element in the momentum representation, $$\langle p'|[x,p]|p'\rangle=i\hbar\langle p'|p'\rangle=i\hbar\delta(0).$$
But the LHS = $$\langle p'|...
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How can difference of two operators produce a constant?
In the canonical commutation relations, $$[\hat{x}, \hat{p}] = \hat{x}\hat{p} - \hat{p}\hat{x} = i \hbar $$ Both $\hat{x}$ and $\hat{p}$ are operators and so are $\hat{x}\hat{p}$ and $\hat{p}\hat{x}$. ...
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A simple question on creation and annihilation operators
We know that the KG solution for a Spin-0 particle has the following Hamiltonian
$$\hat{H}=∫ d^{3}p\frac{ω_{p}}{2}(\hat{a}_{p}\hat{a}^{\dagger}_{p}+\hat{a}^{\dagger}_{p}\hat{a}_{p})\hspace{2cm}[\hat{a}...
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Strange definition of the fermion number operator in Polchinski
In Polchinski's exposition of the RNS formalism for the superstring (String Theory: Volume II, chapter 10), in page 8, he mentions the worldsheet fermion number operator, which he calls $F$. He then ...
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Commutator of annichilation and creation operators [closed]
Let $\phi$ be a real scalar field and $\psi$ a complex scalar field. Therefore, we can expand $\phi$ in terms of $a_{\boldsymbol{p}}, a^\dagger_{\boldsymbol{p}}$ and $\psi$ in terms of $b_{\boldsymbol{...
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Operators depending on the same independent variable but commuting between them [closed]
As far as I understood in quantum mechanics two operators can commute even though they are not functionally independent, which means that they can depend on the same independent variable.
On the other ...
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Rotor wave fomulation of quantum mechanics and the associated canonical commutation relations of position and momentum operators
I've been trying to determine whether it would be possible to formulate non-relativistic quantum mechanics entirely in the algebra of physical space (APS) by using rotor waves instead of complex-...
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Commuting operators and their physical interpretation in QM
I'm studying Quantum Mechanics for the first time at the moment and I have a few questions in mind.
So recently, I saw a proof on that if two operators share the same eigenstates is equivalent to the ...
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Do position and spin commute?
I recently learned that position vectors and spin vectors lie in different spaces, and the complete wave is the tensor product of both. I wanted to know that whether we can talk about commutation of ...
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Product of normally ordered exponentials as a normal ordering of product of exponentials
I want to simplify a product of normally ordered exponentials that are in the following form
$$:e^{x(\hat{a}^\dagger+\alpha_x^*)(\hat{a}+\alpha_x)}:\times :e^{y(\hat{a}^\dagger+\alpha_y^*)(\hat{a}+\...
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Matrix element of two operators, where one does not commute with the operators of the basis kets
If I am considering 2 operators A and B and an initial basis $|F,m_F\rangle$ , where F is the total angular momentum operator of the two systems. While assuming that operator A does commute with $\vec ...
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Variable Dependence of Quantum Operators and Commutator Relationships [duplicate]
EDIT: After doing some digging, I am convinced that the approach taken in this paper was simply an incorrect approach to deriving a quantum version of Hamilton's equations (also related to Ehrenfest's ...
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Showing that time ordering does not matter for the measurement of commuting observables
Suppose I have two observables $R$ and $S$ who are represented by operator $R$ and $S$ which commute (I will hereafter ignore the distinction between observables and the operators representing them), ...
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Time-ordered exponential operator generated by two commuting Hamiltonians
Define a time-dependent Hamiltonian $$H(t) = H_1(t) + H_2(t),\tag{1}$$ where $$[H_1(t), H_2(t)] = 0 ~ \forall t \in [0,T].\tag{2}$$ Is it true that the unitary operator generated by $H(t)$ is a ...
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Why can't commuting operators allow for full state determination?
For the purposes of this question, suppose that the operator $R$ representing the observable $\mathsf{R}$ has nondegenerate eigenspaces.
In discussing state determination (for some given situation, ...
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Why does $[L_z,L_{\pm}]\neq 0$ imply $[J^2,L_z]\neq 0$?
In a lecture about the angular momentum operator, it is stated that the operator $L_z$ commutes with itself, with $L^2$, with all of spin angular momentum operators, but not with $L_{\pm}$, so;
$$[J^2,...
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Angular momentum commutator in QFT/the roles of coordinates
this is my first question on this site, so please correct me if I break any convention.
It arose as part of homework, but I suppose that there is some deeper irritation behind it.
We are supposed to ...
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For hydrogen atom wave function, why eigenstate of Hamiltonian isn't only spherical harmonics if $H$ commutes with square of angular momentum? [closed]
I was reading quantum mechanics and I read about CSCO. So, commuting operators should share common eigen states, does that mean same eigen states? because in hydrogen atom wave function, Hamiltonian ...
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What is the value of the commutator $[L_x, S_y]$? [closed]
I would like to know how to calculate the commutator $[L_x, S_y]$ in quantum mechanics, if $L_x$ is the x-component of the angular momentum operator and $S_y$ the y-component of the spin operator.
I ...
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Integrals and Laplace Operator with a Commutator
If I have a Hamiltonian which looks like:
$H=\int_{}^{} \psi(x)^\dagger \nabla^2_{x} \psi(x)d^3x $
and a total number operator like:
$N=\int_{}^{} \psi(x)^\dagger \psi(x)d^3x $
and I want to ...
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Canonical Commutation relations in gravity
The canonical commutation relations in gravity are sometimes written
$$
[\gamma_{ij}(x),\pi^{kl}(y)]=\frac{i\hbar}{2}(\delta_i^k\delta_j^l+\delta_i^l\delta_j^k)\delta^3(x-y),\tag{0}
$$
where $\gamma_{...
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