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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Definitions of operators and commutativity in quantum mechanics

If $[\hat A,\hat B] = 0$, where $\hat A$ and $\hat B$ are operators, then the operators commute. This also means that, when applied to a wavefunction, that one can measure observables $A$ and $B$ in ...
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Do all well-measured observables effectively commute?

Do all well-measured observables effectively commute with each other? The rest of this long post clarifies what I mean by that simple-looking question. Consider quantum field theory in Minkowski ...
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Canonical commutation relation from Pauli-Jordan function

In Steinmann's book, Perturbative Quantum Electrodynamics and Axiomatic Field Theory is stated that the commutator of two fields $\varphi$, which satisfy the Klein-Gordon equation $$ (\square-m^2)\...
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Dirac field local observables

This is actually a continuation of calculation I've been working on. It is well known that, in the case of Dirac fields $\psi(x)$, they satisfy anticommutatation relationships since they're fermionic ...
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Commutator expectation value in quantum mechanics

Suppose $A$ and $B$ are operators, $A$ is Hermitian, $B$ anti-hermitian, and their commutator is the identity, i.e. $$[A, B] = I \, .$$ Denoting the eigenvectors of $A$ as $\lvert a \rangle$, so that $...
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If the commutator $[x,p]=i$, why does $[x,p^2]=2ip$?

According to Arfken et al. Mathematical Methods p.277 $$[x,p^2]=xp^2 - pxp + pxp -p^2x =[x,p]p + p[x,p]= 2ip \, .$$ According to the text this follows solely from $[x,p]=i$. I'm not understanding ...
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Commutator of normal ordered squared scalar field

The teacher left us prove the following statement. Let $\phi(x)$ be a scalar field $$ \phi(x) = \int \frac{d^3p}{2\pi \sqrt{2\omega_{\boldsymbol{p}}}} \left[ a_{\boldsymbol p} e^{-ip \cdot x} + a^+_{...
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If $\psi$ acting on $a_+$ and $a_-$ operator just moves up and down the ladder, why is $[a_-, a_+] = 1$ and not 0? [duplicate]

If $\psi$ is acted upon by both the operators one by one, it should return the same wave function. Thus order in which you increase or decrease the energy shouldn't matter. Then why is it so that the ...
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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.
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Propagator Causality with commutators all the way

We know that two fields commute - by locality and causality - iff there is spacelike separation $\left[\phi_l^k(x) , \phi_m^{k'}(y)\right] = 0$ for $(x-y)^2<0$ In the canonical quantization of ...
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A problem in three sequential selective measurements in case of incompatible observables

In case (a) the B filter takes care of B's eigenvalues and we sum all of their probabilities to calculate the probability of obtaining $|c'\rangle$. In case (b) B filter is not used and this creates ...
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The creation and annihilation operators in quantum mechanics

What is the result of the commutation relation between the creation operator and a power of the annihilation operators in simple harmonic oscillator problem?
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The commutator of position and momentum operators in three-dimensional Cartesian coordinates

I'm to calculate the commutator of the following operators : $\mathbf{\widehat{r}}=\mathbf{e}_{x}x+\mathbf{e}_{y}y+\mathbf{e}_{z}z$ and $\mathbf{\widehat{p}}=-i\hbar\left ( \mathbf{e}_{x}\frac{\...
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States created by translation operator

Quantum Mechanics Volume One page 188 by Claude Cohen Tannoudji. In $q$ and $p$ state vectov formalism. $QS(\lambda) |q\rangle=(q+\lambda)S(\lambda)|q\rangle$, where $S(\lambda)=e^{-i\lambda P/\hbar}...
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Operating a Commutator as an observable

I came across the Baker-Hausdorff theorem in my quantum mechanics course and wanted to know how an observable commutes with a commutator it was a part of. Well if I have $[p,x]$, it should give me $-...
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How does $[A,B]=0$ imply the possibility of measuring the corresponding eigenvalues simultaneously?

I came across a webpage where they showed $[A,B]=0$ implies that we can measure it's corresponding eigenvalues simultaneously. I don't understand which step of the mathematical proof points to this ...
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Relation between groups and commutators

I am getting introduced into supersymmetry following chapter 11 of Ryder's Quantum Field Theory. I have a question about why $[\delta_1,\delta_2]=\Delta$ being this result independent of the field ...
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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\}=\...
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How to prove commutation relation between charge and current in current algebra?

