Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

0
votes
0answers
25 views

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 ...
0
votes
1answer
58 views

Commutator expectation value in Quantum mechanics

Suppose A and B are operators, A is Hermitian and their commutator is the identity operator: ...
0
votes
2answers
56 views

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 ...
0
votes
1answer
20 views

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^+_{...
-2
votes
0answers
38 views

Bosonic annihilation and creation operators commutators [on hold]

After proving the relations $[\hat{b}^{\dagger}_i,\hat{b}^{\dagger}_j]=0$ and $[\hat{b}_i,\hat{b}_j]=0$, I want to prove that $[\hat{b}_j,\hat{b}^{\dagger}_k]=\delta_{jk}$, however I'm not sure where ...
0
votes
1answer
60 views

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 ...
-1
votes
0answers
28 views

Proving the weak form of BCH formula using Hadamard's Lemma [duplicate]

I am trying to prove the weak form of BCH formula. Now, I know there are plenty threads already open on this topic, but this question is concerned more with the strange formulation of the problem from ...
2
votes
3answers
118 views

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.
0
votes
1answer
43 views

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 ...
1
vote
1answer
45 views

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 ...
0
votes
1answer
27 views

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?
-1
votes
0answers
25 views

Commutator of position and momentum to the $n$th power [duplicate]

I'm trying to prove the following equality: $$[\hat{q}_j,G(\textbf{p})] = i\hbar \frac{\partial G}{\partial p_j},$$ where $\hat{q}_j$ and $\hat{p}_j$ are the $j^{th}$ components of the position and ...
1
vote
1answer
43 views

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{\...
1
vote
1answer
41 views

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}...
0
votes
1answer
52 views

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 $-...
3
votes
2answers
116 views

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 ...
0
votes
0answers
51 views

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 ...
2
votes
1answer
53 views

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\}=\...
0
votes
1answer
39 views

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 ...
0
votes
0answers
23 views

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, ...
2
votes
2answers
128 views

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 ...
5
votes
1answer
67 views

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, ...
0
votes
1answer
63 views

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, ...
0
votes
1answer
50 views

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 ...
4
votes
1answer
44 views

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 ...
1
vote
2answers
73 views

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 ...
2
votes
1answer
93 views

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 ...
0
votes
1answer
78 views

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 ...
0
votes
1answer
34 views

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 ...
0
votes
0answers
29 views

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^{\...
2
votes
2answers
55 views

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 = ...
0
votes
1answer
95 views

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 ...
11
votes
2answers
540 views

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} &\...
0
votes
0answers
61 views

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 ...
1
vote
1answer
53 views

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. ...
0
votes
3answers
112 views

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 ...
1
vote
1answer
69 views

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 ...
1
vote
1answer
175 views

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{...
0
votes
0answers
57 views

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]...
1
vote
1answer
44 views

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\...
1
vote
0answers
116 views

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 ...
1
vote
1answer
70 views

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 ...
0
votes
2answers
45 views

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?
0
votes
0answers
40 views

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}...
0
votes
0answers
40 views

Commutation relation regarding to generators of translation

When we say that linear momentum is the generator for translation, it is implied that $$T(d\mathbf{x}) = 1- i\frac{\mathbf{p}}{\hbar} \cdot d \mathbf{x}.$$ But (correct me if I am wrong), there is ...
2
votes
0answers
32 views

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 ...
0
votes
0answers
47 views

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 $c$ ...
3
votes
2answers
67 views

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) - \...
0
votes
0answers
39 views

Please explain the 2 equalities

why $p^{I}$ can be factored out? what things can be factored out in commutation relations more generally? also in calculating commutation relations does order matter? multiplying from left or right or ...
1
vote
1answer
54 views

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 ...