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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11
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269 views

How to evaluate this sum of coupling coefficients?

I would like to evaluate the following summation of Clebsch-Gordan and Wigner 6-j symbols in closed form: $$\sum_{l,m} C_{l_2,m_2,l_1,m_1}^{l,m} C_{\lambda_2,\mu_2,\lambda_1,\mu_1}^{l,m} \left\{ ...
6
votes
0answers
236 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: ...
5
votes
0answers
84 views
+50

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 ...
4
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0answers
64 views

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, ...
4
votes
0answers
64 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 ...
3
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0answers
89 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
votes
0answers
164 views

Right-angle lever paradox in special relativity

I remember to have read somewhere an interesting special relativity "paradox" considering two perpendicular rods $A$ and $B$ of equal proper length $L$ fixed at point $O$. In the "rest" frame equal ...
3
votes
0answers
192 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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0answers
55 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 ...
2
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0answers
67 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 ...
2
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0answers
42 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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0answers
61 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 ...
2
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215 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
votes
0answers
221 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 ...
1
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0answers
55 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, ...
1
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0answers
46 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 ...
1
vote
0answers
42 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 ...
1
vote
0answers
38 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 ...
1
vote
0answers
67 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 ...
1
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0answers
45 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 ...
1
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0answers
120 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) ...
1
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0answers
183 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 ...
1
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0answers
176 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, ...
1
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0answers
37 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 ...
1
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0answers
143 views

Hamiltonian Nuclear collective model

Let $\mathscr{I}_3$ and $\mathscr{I}$ are the moments of inertia for rotations about symmetry axis 3 and about an axis perpendicular to it , and I is the angular momentum operator with components ...
1
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55 views

Two impulses on body

Say I have a physics simulation with a single body with three degrees of freedom (two axis and rotation). I have two impulses of equal magnitude in opposite directions. When applying these impulses in ...
1
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0answers
104 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 ...
1
vote
0answers
851 views

How is multiplicity given by 2S+1?

Suppose there are two electrons in an atom with $s_1 = \frac{1}{2}$, $l_1 = 1$ and $s_2 = \frac{1}{2}$, $l_2 = 1$. Hence the total $S$ (of the atom) may be +1 or 0. And total $L$ is either $+2$, $+1$ ...
1
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43 views

Calculate Rotational Intertia

If a can of soup, and a can of beans (tightly packed), are set in a race down a rough hill (has friction), the soup wins, because the inside of the can (soup) is not drawing energy from the system. ...
1
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0answers
195 views

Angular momentum confusion

Could somebody please explain what is going on here? We have a system of two indistinguishable spin-1 bosons. We shall adopt the center of mass frame. Let $S$ = total spin $L$ = relative orbital ...
1
vote
0answers
62 views

Wigner $3j$ symbols

I am trying to determine the expansion that requires using $3j$ symbols; however, I am running into some conceptual snags. First, the expansion produces symbols that have m's that do not agree with ...
0
votes
0answers
18 views

Lateral forces on earths surface

If a giant stone sphere was sitting on the earths surface, would it experience a later force due to the earths rotation? For example my intuition tells me that if there was a sphere a mile high ...
0
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0answers
18 views

Precessing gyroscope torque

I've been studying for an upcoming test and am completely stumped. Why is it that the weight of a gyroscope causes the torque to be directed along the horizontal as opposed to the vertical?
0
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0answers
25 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. ...
0
votes
0answers
25 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 ...
0
votes
0answers
18 views

Angular Momentum Expectation in Magnetic Field

I am trying to find the time dependent expectation value for J ($\langle J(t) \rangle$) for a spin 3/2 particle in a uniform magnetic field (in the z direction). My method is as follows: ...
0
votes
0answers
36 views

Angular momentum of 2d harmonic oscillator

So, I have the problem of determining the spectrum of H and L, in terms of creation and annihilation operators of angular momentum... The problem goes along with what is happening on this page. ...
0
votes
0answers
69 views

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 ...
0
votes
0answers
23 views

Spherical Harmonic projection on axis

I am trying to solve for the Spherical harmonics $Y^m_{l=1}$ with a second axis at an angle $\alpha$ with respect to the z axis. Then this can be used to find the probability that a particle with ...
0
votes
0answers
64 views

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 ...
0
votes
0answers
36 views

Transforming the angular momentum operator (from spherical to Euler and the internal valence angle) for a reduced 3-body problem

Let me introduce the problem: In a two electron fixed nucleus problem the "body" is the atom, whose electrons are located relative to the nucleus by the coordinates $r_1$ and $r_2$, and the angle ...
0
votes
0answers
34 views

Angular momentum operator for 2 dimensions?

Recently I get the task to build (2 + 1)-Dirac theory. I wrote corresponding Dirac equation in a form $$ (i\sigma_{0}\partial_{0} + i\sigma_{1}\partial_{1} + i\sigma_{2}\partial_{2} - m)\Psi = 0, $$ ...
0
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0answers
63 views

Intuitive explanation for angular momentum uncertainty?

The basic commutator relation $$[J_1,J_2]=i \hbar J_3$$ of quantum mechanics yields the uncertainty relation $$\Delta(J_1)\Delta(J_2)\ge \frac{\hbar}{2}|\langle J_3\rangle|.$$ However, unlike the ...
0
votes
0answers
36 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 ...
0
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0answers
38 views

What is the difference between orbital angular momentum of photons and their polarization

What is the difference of OAM of photons and their polarization?
0
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104 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 ...
0
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0answers
50 views

Deriving the relationship between change in Energy and change in angular momentum in orbital repulsion

I know that for a test particle in between a planet and its satellite, there is a direct relationship between the change in energy and change in angular momentum, when the particle's orbit nears the ...
0
votes
0answers
46 views

Can a single atom have angular momentum? At what angel does a single atom reflect a single photon?

Groups of atoms, say two of them, can have angular momentum as a group, but only because they individually have linear momentum and are bound together through a force that causes them to pull on each ...
0
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0answers
65 views

Angular Momentum with Upper Index

I am asked to show $[L^2,L_i] = 0 $, but with the definition : $L^2 \equiv L_i L^i$ I tried this: $[L_i L^i,L_i] = L_i [L^i,L_i] + [L_i,L_i]L^i$ We know that : $[L_i,L_i]$ = 0 , so we have, $[L_i ...
0
votes
0answers
73 views

Photons angular momentum / spin

I have a textbook that says that photons have a spin of absolute value $\hbar$ and at some other point, they say that it has angular momentum of absolute value $\hbar$. Now, since they are different ...