I am reading Gauge Theory of Elementary Particle Physics by Tapei Cheng and Lingfong Li. Proceeding equation 5.54, there is a statement which says Then from Lorentz covariance, we can include the ...
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Commutation relation, field and conserved charges

One more question on basic commutation relation for fields. Let $\phi(x)$ be a scalar field and $$ P^\alpha = \int d^3x T^{0\alpha}, $$ where $T^{\alpha\beta}$ is the energy momentum tensor. Now, ...
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Physical meaning of commutation

I was reading the solution to quantum harmonic oscillator by J.J. Sakurai. He uses the annihilation and creation operators and there's a key step (I think) which is $$[a,a^{\dagger}]=1$$ I know we can ...
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Can non-central hamiltonians commute with $\vec{L}^2$?

Central potentials $V(r)$ trivially commute with the operator $\vec{L}^2$ in quantum mechanics because the latter is a function of the angular coordinates $(\theta,\phi)$ only. Non-central potentials, ...
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Find the commutator $[AX+BY,Z]$

The problem I'm asked to solve is on quantum mechanics: Find the commutator $[xH+pH, p^2]$, where $H$, $x$ and $p$ are the Hamiltonian, space and momentum operator respectively A the moment, ...
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Show that $\vec L$ and $\vec S$ commute with each other

It is stated in Griffiths in a hint to a question that $\vec L$ and $\vec S$ commute with each other but no proof is given. $\vec L$ is given in the differential form and $\vec S$ is given in matrix ...
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Momentum operator ambiguous?

In nonrelativistic quantum mechanics, are different operators possible as a candidate for the momentum operator, given that one has fixed one position operator and a hilbert space that this position ...
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How to prove the translation generator commutes with the spinors in SUSY algebra?

I was reading Modern Supersymmetry by John Terning, the book starts with SUSY algebra and says $$ \left[ P_{\mu} , Q_{\alpha} \right] = \left[ P_{\mu} , Q_{\alpha}^{\dagger} \right] = 0 $$ I am ...
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Solving the von-Neumann equation explicitly [closed]

The von-Neumann equation reads: $$\frac{d\rho}{dt} = -\frac{i}{\hbar}[H,\rho]$$ The solution: $$\rho(t)=U\rho U^{\dagger}$$ with $U=e^{-\frac{iHt}{\hbar}}$ is easily obtained when starting from the ...
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Strange question involving finding a relation between a commutator and the time derivative of an operator

In order to get to the parts I am stuck at, I will add the examiners' solutions to each subquestion, which is needed to get to the subquestion that I am querying. The following is a bizarre question ...
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Energy $E_n$ for eigenstates $n$ and position probabilistic distribution

Energy and position operator does not commute. Uncertainty relation and Energy-Position interference So how come "For a particle in a box with given states $n$," and we obtained the $E_n$ for exact ...
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Simmons-Duffin CFT Lecture Notes: Exercise 3.3

Here is the exercise 3.3 in the Simmons-Duffin CFT lecture notes: Show that in $d \ge 3$, $$[Q_\epsilon,T^{\mu\nu}] = \epsilon^\rho \partial_\rho T^{\mu\nu} + (\partial^\rho \epsilon_\rho)T^{\...
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How to change a commutator of SUSY super-charges into an anti-commutator?

I would like to understand an apparently rather simple calculation which checks the closure of the Supersymmetry algebra via the commutator of 2 supersymmetric variations of the type: $$\delta \phi = ...
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Misner, Thorne and Wheeler, Box 9.2 Commutator … doesn't make sense to me

I apologize for the goofy commutator $\left[\left[\_,\_\right]\right]$ notation. MathJax doesn't like my \llbracket \rrbracket notation. And I religiously use $\left[\dots\right]$ for function ...
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What is the mistake in calculating such a commutator? [duplicate]

$B$ is an Hermitian operator in Hilbert space, and $|b\rangle$ is the eigenstate of $B$. We can have $[A, B] = 1$ where A is arbitary operator. Then we can calculate as below: \begin{align} &\...
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In which cases is the Schrodinger representation more useful?

Dirac in his brilliant book derived quantum mechanics using non-commuting operators $\hat{q}$ and $\hat{p}$. He related these to the Schrodinger representation using a wavefunction $\psi(x)$ and the ...
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Is $:A: \; \;= A - \left<0\right|A\left|0\right>$ a correct definition of normal ordering?

My course notes say that normal ordering is defined as $$:A: \;\; = A - \left< 0\right| A \left| 0\right>.\tag{1}$$ This works for $A = aa^\dagger$ and all already normal ordered expressions. ...
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How to see that $[\textbf{p}^2,\textbf{L}^2]=[\textbf{p}^4,\textbf{L}^2]=0$ without doing any messy algebra?

