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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Primary fields in di Francesco's CFT

In the CFT book by Di Francesco et al. they use conventions such that part of the conformal algebra (see eq. 4.19) is $$ [D,P_\mu]=iP_\mu, \\ [D,K_\mu]=-iK_\mu, \tag{1} $$ where $P_\mu$, $D$ and $K_\...
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Prove that every component of angular momentum commutes with $f$

Let $f( \hat {\vec r} , \hat {\vec p} )$ be the any polynomial in variables $r^2, p^2 $ and $(\vec r \cdot \vec p)$. prove that every component of angular momentum commutes with $\hat f$: $[\hat L_k ,...
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Generalized momentum in terms of wavefunction: Is it always $-i\hbar \partial/\partial q$?

I saw this kind of derivation several times in different notes/review/educational articles. (For example https://arxiv.org/abs/1904.06560 or http://wcchew.ece.illinois.edu/chew/course/QMALL20121005....
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Matrix representation for Gupta-Bleuler creation/annihilation operators

I am wondering what would be the closest analogue of the matrix representation for the creation and annihilation operators arising in Gupta-Bleuler formalism, which are defined by $$ [a,a^\dagger] = -...
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Commutation relation between squares of angular momentum [duplicate]

We usually come across the formula $$\vec{L}.\vec{S}=\frac{1}{2}\left[\vec{J}^2-\vec{L}^2-\vec{S}^2\right].$$ Do $\vec{J}^2$, $\vec{L}^2$ and $\vec{S}^2$ commute always, or do they commute under ...
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How to calculate the commutator between Hamiltonian and momentum operator squared?

I want to calculate the commutator $[H,p^2]$, where H is the Hamilton operator in one dimension and $p$ the momentum operator in one dimension. I tried it the same way as for $[H,p]=i\hbar(\frac{d}{dx}...
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Equal-time Canonical Commutation Relation for a scalar field

In chapter 2 of Quantum Field Theory and the Standard Model, Schwartz derives the equal-time commutation relations of the second-quantised field. Using $$ \phi(\vec{x}) = \int \frac{d^3p}{(2\pi)^3} \...
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Normal ordering of an exponential [duplicate]

I would like to recalculate Eq.(2.4) in PRA, 31,4,(1985), which expresses the exponential of operators as a normal ordering form. This equation reads \begin{equation} D=e^{\alpha K_{+} - \alpha^{*} K_{...
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Calculating $\langle p | [x,p] | \psi \rangle $ using Dirac notation

Calculating $\langle p | [x,p] | \psi \rangle $ using Dirac notation. I am aware of the relations $$\langle p|x| \psi \rangle = i \hbar \frac{d}{dp}\langle p| \psi \rangle, \langle x | p|\psi\rangle = ...
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Commutator property proof

I am working through Griffiths, and about a chapter or so ago, I came across the following commutator identity: $$[AB,C] = A[B,C] + [A,C]B$$ I tried to prove this rule by calculating the commutator ...
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Is the Heisenberg uncertainty principle limited to measurements in the same direction?

Heisenberg's Uncertainty Principle follows from the commutational relationship between the position and momentum operators, namely: $[\hat x_i,\hat p_j]=i\hbar\delta_{ij}$. Of course, in one dimension,...
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Why can't $iℏ ∂/∂t$ be used when calculating commutators of $H$? [duplicate]

$[x, \hat{H}]$ or $[\hat{p}, \hat{H}]$ can be computed by substituting $\frac{\hat{p}^2}{2m} + V(x)$ for $\hat{H}$ and doing some simple calculations. e.g $$[x, \hat{H}] = [x, \frac{\hat{p}^2}{2m} + V(...
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Why does the interaction hamiltonian not commute with itself at different times?

If you have a poincare invariant Hamiltonian $H$, then the Hamiltonian must commute with itself at different times and not explicitly depend on time. If the Hamiltonian $H$ can be written as $H$ = $H_{...
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Relation of two commuting operators to other operators

Consider two commuting quantum operators: $$\hat{A} \hat{B} = \hat{B} \hat{A} \quad $$ For any operator $\hat{C} $, how can we prove that: $$\hat{A} \hat{C} \hat{B} \hat{C} = \hat{B} \hat{C} \hat{A} \...
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Commutation relations for ladder operators in photons covariant theory in Mandl Shaw "Quantum field theory "

In the chapter 5, "Photons:covariant theory" at page 78 the book claims that the relations between ladder operators : $[a_{r}(\textbf {k}),a^{\dagger}_{r}(\textbf {k'})] = \...
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Can the operator field Dirac equation be expressed as Heisenberg's equation?

The Dirac equation of the operator spinor field is: $$(i\gamma ^{\mu}\partial _{\mu} -m)\psi =0$$ where $\psi$ is interpreted to be a quantum field. I'm wondering, can this be derived from the ...
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Is that possible to from the commutator relation of angular momentum to derive the coordinate representation?

