# Tagged Questions

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

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### Moyal Product in Non Commutative Quantum Mechanics

Can someone please explain me what is a Moyal product? Also, how does putting $$X_a(\psi) ~=~ x_a\star\psi$$ realise $$[X_a,X_b]=i\theta_{ab}{\bf 1}?$$ Ref: Quantum mechanics on non-commutative ...
My guess it should look something like this: $c_\sigma = ... 0answers 153 views ### QFT basics for Klein-Gordon fields I am teaching myself QFT from Peskin for next years maths course and I have two questions: What is a c-number? Is it a complex number, and if so why does it mean, ... 2answers 529 views ### Quantum commutator I'm given this commutator: $$\left[PXP,P\right]$$ Being$P\psi=-i\hbar\partial_x\psi$, and$X\psi=x\psi$I've solved it in two ways, the first one is just aplying the commutator to some function ... 1answer 669 views ### Quantum mechanical analogue of conjugate momentum In classical mechanics, we define the concept of canonical momentum conjugate to a given generalised position coordinate. This quantity is the partial derivative of the Lagrangian of the system, with ... 1answer 406 views ### Find$\hat{x}$operator given$\hat{p}$operator This is problem$1.2$of Molecular Quantum Mechanics by Atkins, 4th edition. I'm given the momentum operator $$p=\sqrt{\frac{\hbar}{2m}}(A+B)$$ with $$[A,B]=1$$ and I need to find$x$in this ... 1answer 494 views ### Klein-Gordon Canonical Commutation Relation (CCR) In the complex Klein-Gordon field we regard as dynamical variables the field$\phi$, the complex conjugate of the field$\phi^*$, and the momenta$\pi$,$\pi^*$. I can't see how should arise the ... 1answer 236 views ### State space of QFT, CCR and quantization, and the spectrum of a field operator? In the canonical quantization of fields, CCR is postulated as (for scalar boson field ): $$[\phi(x),\pi(y)]=i\delta(x-y)\qquad\qquad(1)$$ in analogy with the ordinary QM commutation relation: ... 1answer 781 views ### Evaluate Commutator with Partial Derivatives I need to evaluate the following commutator...$[x(\frac{\partial}{\partial y})-y(\frac{\partial}{\partial x}),y(\frac{\partial}{\partial z})-z(\frac{\partial}{\partial y})]$i tried applying an ... 2answers 569 views ### Causality and Quantum Field Theory I have a problem with proof of causality in Peskin & Schroeder, An Introduction to QFT, page 28. To avoid confusion I use three vectors notation, rewriting the Eq. (2.53) for$y=0$as follows: ... 1answer 455 views ### What conservation law corresponds to this local$U(1)\$ symmetry of the CCR?
In relativistic quantum field theories (QFT), $$[\phi(x),\phi^\dagger(y)] = 0 \;\;\mathrm{if}\;\; (x-y)^2<0$$ On the other hand, even for space-like separation $$\phi(x)\phi^\dagger(y)\ne0.$$ ...