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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Coherent photon scattering on a 2-level system: How can the commutator of the Electric field and the dipole operator be nonzero?
Girish Argawal writes in his book Quantum statistical theories of spontaneous emission and their relation to other approaches (1974) at the end of chapter 7:
It may appear from (7.13) that the equal-...
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Commutation of $L_x$ and $x$ is zero. How is it possible according to uncertainty principle? [closed]
[ Lx x] = 0 But[ px, x ]≠0. WHY?
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What is the motivation for the Self-Dual Canonical Anticommutation Relation (CAR) algebra?
What exactly is the motivation for the use of the Self-Dual Canonical Anticommutation Relation (CAR) algebra in the context of infinite lattice systems? Why not remain with the CAR algebra, as both ...
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Show that a simultaneous eigenfunction of $\hat{L}_x$ and $\hat{L}_y$ leads to a contradiction [duplicate]
It is easy to show via the Canonical Commutation Relation $ [\hat{x}, \hat{p}_x] = i\hbar $ that the existence of a simultaneous eigenfunction of $\hat{x}$ and $\hat{p}_x$ leads to a contradiction, ...
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Exponential of the sum of two non-commuting operators where their commutator is proportional to one of them
I was shown the following property:
Given two operators $A$ and $B$, and $$[A,B]=-\frac{k}{2}B,$$ being $k$ an arbitrary constant, then:
$$ \exp(A+B)=\exp(A)\exp\left(\frac{-2}{k}B \left(1-e^{\frac{-...
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How to approximate $\exp(-t(A-B))$ for diagonal $A$ and rank-1 $B$? [migrated]
I have $d\times d$ diagonal matrix $A=\operatorname{diag}a$ and rank-1 $B=b b^T$ and need to efficiently approximate the following function:
$$f(t)=\langle\exp(-t(A-B))\rangle$$
Here $\langle M \...
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Quantum fidelity commutativity proof
I am looking for a proof that the quantum fidelity $$F(\rho, \sigma) = \left(\text{tr} \sqrt{\sqrt{\rho}\sigma\sqrt\rho}\right)^2$$ is commutative, i.e. $F(\rho, \sigma) = F(\sigma, \rho)$.
I have ...
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If an operator commutes with the Hamiltonian then do all of its projectors commute also?
Suppose I have an (self-adjoint) operator (representing an observable) $O$ which commutes with the system Hamiltonian $H$. Does it follow that $[P_O(\Delta),H] = 0$ where $P_O(\Delta)$ is an arbitrary ...
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Evaluating the commutator of derivative and position [duplicate]
In Zettili's book on quantum, the fully worked problem 2.6 asks to show
$$
\hat{A} = i(\hat{X}^2+1)\frac{d}{dx} + i\hat{X}.
$$
Is Hermitian. Where $\hat{X}$ is the position operator. I took the ...
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Query in the proof of Wick's theorem
I am looking at the proof of Wick's theorem in the notes here:
https://www.imperial.ac.uk/media/imperial-college/research-centres-and-groups/theoretical-physics/msc/current/qft/handouts/...
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Connection between definitions of "conjugate momentum density" as "generator of displacement of the field" and as "Lagrangian partial derivative"
I am reading Jakob Schwichtenberg Physics from Symmetry where in 5.2 conjugate momentum density $\pi(x)$ is defined as generator of displacement of the field itself (1):
$$
\pi(x) = −i\hbar\frac{\...
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Is the evolution of a density matrix linear with respect to density matrices?
The evolution of a density matrix in quantum mechanics is given by
$$i\hbar\dot\rho=[H,\rho]=H \rho-\rho H.$$ If it is linear, can be written the rhs as $A \rho$ for a linear operator $A$?
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Commutator of two Lorentz charges/angular momenta
In Barton Zwieback's book "A first course in string theory" page 261, we calculated a Lorentz charge/angular momentum $M^{-I}$ of the open bosonic string in the light-cone formulation to be;
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Why can the commutator of a general expression be replaced by the anti-commutator in the $bc$ CFT theory?
Polchinski states in his equation 2.6.14 (in his book String Theory Vol. 1, Introduction to the Bosonic String) that for charges $Q_1$ and $Q_2$ the following equation holds, where $j_i$ is the ...
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How can I demonstrate that, in a degenerate system, if $[H_0,\lambda H_1] = 0$ then $ H_1$ is already diagonalized?
In the context of degenerate perturbation theory, for a perturbed Hamiltonian $H_0 + \lambda H_1$, I've heard of a very useful tool:
"If $[H_0,\lambda H_1] = 0$ holds (both parts of the ...
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Commutation of kinetic energy operator with Hamiltonian
I am basically trying to calculate current energy operator $\hat{\mathbf{J}}_E(\mathbf{r})$ by using Heisenberg equation of motion as
$$
-\nabla\cdot \hat{\mathbf{J}}_E(\mathbf{r})=\frac{i}{\hbar}[H,\...
