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 equal time commutation relations in QED

I understand that to quantize the classical electromagnetic field one needs to impose commutation relations and express the field in terms of creation and annihilation operators. I notice that the ...
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328 views

Schroedinger field operators and their commutation relations

I've got several questions regarding the so called second quantization of the Schroedinger equation. My professor introduced the field operators for the Schroedinger field by simply stating them as ...
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What is the commutator? [closed]

$e$ and $f$ are unit vectors, $L_e$ is defined by $L_e=eL$, where $L$ is of course the angular momentum operator. A similar definition for $L_f=fL$ The commutator that I can't solve: ...
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178 views

Commutators with function

I have following exercise: If $[C,D]$ is a c-number and $f(x)$ is a well-behaved function (i.e. all derivatives exist and are finite), show that: $$[C, f(D)]=[C,D]f'(D)$$ where $f'(D) = ...
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1answer
221 views

Observables still commute even if fields only anti-commute

In Peskin & Schroeder page 56, after introducing anti commutation relations for the fields instead of commutation relations (in order to fix the negative energy problem as well as to have proper ...
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1answer
318 views

Quantizing the Dirac Field: which commutation relations are more fundamental?

When quantizing a system, what is the more (physically) fundamental commutation relation, $[q,p]$ or $[a,a^\dagger]$? (or are they completely equivalent?) For instance, in Peskin & Schroeder's ...
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1answer
101 views

Commutator evolution operator and position operator

Let $H= \frac{p^2}{2m}$, then I am supposed to calculate $[x,e^{-iHt}]$. My idea was to use $[x,p^n]=i \hbar n p^{n-1}$ and so I ended up by using the series for the exponential function with ...
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754 views

Is uncertainty principle a technical difficulty in measurement? [duplicate]

Is the uncertainty principle a technical difficulty in measurement or is it an intrinsic concept in quantum mechanics irrelevant of any measurement? Everyone knows the thought experiment of measuring ...
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556 views

How does the proof of operator commutativity work with non-continuous operators?

In some books, a proof that if two self-adjoint operators $A$ and $B$ share a common eigenbasis $\{\phi_n\}$, then they commute is given as follows : For any $\phi_n$, $$AB\ \phi_n = a_n\ ...
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1answer
76 views

How does the following commutator for measured observables and this operator relation imply the following relation?

$$ \hat{\Omega}_j{(\tilde{q}_j)}=\Omega_j(\tilde{q}_j-\hat{q}_j) $$ $$ [\hat{q}_j,\hat{q}_l]=ik_{jl} $$ Implies $$ [\hat{q}_j,\hat{\Omega}_l]= ...
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1answer
140 views

Product of position eigenvectors at different times

I've been thinking about this, and it might sound like a stupid question, but I can't seem to find an answer anywhere, here goes: Whenever we calculate expecation-values between two position ...
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Commutator $[\hat{p},F(\hat{x})]$ of Momentum $\hat{p}$ with a Position dependent function $F(\hat{x})$?

I heard from my GSI that the commutator of momentum with a position dependent quantity is always $-i\hbar$ times the derivative of the position dependent quantity. Can someone point me towards a ...
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1answer
699 views

A few simple questions about Grassmann numbers: commutation relations and derivatives

I'm trying to learn about Grassmann numbers from the book "Condensed Matter Field Theory" by Altland and Simons, but I am currently encountering some difficulties. I have several smaller questions ...
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68 views

How does linearity of a measurement imply that the commutator of all measured observables are $c$-numbers?

I really don't understand with the linearity conditions I have where this comes from.
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1answer
281 views

Why does $[xp_{y},x]$ commute?

I'm looking at a solution in my book that says $[xp_{y},x]$ commutes. Does bracket notation imply: $[A,B]=AB-BA$ so that $[xp_{y},x]=xp_{y}x-xxp_{y}$ Taking the comment from Max Graves and ...
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1answer
898 views

What exactly is the connection between the Jacobi and Bianchi identities

While reviewing some basic field theory, I once again encountered the Bianchi identity (in the context of electromagnetism). It can be written as $$\partial_{[\lambda}\partial_{[\mu}A_{\nu]]}=0$$ ...
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1answer
105 views

Fock state and corresponding relations for continuous momentum label

In Wikipedia I found following relation for Fock state: $$ \hat {a}_i| \{n_j\}_j\rangle ~=~ \sqrt{n}_i| \{n_j-\delta_{ij}\}_j\rangle, $$ where $n_j$ refers to the number of $j$'th particles. This ...
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Quantum Commutator Identities

Question: Prove that $p^2$ and ${\bf r}\cdot {\bf p}$ commute with every component of ${\bf L}$ using the identity $$[{\bf p},{\bf e}\cdot {\bf L}]=i\hbar\, ...
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3answers
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Why Don't the Ladder Operators Commute?