From $[x,p]=i$, one could somehow show the coordinate representation of the momentum operator, e.g., in Dirac's principles of quantum mechanics section (22), as $p_x = i \frac{ \partial }{\partial x} +...
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Why implementing $\partial_\mu A^\mu=0$ as an operator equation is not valid?

This questions relates to equation (4.43) in Timo Weigand's QFT lecture note (page 108). Weigand makes the claim that $\partial_\mu A^\mu=0$, interpreted as an operator equation, causes issues due to ...
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Commutator of gauge field and the scalar field in the Stueckelberg Lagrangian with gauge-fixing terms

I was trying to add a gauge fixing term to Stueckelberg Lagrangian and cancel the mixing term between scalar $\chi$ field and vector $A_\mu$ field. $${\cal L}_{Stueckelberg} = -\frac{1}{4}V_{\mu\nu}V^{...
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Can always find compatible observables removing all degeneracies? [duplicate]

When dealing with a Hermitian operator $A_1$ with degeneracies, it is a common practice to figure out all given quantum states, that finding other compatible operators, say $A_i$s, so that $[A_i,A_j]=...
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How does this author solve commutation relation of Hamiltonian and a function?

In this book Modern Theory of Thermoelectricity by Veljko Zlatić and René Monnier the author gives the following commutation relation in Eq $(11.11)$. I want to understand how they do it. The ...
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What is the justification of imposing the commutation relation between $a$ and $a^\dagger$ in the quantized electromagnetic field?

From this commutation relation, many things are proved such that $a$ and $a^\dagger$ are creation and annhilation operators in some basis (the eigenstates of $n$). But is there a justification to this ...
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Do quadratures in quantum optics have a physical meaning?

Do quadratures in quantum optics have a physical meaning? Or are they just defined as such to satisfy the same commutation relation as the position and momentum operators ?
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A condition for two operators to commute, help with a proof [closed]

In my quantum mechanics textbook (chapter regarding central potentials), it is said that: if $L$ (angular momentum operator, but could by any operator) acts only on the variables $ \theta, \phi $, and ...
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Can any pair of operators $(a,a^*)$ defined as such be seen as ladder operators?

The ONLY thing that we know about $p$ and $q$ is their commutation relation $[q,p]=2i$. In other words, can we find some Hilbert basis such that one of them has the form I am asking this because it ...
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Can any pair of operators $(a,b)$ that has the same commutation relation as the ladder operators $([a,b]=1)$ be seen as ladder operators?

In other words, can we find some Hilbert basis such that one of them has the form
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On minimal coupling in Coulomb gauge

The Hamiltonian for a non-relativistic particle in a uniform external magnetic field is given in its simplest form by: $$ \mathcal{H} = \frac{\left|\mathbf{\hat{p}}-\frac{q}{c}\mathbf{\hat{A}}\right|^...
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Find commutator $[P_\mu,K_\nu]$ in conformal group

We have conformal group with next element of this group: $$U=e^{i(P_\mu\epsilon^\mu-\frac{1}{2}M_{\mu\nu}\omega^{\mu\nu}+\rho D+\epsilon_\mu K^\mu)},$$ where $D$ is dilatation operator $$x^\mu\...
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Does the commutator between the total derivative term of a symmetry generator and a quantum field always vanish?

I am trying to understand the following derivation in Schwartz section 28.2 as to how Noether Charges can be thought of as symmetry generators. We start with the definition of $Q$ (for simplicity let'...
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Fermionic commutator with summation

Is there any way to calculate a commutator of this type? $ [ \sum_{i,j,k,l}a^{\dagger}_{i}a^{\dagger}_{j}a_{k}a_{l} ,\sum_{i,j}a^{\dagger}_{i}a_{j}]= \sum_{i,j}[ \sum_{k,l}a^{\dagger}_{i}a^{\dagger}_{...
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Is there a Stone-von-Neumann theorem-like result for the canonical anti-commutation relations (CAR)?

The canonical commutation relation (CCR) $$[\phi(x), \pi(y)] = i\hbar\delta(x-y)$$ is kind of the key to basically any bosonic quantum theory. This is due to many different remarkable properties: By ...
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Commutator between covariant derivative and a field

I have field as an element of a Lie algebra as $\Phi = \phi^at^a$ and I want to calculate the commutator $$[D_{\mu}, \Phi],$$ with $$D_{\mu} = \partial_{\mu} + igA^a_{\mu}t^a,$$ the covariant ...
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Diagonalizing Operators Simultaneously [duplicate]

Suppose we have a Hamiltonian operator $\hat{H}$ and another operator $\hat{A}$ such that $[\hat{H},\hat{A}]=0$. Then, if the spectrum of $\hat{H}$ is non-degenerate, from my understanding the ...
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Quantum commutation relation $[e^{-x^{2}},e^{\alpha i p}] = ?$

I have been trying for find a closed form solution, or at least something neat for the commutation relation $$[e^{-x^{2}},e^{\alpha i p}] = ?$$ (where $[x,p] = i\mathbb{I}$) but have had little luck. ...
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Which of these commutation relations are correct? [closed]

I saw, in two different references, the following two commutation relations for the fermionic field operator: and which one of them is correct? 1 "Stefanucci, Gianluca, and Robert Van Leeuwen. ...
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How are these Covariant Derivative Identities found?