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Understanding where vacuum energy comes from in QFT From Peskin & Schroeder
I am studying from Peskin And Schroeder's QFT book, and I have managed to understand all the way to page 21, where I proved that the Klein-Gordon Hamiltonian can be written as (eq 2.31):
$$H=\int \...
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The discrepancy between first and second quantised theories (Non-relativistic) in energy and position measurements
I realize I've asked a similar question before. In this question, I really want to focus on non-relativistic QM.
Energy and position measurements are straight-forward in the first quantised theory. ...
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How to evaluate the commutation relations in the Heisenberg equation of motion (Polarisation)
For a Hamiltonian in the following second quantisation form:\begin{equation}
H=\begin{aligned}
& \sum_{\vec{k}, s} E_{c, s}(\vec{k}) c_s^{\dagger}(\vec{k}) c_s(\vec{k})+\sum_{\vec{k}, s} E_{v, s}(\...
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Is this operator Hermitian? Commutator of non-Hermitian operators [closed]
In the derivation of a Master Equation, I am left with two additional terms:
$$ \sigma_j [\sigma^{\dagger}_k,\rho] - [\rho, \sigma_k]\sigma_j^{\dagger} \quad ,$$
where $\sigma_j = |g\rangle \langle e|...
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Why is the commutator of ladder operators non-zero?
Griffiths states that the "ladder" of stationary states for a harmonic oscillator should be unique. That should mean that for one particular energy level, there exists only one energy state. ...
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How can the commutator of two observables be realized in an experiment?
Let's say we have two observables $A$ and $B$, and that we have an experimental setup in which we are able to measure both observables (not necessarily simultaneously). My question consists of two ...
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Contour integral for commutator of fermionic fields
Suppose we have primary fields $A$ and $B$ which have the OPE,
$$A(z) B(w) = \frac{1}{z-w} = -B(z)A(w), \quad |z| > |w|,\tag{1}$$
so they have fermionic statistics. Now I was curious how this would ...
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A theorem in Sakurai's QM book (section 1.4)
I was trying to understand the theorem 1.2 in Sakurai's Modern QM, it is on page 29 (the second edition):
Suppose that $A$ and $B$ are compatible observables, and the eigenvalues of $A$ are ...
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Product of Pauli-matrix exponentials [closed]
Given Pauli matrices $\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ and $\sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$, can one write $e^{\alpha \sigma_z} e^{\...
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Vector field coordinate transformation
On Carroll's spacetime and geometry book, page 67, the book gives the component form of vector field commutator
$$[X,Y]^\mu=X^\lambda\partial_\lambda Y^\mu- Y^\lambda\partial_\lambda x^\mu \tag{2.23}$$...
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Paul Dirac and Matrix mechanics
Did Paul Dirac in 1925 derive the Heisenberg's matrix mechanics from the Newtonian mechanics and canonical commutation relation? That is, from the expressions $$P=m\frac{dX}{dt}, \space \space \space ...
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Condition of the product of 2 operators being an observable [duplicate]
I'm trying to understand a bit the conditions of operators commuting, or themselves being an observable.
Here I have the operator $\hat{A}$ which has Eigenvalues $-1,+1$ and Eigenstates $|u_1\rangle, |...
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Inconsistency Regarding Commutators and Integration?
I have this following confusion regarding the ordering of integration and commutator.
Consider an operator $\mathcal{O}$ defined as
$$
\mathcal{O}(t) \equiv \int d^3 x' \ \phi(t, \vec{x}') \nabla'^2 \...
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Commutativity of $\rho_{AB}$ and $\mathbb{I}_A\otimes\rho_B$
Suppose the density operator of a composite system $AB$ is given by $\rho_{AB}$ and $\rho_B =\mathrm{Tr}_A(\rho_{AB})$ the marginal density operator of the sub-system $B$. I have some doubt whether $\...
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Is there a physical interpretation of the quadrature operators of the quantised EM field in a cavity?
I am considering a cavity setup using two mirrors perpendicular to the $z$-axis separated by a distance $L$, as seen here
Assuming the electric field is polarised along the $x$-axis and uniform in ...
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Fock vs Schrödinger Representation of QHO
Consider the harmonic oscillator hamiltonian
$$\hat{H} = \frac{1}{2}(\hat{x}^2+\hat{p}^2)$$
where as usual $[\hat{x},\hat{p}] = i$ (with $\hbar=1$).
We could use the ladder ops
$\hat{a}=\frac{1}{\sqrt{...
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How to take hermitian conjugate of an operator containing multiple elements?
The annihilation operator in the Dirac field could be written as
$$
a_p^s = \frac{e^{iE_pt}}{\sqrt{2E_p}}u^s(p)^\dagger\int d^3xe^{-ipx}\psi(t,x)
$$
Where
\begin{equation*}
\begin{split}
\psi(t, x) = \...
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"Physically distinct" Hermitian operators with the same eigenspaces
I understand that Hermitian operators can be decomposed in terms of their eigenbasis:
\begin{equation}
H = \sum_i\lambda_i|i\rangle\langle i|
\end{equation}
where the $\lambda_i$ are all real. I've ...
