A mathematical construct used to study the effect of applying two operators in succession.

learn more… | top users | synonyms (1)

3
votes
2answers
164 views

Commutator not transitive

I noticed the following: $$[L_{+},L^2]=0,\qquad [L_{+},L_3]\neq 0,\qquad [L^2,L_3]=0.$$ This would suggest, that $L^2,L_+$ have a common system of eigenfunctions, and so do $L^2,L_3$, but $L_+,L_3$ ...
1
vote
1answer
154 views

Tricky operator identity: $[L^2,[L^2,\vec{r}]]=2 \hbar ^2 \{ L^2, \vec{r}\}$?

This operator identity showed up in a course I was taking, and it was given without proof. $$[L^2,[L^2,\vec{r}]]=2 \hbar ^2 \{ L^2, \vec{r}\}$$ The curly brackets denote the anticommutator, $AB+BA$. ...
0
votes
1answer
140 views

Commutators in bra-ket notation

2-d Hilbert space, with 2 (orthogonal) kets $|a\rangle$ and $|b\rangle$ Operator $A=|a\rangle\langle b| + |b\rangle\langle a| $ Operator $B=-i|a\rangle\langle b| +i|b\rangle\langle a| $ Commutator ...
0
votes
0answers
88 views

Change of QM Momentum operator under coordinate transformation

Can any one please let me know what is the general procedure to construct the momentum operator under some coordinate transformation? For example, I understand that if ...
3
votes
2answers
299 views

Commutator of $L^2$ and $X^2$, $P^2$

In our quantum mechanics script, it states that $[L^2, X^2] = 0$ and $[L^2, P^2] = 0$, therefore for the following Hamiltonan $$H = \frac{P^2}{2m} + V(X^2)$$ it is that $[H, L^2] = 0$ therefore $H$ ...
2
votes
1answer
180 views

Can the quantum angular momentum operator be derived from its commutation relations with position and momentum?

Exercise 12.2.2 in Shankar's Principles of Quantum Mechanics asks to derive the expression for the angular momentum operator $L_z$ \begin{equation} L_z = XP_y-YP_x \end{equation} using its ...
2
votes
1answer
91 views

Does this commutation relation hold?

I was wondering whether it is true that $[L_x^2,x^2+y^2+z^2]=0$. I could not find it in the internet and therefore I wanted to ask here whether anybody here knows that this is true or false.
5
votes
0answers
83 views

Commutator as a time-ordered product

I'm reading through Seiberg and Witten's paper "String Theory and Noncommutative Geometry," and one part in $\S$2.1 isn't quite clear to me. (Sorry, in advance, for the length.) My question is about ...
5
votes
3answers
260 views

Evaluating commutator of $[\operatorname{sign}(X),\, \operatorname{sign}(P)]$

I wish to evaluate the following commutator: $[\operatorname{sign}(X),\, \operatorname{sign}(P)]$. Is there a general method for evaluating $[\operatorname{f}(X), \operatorname{f}(P)]$? I thought of a ...
1
vote
2answers
314 views

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 ...
2
votes
3answers
193 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 ...
-3
votes
1answer
63 views

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: ...
2
votes
0answers
80 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) = ...
1
vote
1answer
110 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 ...
4
votes
1answer
170 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 ...
1
vote
1answer
69 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 ...
4
votes
4answers
360 views

Is uncertainty principle a technical difficulty in measurement?

I have searched for an answer to this question on physics SE but I have not seen a question in which it is addressed properly. Please let me know if there is an answer already. My question briefly ...
6
votes
2answers
246 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\ ...
1
vote
1answer
63 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]= ...
2
votes
1answer
105 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 ...
1
vote
1answer
982 views

Commutator of Momentum with a Position dependent function

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 ...
1
vote
1answer
207 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 ...
1
vote
0answers
59 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.
1
vote
1answer
155 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 ...
3
votes
1answer
311 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 $$\partial_{[\lambda}\partial_{[\mu}A_{\nu]]}=0$$ ...
2
votes
1answer
72 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 ...
0
votes
2answers
560 views

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\, ...
2
votes
3answers
721 views

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 ...
5
votes
2answers
154 views

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 ...
3
votes
2answers
407 views

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

How to prove that $i\hbar\frac{\partial}{\partial \mathbf{p}}$ is the operator of $\mathbf{x}$ in momentum space?
0
votes
0answers
99 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, ...
1
vote
3answers
389 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 ...
1
vote
0answers
119 views

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 ...
1
vote
2answers
657 views

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 ...
1
vote
3answers
267 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
votes
2answers
231 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 ...
1
vote
1answer
164 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 ...
5
votes
1answer
318 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 ...
4
votes
2answers
340 views

Heisenberg picture of QM as a result of Hamilton formalism

Let's have formula of full time-derivative of physical value in Poisson's formalism: $$\tag{1} \frac{df}{dt} = -[H, f]_{P. br.} + \frac{\partial f}{\partial t}, $$ where $[A, B]_{P. br.}$ is Poisson's ...
4
votes
0answers
80 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: ...
1
vote
0answers
55 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 ...
2
votes
0answers
76 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.
1
vote
2answers
321 views

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 ...
8
votes
3answers
742 views

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 ...
4
votes
2answers
293 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 ...
0
votes
2answers
129 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 ...
3
votes
2answers
294 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
votes
1answer
134 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} ...
3
votes
1answer
476 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, ...
2
votes
0answers
194 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 ...