The Hamiltonian of a particle moving under the influence of a central potential $V(r)$ given by $$H=\frac{\textbf{p}^2}{2m}+V(r)$$ commutes with $\textbf{L}^2\equiv L_x^2+L_y^2+L_z^2$. Without doing a ...
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How to prove a set of matrices form a representation of Lie algebra?

When reading Paul Langacker's The Standard Model and Beyond, I am quite confused on equation 3.29, which says with a set of fields $\Phi _a$, where $a$ goes from 1 to $n$, are chosen to be transformed ...
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No sense in the expression $\hat{x}| 1\rangle=\sqrt{\frac{2}{a}}\int_{-\frac{a}{2}}^{\frac{a}{2}}x\cos\left(\frac{\pi}{a}x\right)dx=0$

I am considering a particle of mass m in a symmetric infinite square well of width a in the fundamental state. $$V(x)= \begin{cases} 0 & \mbox{$|x|<\frac{a}{2}$} \\ \infty & \mbox{...
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Computing the commutator of the potential and angular momentum

Assume the potential $V$ is not just a function of position. I'm trying to compute $[V, L_i]$. This is what I have so far: $$ [V, L_i] = [V, \epsilon_{ijk}x_jp_k] = \epsilon_{ijk}(x_j[V,p_k]+[V,x_j]...
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Result of the measurement of operators $A$ and $B$ on same state $|\psi\rangle$ if $[A,B]=0$

Consider a 3-dimensional Hilbert space spanned by the normalized eigenstates $|1\rangle,|2\rangle,|3\rangle$ of an operator $A$. Consider a normalized superposition, $|\psi\rangle=c_1|1\rangle+c_2|2\...
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Tensor operator of rank 2

Statement of the problem: Given that a second-rank tensor operator transform as $$T'_{jk} = R_{jm}R_{kn}T_{mn}$$ where R is the three-dimensional rotation matrix, I need to find the commutation ...
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Why do we say spin/angular momentum is observable even though its components can't be determined simultaneously?

Why do we say spin or angular momentum of a particle is observable even though all of its components can't be determined simultaneously? For example, we can measure the $\hat{L_x}$ of a particle's ...
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Is the right-hand side of the canonical commutation relation an operator?

Is $i\hbar$ in canonical commutation relation, $$[x,p]=i\hbar,$$ an operator? like the result of $[L_x,L_y]$. If not, why?
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Defining the propagator using the positive and negative frequency of the field

I am currently reading some notes on QFT and in the notes it defined that $$ \phi(x)=\phi^+(x)+\phi^-(x) $$ where $$ \phi^+(x)=\int\frac{d^3k}{(2\pi)^32E}a_ke^{-ikx} $$ and $$ \phi^-(x)=\int\frac{d^3k}...
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Dyson series for Hamiltonian with $c$-number commutator

I am trying to derive the evolution operator for a time dependent Hamiltonian which satisfies the commutator $$[H(t_1), H(t_2)]=I f(t_1,t_2)$$ Where $I$ is the identity operator, and $f(t_1,t_2)$ is ...
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Problem with Wick's theorem (normal ordering of a contraction)

Taking the example of two bosonic fields, Wick's theorem is \begin{equation} T\{\phi(x_1)\phi^\dagger(x_2)\} = N\{\phi\phi^\dagger\} + N\{(\phi\phi^\dagger)_c\} \end{equation} where the subscript $c$ ...
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Radial ordered commutation relation

In the book Conformal Field Theory of Francesco, Mathieu and Sénéchal, in Sec. 6.1.2, the authors state that the integral $$ \oint_w \mathrm{d}z~ a(z)b(w) ~=~ \oint_{C_1} \mathrm{d}z~ a(z)b(w) - \...
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Commutation relation coherent states

I am reading p. 159, chapter 4 of Condensed Matter Field Theory and I don't really get this commutation relation: They want to show that $\left[\hat{a}_i,\hat{a}_j^\dagger\right] = \delta_{ij}$. The ...
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Closed formula for $[\hat{H},[\hat{H},…[\hat{H},\hat{c}^\dagger_{\mu,\sigma}]]…]]$

Given the interaction part of a general many-body hamiltonian, $$\hat{H}=\sum_{\alpha, \beta,\gamma,\delta,\sigma,\sigma^\prime}O_{\alpha,\gamma,\sigma}^{\beta,\delta,\sigma^\prime}\hat{c}_{\alpha,\...
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Ladder functions and operators - commutation and product

I have been studying An introduction to QFT by Peskin and Schroesder, in my free time, to learn about the fascinating subject of QFT. I have a couple of basic ...