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$ℏ$ in the canonical commutation relation
I am wondering what the physical meaning of the introduction of a "new" constant $\hbar$ in the CCR $[\hat{x},\hat{p}]=i\hbar$ is if you compare it to the classical Poisson-bracket $\{x,p\}=...
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Do commutation relations stay the same when we use different definitions to quantize a free scalar field?
I saw different ways to write the scalar field. For example (Tong p.23):
$$
\phi(x) = \int\frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2E_p}}
\left[a_pe^{ipx}+a_p^\dagger e^{-ipx}\right].\tag{2.18}
$$
And we ...
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Calculating the contraction of a field operator and a creation operator
In Folland's Quantum Field Theory (section 6.4) he considers a field:
$$\phi_\pi = \sum_\tau \int f(\textbf{q})\big[u(\textbf{q}, \tau, \pi)a(\textbf{q}, \tau, \pi) e^{-iq_\mu x^\mu} + v(\textbf{q}, \...
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Is the operator identity $[X,F(P)]=[X,P]F'(P)$ always true?
I came up with the operator identity in my QM textbook
$$
[X,F(P)]=[X,P]F'(P)
$$
where $X,P$ are Hermitian operators whose commutator commutes with them: $$[X,[X,P]]=[P,[X,P]]=0.$$ $F(x)$ is some well-...
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Confused with computing causality for Dirac field
In Peskin and Schroeder's QFT book, P.56 Eq.(3.95) mentions that
$$\begin{align}
\langle 0|\bar\psi(y)_b\psi(x)_a|0\rangle = (\gamma \cdot p -m)_{ab}\int\frac{d^3p}{(2\pi)^3}\frac{1}{2E_p}Be^{ip(x-y)}\...
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How to derive the formal solution of Heisenberg's equation? [closed]
In the book Introductory to Quantum Optics https://ostad.hormozgan.ac.ir/ostad/UploadedFiles/386042/386042-1758823246346514.pdf, we have that for an arbitrary operator $\hat{O}$ having no explicit ...
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Commutation $[H_0, \phi_0(\vec{x},t)]$ in the Heisenberg picture [closed]
Studying from Schwartz "Quantum Field Theory and the Standard Model" p. 23, I got to the part where he discusses time dependence of the field operator $\phi$ and the annihilation/creation ...
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Why do we choose to use $L_z$ and $L^2$ to define a quantum state with angular momentum?
Classically, we need to specify all three $(l_x, l_y, l_z)$ to define the vector $\vec{l}$. I understand that in Quantum mechanics we cannot define states corresponding to the operators $\hat{{L}}_x$, ...
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What is the physical meaning of non-commuting tetrads?
I'm reading about the tetrad formalism in GR and one main difference between the coordinate and the tetrad frame is that coordinate derivatives commute $\partial_\mu \partial_\nu = \partial_\nu \...
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Is there a connection between $\hbar$ being in the path integral and being the unit of spin? [duplicate]
On the one hand, Planck constant $\hbar$ seems to fundamentally enter quantum theory via the path integral, in the factor $e^{iS/\hbar}$. Or via the Schrodinger or Dirac equations, in those versions ...
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Commutator Constant
I have seen a lot of commutators in quantum mechanics having a constant factor $i\hbar$. I have read about Dirac supplanting Poisson Brackets with commutators having a constant $i\hbar$.
I want to ...
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Why do we care about the canonical commutation relations?
Suppose $\hat{x}$ and $\hat{p}$ are the position and momentum operators, it can be shown that
$$[\hat{x}, \hat{p}] = i\hbar\mathbb{I}.$$
The Stone-von Neumann theorem tells us that that the above is ...
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On the infinite potential well: an apparent paradox [duplicate]
$$
V(x) = \left\{
\begin{split}
0&\quad \operatorname{in}\ [0, a] \\
+\infty&\quad \operatorname{elsewhere}
\end{split}
\qquad a>0,
\right.
$$
The Schrödinger equation for stationary ...
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Unitary operators that implement the same canonical transformation
In quantum mechanics a transformation of the spatial coordinate operators and conjugate momentums of the type: $$(q_1,\dots,q_n,p_1,\dots,p_n) \to (Q_1,\dots,Q_n,P_1,\dots,P_n),$$ is called canonical ...
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On the commutation with functions of operators in quantum mechanics
Given any complex analytic function $f$ defined on an open subset $D\subset\mathbb{C}$ of the complex plane containing the origin, for every linear self-adjoint operator $A\in{\mathcal{B}}(\mathscr{H})...
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Proving the relation $\frac 1 2 \left[\nabla^2,r \right] = \frac 1 r + \frac \partial {\partial r}$ (quantum mechanics exercise) [closed]
I'm trying to prove this relation in my quantum mechanics exercise book
$$\frac 1 2 \left[\nabla^2,r \right] = \frac 1 r + \frac \partial {\partial r}.$$
Here's my attempt:
Expand the Laplacian ...
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