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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266 views

Coupling Coefficients in SO(4)

I have two equations (from two distinct authors) for the decomposition of a coupling coefficient of SO(4) (i.e. Wigner 3j-symbol for SO(4)). In the first: ...
4
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127 views

Spin via Change of Phase

Thinking of spin as arising from a change in the phase of a wave function: The angular momentum is defined by the change of the phase of the wave function under rotations, which may come from the ...
4
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120 views

Interchange symmetry for states with identical particles

I was reading this web page about interchange symmetry for states with identical particles here: http://quantummechanics.ucsd.edu/ph130a/130_notes/node317.html The article states that the highest ...
4
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95 views

On an Uncertainty Relation for Angular Variables

I'm looking for a proof of the Angular Momentum - Angle uncertainty relation $$\frac{\Delta L \Delta \theta}{1-(3/\pi^2)\Delta \theta^2} \geq \frac{\hbar}{2}$$ which does not involve solving the ...
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50 views

What is the relationship between the angular speed and the diameter of the eye of a water vortex?

In a water vortex formed in a plastic bottle, what is the relationship between the angular speed of the water and the diameter of the hole in the cap? I would have expected that according to the ...
3
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59 views

When Did Structures Begin to Orbit in the Universe?

In Big Bang Cosmology, I am familiar with the radiation era, and the moment of last scattering, but I am just not familiar with the physics of orbits, or of the “falling outward,“ if you will, that ...
3
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94 views

Why are some things attracted to you but others repelled by you in rotating reference frames?

Note that my understanding of general-relativity is rudimentary. If I understand right, it means that basically any reference frame can be considered stationary, but there may be random gravitational ...
3
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107 views

What are the assumptions behind “term symbols”?

In multi-electron atoms, the electronic state of the optically active "subshell" is often expressed in "term symbols" notation. I.e. $^{2S+1}L_J$. This presumes that the system of electrons has ...
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189 views

3 Axis Gryroscope with forced Precession and Limits of Motion

I am working a problem concerning a 3 axis gryoscope, the spinning mass is a magnet (dipole). This is part of a optical sensing device. The inner gimbal is for pitch rotation, and the outer gimbal is ...
3
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257 views

What is the Landé g factor?

What is the Landé g factor? I know that it gives the relation between magnetic moment and angular moment, but i wanted to know why are those magnitudes related to each other and why is the magnetic ...
2
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32 views

Atomic physics, determining levels and terms

In atomic physics I understand there a configurations, terms and levels. I think levels for instance appear because of spin-orbit interactions, so that terms are split. But I'm confused about the ...
2
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25 views

Simplification of matrix-element given the Wigner-Eckardt theorem and Clebsch-Gordon coefficients of a 1,1/2 system

How can I simplify the following matrix-elements $$\left\langle 1,1/2;m_1,m_2\left| S \right| 1,1/2;m_1^{'},m_2^{'} \right\rangle$$ given the Wigner-Eckard theorem $$\left\langle j,m|S|j^{'},m^{'} ...
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64 views

How to get from momentum to force

Situation: I have a solid object (black) attached to a rod (blue) as shown below: The rod is fixed at the top. The solid object is a cylinder as shown, with a rate of rotation ...
2
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0answers
69 views

Protoplanetary disks, angular momentum and prograde orbits

So you've got a protoplanetary disk and you're going to gravitate yourself some planets together. The disk is made up of the usual planetary system stuff, dust and gas and whatnot, orbiting a common ...
2
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0answers
95 views

Lorentz transformation - Bjorken & Drell

I'm trying to derive (14.25) in Bjorken & Drell (B&D) QFT. This is $$\tag{14.25}U(\epsilon)A^\mu(x)U^{-1}(\epsilon) = A^\mu(x') - \epsilon^{\mu\nu}A_\nu(x') + \frac{\partial ...
2
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62 views

Do coupled angular momentum eigenstates depend on angle between particle positions?