I have two problems with ladder operators. The first is that I feel they should somehow result in measurable things. The asymmetry of applying the plus operator versus the minus operator is very ...
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Do we need to expand the potential in a power series to show $[x, V(x)] = 0$?

Today in class (Intro to QM) we went over a couple of commutators. Among them was $[x, V]$, where $V=V(x)$ is a potential. What the teacher said to prove this is zero was: let's assume $V$ is analytic ...
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Proving that $i\hbar\frac{\partial}{\partial \mathbf{p}}$ is the operator of $\mathbf{x}$ in momentum space

How can I prove that $i\hbar\frac{\partial}{\partial \mathbf{p}}$ is the operator of $\mathbf{x}$ in momentum space?
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161 views

Quantum Hamiltonian commuting with the Pauli-Runge vector

I have to prove that $[A_j, H] = 0$, with; $$\vec{A} = \frac{1}{2Ze^{2}m}(\vec{L} \times \vec{P} - \vec{p} \times \vec{L}) + \frac{\vec{r}}{r}$$ $$H = \frac{p^2}{2m} - \frac{Ze^2}{r}$$ And, $Z, e, ...
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780 views

Simple Quantum Mechanics Question about The Commutator of Translation Operators

Say there is $\hat{J} = \exp[-i \hat{p} l/ \hbar]$ and $\hat{U}= \exp[-i\hat{H}t/ \hbar]$, where $\hat{H}$ is time-independent. Can we say anything about $[\hat{J},\hat{U}]$? Is it zero? How do we ...
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Commutator problem

I am trying to calculate the following commutator $$[\mathcal{H}_0(r',t'),\psi(r,t)]_-$$ where $\mathcal{H}_0 = (\frac{1}{2m}\nabla^2 + e\mathbf{A}(r',t'))^2 + e\phi(r',t') - \mu$, and $\mu$ is the ...
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Prove $[A,B^n] = nB^{n-1}[A,B]$

I am trying to show that $[A,B^n] = nB^{n-1}[A,B]$ where A and B are two Hermitian operators that commute with their commutator. However, I am running into a little problem and would like a hint of ...
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3answers
416 views

$\mathrm{Tr}(XY) = \mathrm{Tr}(YX)$?

I'm trying to understand the Dirac notation to understand quantum mechanics better. I'm trying to show the above relation using the Dirac notation. Given $$\mathrm{Tr}(X)~=~\sum_j\langle ...
4
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2answers
297 views

Star product of two commuting spinors

Ok so this might be a very stupid and trivial question but I have spent a couple of hours on this little problem. I am trying to derive a simple formula in a paper. We have a real commuting spinorial ...
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1answer
256 views

Spinor irreducible reps of the Lorentz group and their algebra

Antisymmetric tensor of rank two can be connected with spinor formalism by the formula $$ M_{\mu \nu} = \frac{1}{2}(\sigma_{\mu \nu})^{\alpha \beta}h_{(\alpha \beta )} - \frac{1}{2}(\sigma_{\mu ...
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1answer
614 views

Do mutual eigenkets imply commutation of two operators?

I have been working on this question. I have solved it, and I would like to check whether my line of reasoning is right or wrong Question: Prove that if there exists a mutual complete set of ...
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546 views

Heisenberg picture of QM as a result of Hamilton formalism

Consider the formula for the total time-derivative of a physical value in Poisson's formalism: $$\tag{1} \frac{dA}{dt} = -\{H, A\}_{P.B.} + \frac{\partial A}{\partial t}, $$ where $\{A, B\}_{P.B.}$ is ...
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91 views

Do commutation relations exist between superfields?

To quantize a theory, Klein gordon field for example, commutation relations are stablished. Or anticommuting ones in the fermionic case. If I have the Wess.Zumino model or the free model: ...
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59 views

commutators in an uncertainty relationship derived from a partition function?

The maximum information principle for the discrete case gives rise to a partition function (>>> see details here) $$Z(\lambda_1,\ldots, \lambda_m) = \sum_{i=1}^n \exp\left[\lambda_1 f_1(x_i) + \cdots ...
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82 views

commutator to entropy in an uncertainty relationship?

Question: Does there exist a commutator to entropy in an uncertainty relationship? Similar Energy and time for instance.
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Full time-derivative of a function and Schrodinger equation

From Hamiltonian formalism there is well known equation, $$ \frac{d F}{dt} = \frac{\partial F}{\partial t} + \{F, H\}_{PB}, $$ where $ \{H, F\}_{PB}$ is the Poisson bracket. After using Hamiltonian ...
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Eigenstate of position+momentum?

