# Tagged Questions

The conserved quantity arising from a rotational invariance. Combine with rotational-dynamics for the classical mechanics approach and quantum-mechanics for the QM interpretation

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### Angular Momentum in Quantum mechanics

In Gasiorowicz's Quantum Physics, we determined the relation: $$L_z | l,m\rangle= \hbar m | l,m \rangle$$ I would like to determine: $\langle l,m_1 | L_x | l,m_2 \rangle$ I thought about expressing ...
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### Angular momentum of a black hole

I recently read this Phys.SE post and, since I didn't know that black holes had a spin, a question came to my mind: how can I calculate the spin velocity of a black hole? Does mass or radius affects ...
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### Collision of Discs and Snooker Kicks

I woke up this morning thinking about spinning discs. Could someone verify whether my reasoning below is correct? Problem 1 Suppose have two identical uniform discs constrained to move in a plane. ...
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How to prove that $$[\hat{L}^2,\hat{\textbf{p}}] = i\hbar(\hat{\textbf{p}}\times\hat{\textbf{L}} - \hat{\textbf{L}} \times \hat{\textbf{p}})$$ I tried to expand $\hat{L}^2$: [\hat{L}^2,\hat{\... 2answers 1k views ### Describing a motion of gyroscope with gimbal Can you tell be how to set the equations to describe the motion of this machine in movie "Contact": https://www.youtube.com/watch?v=TSaO9VGjLXc This is gyroscope with gimbal, am I right? 1answer 239 views ### Angular momentum of two rotating spheres I am trying to calculate an instantaneous merger of two rotating spheres into one. Two spheres each rotating around their own axis of rotation (which are generally not aligned) and moving relative to ... 1answer 84 views ### Where do L_+ and L_- live, if not in \mathfrak{so(3)}? This question is continuation to the previous post. The lie algebra of  \mathfrak{so(3)}  is real Lie-algebra and hence,  L_{\pm} = L_1 \pm i L_2  don't belong to  \mathfrak{so(3)} . However, ... 3answers 2k views ### Instantaneous angular momentum of a disc Suppose we have a disk of radius r and mass m travelling at velocity v. I want to calculate the instantaneous angular momentum with axis through the edge of the disc (on the circumference). ... 0answers 354 views ### Calculating the J value for atomic terms, having a lot of trouble with this. Already attempted I am trying to understand this, and want to be very very clear. This is a homework question but I already attempted to answer it, so please don't put this question on hold. The question What ... 3answers 4k views ### What's is the origin of Orbital Angular Momentum of electrons in atoms? Consider the Hydrogen 1s electron. We know that, in the quantum picture, the electron isn't orbiting or rotating at all, rather we simply state that the electron is spread over the entire space with ... 1answer 116 views ### Eigenstates of coupled Angular Momentum SO I have a hamiltonian:H=\alpha J_1\cdot J_2$$And I am asked to find the eigenstates and eigenvalues of this Hamiltonian in terms of products of the eigenstates of the z components of the ... 0answers 106 views ### Angular momentums addition in QM We know that the spatial inversion parity for eigenfunctions of \hat {L}_{z} operator (spherical functions) is (-1)^{l}, where l refers to angular momentum. So for product of two eigenfunctions ... 1answer 1k views ### Diagonal Hamiltonian of 3 Spin 1/2 Particles I have three Spin 1/2 Particles and a Hamiltonian given by$$H=A(S_1\cdot S_2)+B(S_2\cdot S_3+S_1\cdot S_3)$$In order to find the energy spectrum, I want to diagonalize H in terms of (S_1+S_2+S_3)^2... 2answers 1k views ### Angular Momentum Operator in Quantum Field Theory I'm following along with David Tong's QFT course and am trying to derive the result shown in question 6 on his 2nd problem sheet, but am running into problems when applying it to the free real scalar ... 3answers 725 views ### Addition of spin angular momentum for massless particles How do I add the spin angular momentum of massless particles, like photons, where only the transverse polarizations are allowed? If all three polarizations were allowed, this would be an easy ... 1answer 278 views ### Why have n, \ell, m_\ell, m_s been picked as quantum number symbols \mathbf{\text{in this order}}? I’m learning about electron configurations and don’t quite understand why n, \ell, m_\ell, m_s have been picked as symbols for the quantum numbers. As far as I understand it, the principal ... 1answer 596 views ### Finding angular momentum about the center of mass? If we have a couple of particles of an equal, unknown mass:$$r_{+} = (c + e^{-Bt} \cos({\theta}))\textbf{x} + (d + e^{-Bt} \sin({\theta}))\textbf{y}r_{-} = (c - e^{-Bt} \cos({\theta}))\textbf{...
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I am tring to find the eigenvectors of a two spin system, with $j_1=3/2$ and $j_2=1/2$. To start, $$m_1 =-3/2,-1/2,1/2,3/2$$ $$m_2=-1/2,1/2$$ For $j_1$, there are 4 possible states, and 2 possible ...
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### Difference between expectation values of $L^2$, $L_z$ and measuring $L^2$, $L_z$

I was given with this hydrogen radial wavefunction $$R_{21} =\left(\sqrt{\frac{1}{3}}Y^0_1 + \sqrt{\frac{2}{3}}Y^1_1\right)$$ and was asked to find a) What are the expectation values of the $L^2$,...
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### How does $\bar{r}\times(\bar{\nabla}\times) - \bar{\nabla}\times(\bar{r}\times)$ relate to the orbital angular momentum operator?

When I attempted to calculate the following by hand $$\bar{r}\times(\bar{\nabla}\times\bar{F}) - \bar{\nabla}\times(\bar{r}\times\bar{F}),$$ I noticed some of the terms I extracted looked similar to ...
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### Construction of Angular Momentum eigenvectors

I have a problem that asks (verbatim) Carryout the construction of the eigenvectors of total angular momentum and its z component for $j_1$=3/2 and $j_2$=1/2 I am not completely sure where to ...
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### What makes a wrist-energized gyroscope rotate faster?

I'm considering a wrist-energized gyroscope, shown below (after my daughter let it fall and it broke open). That one was sold as Roller Ball, but variants are known as Powerball, DynaBee, Dynaball, ...
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### What is $\langle \sigma_\mu \rangle$ $\langle \sigma_\mu \rangle$ for the Pauli Matrices?

What is \begin{align} \sum_{\mu=0}^{3} \langle \sigma_{\mu} \rangle^2 = ? \end{align} $\sigma_{\mu}$ are the Pauli matrices. The Bra-Ket notation is used in this question: \begin{align} \langle \...
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### Thrust to Weight ratio in Space with an off set CoM

With regards to this thread, Thrust center in space My question is, if the thrust to weight ratio was increased so that it was much higher than the weighted mass of the sphere (ship), would the ...
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### Imagine a 50-mile tower spanning from desert floor to the Karman Line

CORRECTION: The structure weighs 1.568E15 kg.s Does the structure effect the equilibrium of earth's rotation? Would momentum from Earth's rotation apply lateral force to the structure? What else do ...
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### How can a singularity in a black hole rotate if it's just a point?

I guess nobody really knows the true nature of black holes, however, based on everything I know about black holes, there is a "singularity" at their center, which has finite mass but is infinitely ...
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### QM: How to compute position/momentum relation in polar coordinates

So if we are working in one dimensional space, we have the formula: $$\langle x|p\rangle = \frac{1}{\sqrt{2\pi\hbar}} e^{ipx/\hbar}$$ Suppose instead we are confined to a circle of radius $R$ so that ...