Suppose we have two particles with positions $\vec r_1$ and $\vec r_2$. The corresponding coupled angular momentum eigenstates would be expressed via Clebsch-Gordan coefficients as (according to this ...
2
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119 views

Para and ortho hydrogen angular momentum values

In Wikipedia, it is said that: Orthohydrogen, with symmetric nuclear spin functions, can only have rotational wavefunctions that are antisymmetric with respect to permutation of the two protons. ...
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149 views

The relationship between angular and linear momentum

Why is orbital angular momentum not 0 when spin and linear momentum are not collinear? Why can it be 0 when spin and linear momentum are parallel? Like in the example of a scalar field at rest ...
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68 views

Angular momentum in an accretion disk

I need to plot the time evolution of the total angular momentum in an accretion disc. This confuses me because I thought this should be constant, since angular momentum has to be conserved? I'm given ...
2
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165 views

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. ...
2
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71 views

Stern Gerlach experiment - only two discrete beams?

The Stern Gerlach experiment was meant to prove the orbital quantization of electrons where there should be +ml,0,-ml states. So for l=2, there should be 5 beams. But they saw 2 beams, which was ...
2
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93 views

Spin of a decay product

A particle A decays into particles B, C and D. The spin of A, B and C particles is 1/2 each. What are the possible spins of particle D? My attempt is the following: Since B and C have spin 1/2 ...
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122 views

Closed-form equation for orientation and angular velocity over time

If a rigid body, rotating freely in 3d, experiences no friction or other external forces and has an initially diagonal inertia matrix $\mathbf{I}_0$ (with $I_{11}>I_{22}>I_{33}>0$) and ...
2
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344 views

Conservation of Angular Momentum: atomic transitions vs exciton decay

I have a question about the role of photon angular momentum in two different sets of selection rules: In atomic transitions within the dipole approximation, I've seen the selection rule as: $\Delta ...
2
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276 views

Why do control moment gyroscopes exhibit “torque amplification”?

There are a number of articles that describe the benefits of using control moment gyroscopes (CMGs) over reaction wheels in inertial navigation applications. One of the primary benefits of using a CMG ...
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73 views

What is this nested bracket notation?

The following is an excerpt from K. Varga's paper, Precise solution of few-body problems with stochastic variational method on correlated Gaussian basis: ...The function $θ_{LM_L}(\mathbf{x})$ in ...
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24 views

Expectation value in spin-orbit coupling

So I was just trying a question where it asked to find the Energy shift due to a spin-orbit coupling Hamiltonian to first order using perturbation theory. The Hamiltonian is $$H_{LS} = ...
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39 views

how to rotate scaled-vector (orientation) by scaled-vector (rotation)

Recently I seem to have gotten the physics-engine portion of my 3D simulation/game engine [apparently] working correctly. The most convenient way to store and compute position and orientation are in ...
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17 views

How to count total spin degeneracies for many spin one half particles?

Given the spin operator for particle $j$ \begin{align} \bar{S}_{j} = \left( \bigotimes_{k=1}^{j-1} I_{k} \right) \otimes \left(\tfrac{\hbar}{2}\bar{\sigma}\right)_{j} \otimes \left( ...
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41 views

Do the norms of the total and the orbital angular momentums commute? If yes, why is there a problem with 2p_{1/2}?

Question: For $\vec L$ the orbital angular momentum of an electron, $\bar S$ its spin, and $\vec J:=\vec L+\vec S$ the sum, do $\vec J^2$ and $\vec L^2$ commute? I assume it does: $[\vec J^2,\vec ...
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30 views

Dynamics inside a rotating spaceship

I have been thinking about this question for a long time and I din't find an explanation in the web for it lets say that we a rotating spaceship in the shape of cylinder, that is rotating in an ...
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63 views

Determining Parity of Decaying Quantum System

Show that a particle of spin $1$ cannot decay into two identical particles of spin $0$. The $\rho$-meson has spin $1$ and can decay into two spinless (spin-$0$) $\pi$-mesons, or pions, with ...
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69 views

Can spin angular momentum be understood as orbital angular momentum in extra dimensions?