I'm studying Quantum Mechanics on my own, so I'm bound to have alot of wrong ideas - please be forgiving! Recently, I was thinking about the quantum mechanical assertion (postulate?) that states with ...
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690 views

What is the most general expression for the coordinate representation of momentum operator?

I have a question about deriving the coordinate representation of momentum operator from the commutation relation, $[x,p]= i$. One derivation (ref W. Greiner's Quantum Mechanics: An Introduction, 4th ...
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226 views

Operator on Function of Momentum (QM)

I have exactly 0 clue on how to start this problem, but I would be forever grateful for a hint in the right direction. Given the operators $\hat x=x$ and $\hat p=-i\hbar \frac{d}{dx}$, prove the ...
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2answers
565 views

Unitary spacetime translation operator

Srednicki writes: We can make this a little fancier by defining the unitary spacetime translation operator $$ T(a) \equiv \exp(-iP^\mu a_\mu/ \hbar) $$ Then we have $$ T(a)^{-1} \phi(x) T(a) = ...
4
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1answer
238 views

Time-ordered Derivative and Equal-time Commutator

In Green, Schwarz & Witten Superstring theory, Vol. I, page 141, I don't understand how pulling the derivative inside the Time-ordered product can give an Equal-time Commutator: $$\tag{3.2.44} ...
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1answer
892 views

What physical significance has the Heisenberg Group?

I read that the canonical commutation relation between momentum and position can be seen as the Lie Algebra of the Heisenberg group. While I get why the commutation relations of momentum and momentum, ...
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307 views

The correspondence between Poisson bracket and Commutators in Quantum Mechanics

I don't understand canonical quantization. In passing from classical to quantum, one replaces the Poisson brackets with the commutators. I don't really understand this. How can we generally show that ...
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1answer
921 views

Commutator of Lorentz boost generators : visual interpretation

I have always struggled to visualize the correctness of the commutation relation for the generators of the boost in the Lorentz group. We have $$[K_i,K_j] = i \epsilon_{ijk} L_k$$ I fail to picture ...
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559 views

Commutator with a square root

How to find the commutator $[a, \sqrt{a^\dagger a}]$? Here $a$ is a usual bosonic annihilation operator, and $[a, a^\dagger] = 1$. The first thing I tried is $$ [x,A] = [x, \sqrt{A}]\sqrt{A} + ...
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672 views

Imposing anti-commutation relations on fermionic quasi-particles

In many theories of CMT, we assume the nature of quasi-particles (without giving proper justifications). For example, we assume nature of quasi-particles to be fermionic in case of a interacting ...
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201 views

Are higher order mixed partial derivatives of wave function with different ordination equal?

For example, given two operators: $$A = \frac{\partial}{\partial x}+\frac{\partial}{\partial y},$$ $$B =\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2} + 1.$$ Deriving commutator ...
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Deriving the Angular Momentum Commutator Relations by using $\epsilon_{ijk}$ Identities

I've been trying to derive the relation $$[\hat L_i,\hat L_j] = i\hbar\epsilon_{ijk} \hat L_k $$ without doing each permutation of ${x,y,z}$ individually, but I'm not really getting anywhere. ...
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2answers
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Proof $\left[ {\hat H,{{\hat p}_i}} \right] = - \frac{\hbar }{i}\frac{{\partial \hat H}}{{\partial {{\hat q}_i}}}$ [closed]

I have a problem with the Hamiltonian, I don't think anything to solve it!! So could you give me some hints! Knowing that: $$\left[ {{{\hat p}_i},{{\hat q}_k}} \right] = \frac{\hbar }{i}{\delta ...
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1answer
160 views

Energy eigenvalues of a Q.H.Oscillator with $[\hat{H},\hat{a}] = -\hbar \omega \hat{a}$ and $[\hat{H},\hat{a}^\dagger] = \hbar \omega \hat{a}^\dagger$

I just finished deriving the commutators: \begin{align} [\hat{H}, \hat{a}] &= -\hbar \omega \hat{a}\\ [\hat{H}, \hat{a}^\dagger] &= \hbar \omega \hat{a}^\dagger\\ \end{align} On the ...
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2answers
715 views

Proof for commutator relation $[\hat{H},\hat{a}] = - \hbar \omega \hat{a}$

I know how to derive below equations found on wikipedia and have done it myselt too: \begin{align} \hat{H} &= \hbar \omega \left(\hat{a}^\dagger\hat{a} + \frac{1}{2}\right)\\ \hat{H} &= ...
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2answers
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Translator Operator

In Modern Quantum Mechanics by Sakurai, at page 46 while deriving commutator of translator operator with position operator, he uses $$\left| x+dx\right\rangle \simeq \left| x \right\rangle.$$ But for ...