In David Tong's Gauge Theory notes on page 137 near eq. (3.30) he makes use of the following expressions for the covariant derivative $D_{\mu}$ $$\frac{1}{2}[\gamma^{\mu},\gamma^{\nu}]D_{\mu}D_{\nu}=\...
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Eigenbasis of Hamiltonian and momentum operators

I was taught that, if two Hermitian operators commute, they share the same eigenbasis. Since the Hamiltonian and momentum operators commute, am I right in concluding that they share the same basis of ...
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How does one arrive at the relation of commutator $\left[M^{-i}, M^{-j}\right]$ of Lorentz generators $M^i$ in terms of the string modes $\alpha_n^i$?

I am reading the book "String theory demystified" by David McMahon. On page 149, the author discusses the "critical dimension" for superstrings. the number of spacetime dimensions ...
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Is $[\hat x, \hat p_x] = i\hbar\, \mathbb{I}$ contradicting a fact about commutators?

My background in quantum mechanics is minimal, and I had seen the canonical commutation relation $$[\hat x, \hat p_x] = i\hbar\, \mathbb{I}$$ in a course, I took about two years ago. I'm doing pure ...
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A derivation of the canonical commutation relations (CCR) written by Dirac?

Dirac in his book fundamental of quantum mechanic used the following derivation: Is this a derivation of the canonical commutation relations (CCR) in quantum mechanics?
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How to prove commutator relation harmonic oscillator (Quantum mechanics)? [duplicate]

How to prove $$[â^{b},â^{\dagger}]=bâ^{b-1}$$ with the hint that $\hat{n}$=$â^{\dagger}â$ is the number operator of the harmonic oscillator expressed trough the creation and annihilation operators. I ...
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Showing $[\hat{A},\hat{B}] = i\mathbb{1} \Leftrightarrow [\hat{A},\hat{B}^{\textstyle n}] = i\,n\,B^{\textstyle n-1}$ [closed]

Actually it is self redundant to show having in mind $[\hat{A},\hat{B}^{\textstyle n}] = \,n\,B^{\textstyle n-1}\,[\hat{A},\hat{B}]$ but I suppose it is not supposed to solve it this way. Instead I ...
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How do Lieb-Robinson Bounds talk about locality without the position operator?

So we know when one goes from QM to QFT Lieb Robinson bounds become micro causality. But micro causality is a statement on the commutators assuming they are space-like, time-like or light-like. ...
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Asymptotic States, Propagator and Commutation Relations

Following Fradkin's discussion in the book QFT Integrated Approach, the commutation relation for asymptotic states satisfies $$\left\langle 0\left|\left[\phi(x), \phi\left(x^{\prime}\right)\right]\...
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Separability of an Hamiltonian with spin

I'd like to know if this Hamiltonian $\hat{H}=\frac{p^2}{2m}+\frac{1}{2}m\omega^2r^2+\frac{A}{\hbar^2}(J^2-L^2-S^2)$ is separable into two parts $H_1=\frac{p^2}{2m}+\frac{1}{2}m\omega^2r^2$ and $H_2=\...
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Criterion for stationary density matrix

A density matrix $\rho$ is time independent iff it commutes with the Hamiltonian $H$. I am wondering if there is a criterion to test whether $[\rho, H] =0$ using some trace condition. Specifically, I ...
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$[(\hat{a}^{\dagger})^2, \hat{a}] = -2\hat{a}^{\dagger}$?

I'm confused by a line in the following wikipedia article on the squeeze operator in deriving the action of the squeeze operator on Heisenberg basis, the article seems to imply that $$[(\hat{a}^{\...
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Polar decomposition of a complex scalar field theory

In the text I am referring to, the field was substituted in terms of a number density and phase: $$\psi(x) = \sqrt(ρ(x))e^{iθ(x)}.$$ While quantizing the field, a commutation relation was imposed: $$[\...
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Uncertainty principle when the expectation value of commutator is zero

I'm reading section 4.3 of Introduction to quantum mechanics written by David Griffits. The book states that the product of the standard deviation of two components of angular momentum is greater than ...
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Resonance level model: Commutator

As a small part of an exercise on the resonant level model (all fermionic (field-)operators, $\Psi(\vec{x}) = \sum\limits_{\vec{k}}e^{i\vec{k}\vec{x}}c_{\vec{k}} $, $V$ is a constant, $d$ and $c$ ...
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