It is to my understanding that in Kaluza-Klein theories the mass of particles can be understood as linear momentum in the extra dimensions. Let's consider in $\mathbb{R}^{1,3}\times{}B$ space-time a ...
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35 views

Rotational wave funtion of a nucleus

The rotational hamiltonian of an axially symmetric rotor is, in the intrinsic frame of the body, where the moment of inertia is diagonal, $$\mathcal{H} = \frac{\hslash^2}{2I} \left(J^2 - I_3^2\right) ...
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11 views

Pivotal door - how is the load distributed?

A pivotal door, where instead of the door hung or cantilevered from the hinges screwed to the frame, the door is hung using a top and bottom pivot. The bottom pivot assembly's the floor-spring is ...
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114 views

Can we prove Conservation of Angular Momentum without assuming internal forces are central?

The only proof I've seen of Conservation of Angular Momentum assumes that the internal forces of the system act along the line joining particles i and j (i.e. - all internal forces are central.) Can ...
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39 views

Asymptotics of the Wigner 6j Symbol

So, in doing some numerical computations in QFT, I've run into the following Wigner 6j-Symbol: $ \left\{ \begin{array}{ccc} x & J_1 & J_2 \\ \frac{N}{2} & \frac{N}{2} & \frac{N}{2} ...
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215 views

What is the relationship between Coriolis forces and Angular momentum?

Recently I did an assignment about helicopters, and during the oral exam we talked about Coriolis forces. I know where it comes from, and I've seen the derivation: Example In the case of helicopters, ...
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109 views

Spin 1/2 particles hamiltonian, addition of angular momentum confusion

Suppose I want to compute $S^{1}_z -S^{2}_z$ on a singlet state $|0,0>$. (where $S^{i}_z$ are two particles' spin operators). $$|0,0> = \frac{1}{\sqrt{2}} (|\frac{1}{2},-\frac{1}{2}> - ...
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68 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 ...
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67 views

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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182 views

Orbital angular momentum selection rules for three identical particles

I'm trying to figure out if there are selection rules for the total orbital angular momentum for a system of three identical particles, say bosons. For two identical bosons one can argue that the ...
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238 views

Can (quantum) angular momentum $L$ be zero?

I am trying to calculate the orbital magnetic moment, $\bar{\mu}$, for Sodium, which has an electron configuration of $1s^2 2s^2 2p^6 3s^1$. The full shells do not contribute to $\bar{L}$ and ...
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140 views

Rolling in 3D using Torque, Angular Momentum/Velocity

I'm stuggling to get a simulation working correctly. Below you can see what I am attempting to do. You can also view it here. Can you spot where I'm going wrong? (Calculated at start) 1) Inertia ...
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56 views

Larmor Precession - Determing frequency

Every time I go through some literature about Larmor Precession, i.e. the precession of orbit charged particle in the presence of a Magnetic Field. It doesn't give convincing arguements in calculating ...
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239 views

How does the expectation value of the square of angular momentum transform under translations?

In quantum mechanics the angular momentum operator is defined as $$ \mathbf{\hat L}=\mathbf{\hat x} \times \mathbf{ \hat p} $$ This definition explicitly depends on the choice of the origin of the ...
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257 views

Is total angular momentum conserved in particle interaction?

Imagine that two electrons interact by exchanging a virtual photon. I know that the total energy and linear momentum of the two electrons is conserved by the interaction. Is the total (orbital) ...
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254 views

Wave functions for 2D potential with spin interactions

So consider a 2D system with a circular potential and a spin-orbit interaction: $V(r) = V_0 \theta(r_0 - r) + c r_0 V_0 L_z S_z \delta(r-r_0)$ where $\theta$ is step function. So the operators ...
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252 views

Angular Momentum Conservation in Gravitational Interaction

thanks for any help. I'm trying to show that in a 2body problem, angular momentum is conserved given that $\dfrac{dp}{dt}=\dfrac{-GMm(rv)}{r³}$, where p is momentum, t time, G gravitational constant, ...
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40 views

QM -group reps and transforming wavefunctions

QM texts seem to have many ways of motivating the angular momentum operators and deriving the l and m quantum numbers . But the connection between physical rotaions in 3 dim space and an operator in ...