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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What is the physical importance of the commutation relations of angular momentum?

What is the physical meaning of these commutation relations: $$[L_{z},L_{\pm}]=\pm\hbar L_{\pm}\tag{1}$$ and $$[L_{+},L_{-}]=2\hbar L_{z} ~?\tag{2}$$
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37 views

Is spin an observable quantity for fundamental particles? [duplicate]

How we know that the spin of fundamental particles? For example spin-0, 1/2, 1,2.. What is the experimental facts about spin?
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1answer
58 views

Angular momentum of the electric field of a point-like electric charge and the magnetic field of a monopole

I am currently reading "Magnetic Monopoles" of Ya. Shnir. My problem is I can not retrieve a result the author provides in the first chapter of the first part. In this chapter, he studies the ...
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2answers
74 views

If a ball spinning on a rod hits another ball, what is conserved linear or angular momentum?

Suppose a 1-kg ball A is fixed to a spoke 0.2 m long, which is attached to an axle so that the ball can rotate (v=10m/s, KE=50J, $\omega$=50 rps, L=2, p=0) Now, there is a second ball B (m=1kg), ...
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28 views

Gravitational force and time dilation [closed]

Suppose the radius of the earth is reduced by half but the mass is same, then how long will it take to complete one rotation, 24, 48, 12 or 6 h.? please give the mathematical relations and solution. ...
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63 views

How to get rotation speed after disk-disk collision?

Assume two circular disks A and B collide. They have both initial linear momentum and angular momentum. If their surface has no friction, their angular velocity does not change after collision, so I ...
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74 views

Spin-½ and beyond: Measuring spin components other than ± ħ / 2: How to formulate the probability function?

It is my understanding that in quantum mechanics (for 1/2 spin particles) the probability function that describes the direction of a particle's spin state is proportional to the overlap of the ...
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40 views

How do I calculate integer and half integer spin? [closed]

How do I calculate integer and half integer spin, and how do I use the calculations?
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148 views

Why do we look at the representations of $SO(3)$ in QM?

I have a bit of an understanding issue why the representations of $SO(3)$ are so important for Quantum Mechanics. When looking at its Irreps one gets the Spin and Angular Momentum operators and thus ...
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2answers
58 views

Why do rotations of a multicomponent state function take this form?

I am reading Leslie Ballentine's Quantum Mechanics, section 7.2, which is all about the explicit form of the Angular Momentum operators. I understand how he gets the form for the single component ...
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77 views

How does angular momentum transfer between a planet and its moon?

Could you explain how a moon draws angular momentum from a planet? I know that the gravitational force transfers momentum, but I don't understand the mechanics behind it.
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136 views

Angular momentum eigenstates

My textbook says that if $L^2$ is the square of the angular momentum and if it's eigenstate is $|\alpha,\beta>$ then its eigenvalue is $\hbar^2\alpha$ i.e. ...
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111 views

Moment of inertia of a cylinder about its base

I've tried to find the moment of inertia of a cylinder rotating about an axis parallel to its base (i.e about the 'End diameter') as one can see here . But when I checked my results with different ...
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2answers
96 views

Why angular momentum about three independent axes?

The generic commutation relations for the angular momentum operator are $[J_x, J_y] = i \hbar J_z$, where the $J_i$, $i = x,y,z$ are the components of the angular momentum vector operator, $\mathbf ...
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123 views

Why don't we talk about angular momentum at all in fluid mechanics?

People usually talk about similar (or maybe not?) things like vorticity or enstrophy in fluid mechanics, but no one talks about angular momentum, why?
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42 views

Planar motion in central forces

In a two body problem under central force, corresponding to a potential $V(r)$(assume one body is massive compared to the other so that its motion is negligible), conservation of angular momentum ...
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86 views

Given eigenvalues of $\vec l^2$ and $\vec s^2$, calculate the eigenvalue for $\vec j^2$

There was an exam question that read approximatly: Let $\vec j = \vec l + \vec s$. Given eigenvalues of $\vec l^2$ and $\vec s^2$, calculate the eigenvalue for $\vec j^2$. We came up with $$\vec ...
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1answer
67 views

What are the proper domains of the position and squared angular momentum operator?

I am looking at the position operator on a compact set $K \subset \mathbb{R}^n$ and the squared angular momentum operator (so essentially the Laplace-Beltrami operator where I just look at the angular ...
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3answers
202 views

Where does the electron get its high magnetic moment from?

I have always found the concept of spin a little weird. I had read somewhere that for the charge or size of electrons, their magnetic field is very high. In order to produce such fields, they must be ...
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2answers
194 views

Weighing head by angular momentum

A popular Phys.S.E question asks how can I measure the weight of my head. One of the answers suggests measuring the moment of inertia. My suggestion was to construct an apparatus that places the ...
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1answer
322 views

Principle Axes of inertia and moments of inertia

Principal axes of inertia is defined as the eigenvectors of the inertia tensor matrices. I understand that a diagonalised tensor can yield these axes, but why are they necessarily form the axes of a ...
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52 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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1answer
56 views

Formalism and representation in Quantum Mechanics

I am just curious about the formalism of basic Quantum Mechanics. Lets take for instance the system of a spin-$\frac{1}{2}$ particle. The state of the particle is described by a vector in an abstract ...
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1answer
41 views

Are galaxies “disk” shaped?

When you look a sphere from a fixed observation point, you can easily mistake it for a circle, so I was wondering: are galaxies really "disk" shaped or we just don't have the means to detect the ...
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2answers
98 views

Angular momentum for 3D harmonic oscillator in two different bases

I know that the energy eigenstates of the 3D quantum harmonic oscillator can be characterized by three quantum numbers: $$ | n_1,n_2,n_3\rangle$$ or, if solved in the spherical coordinate system: ...
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How can the Earth keep spinning with a liquid core?

In regards to the 'conservation of angular momentum' being the explanation of why celestial objects spin... If you fill a ball or any other container with a liquid and try to spin it, you will not ...
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85 views

Contribution to angular momentum $ L_z$ - due to rotation of probability fluid?

I'm doing a course on QM and this concept is entirely new to me: "The eigenvalue $m\hbar$ of $L_z$ can be understood as the result from the rotational motion of probability fluid around the z-axis. " ...
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23 views

A question on lowering the total spin

Is there a way to lower the total spin of the state and fixing the $S_z$ rather than lowering the $S_z$ by spin ladder operator? Or in other words, how to connect the $S=1$ state with $S=2$ or $S=0$ ...
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58 views

In Orbital Mechanics what is the quantity described below called?

I seem to recall that $r^2 \dot{\theta}$ is a conserved quantity in orbital mechanics, which I just proved using the Euler-Lagrange equations. Namely via: $ \mathcal{L} = \frac{m}{2} (\dot{r}^2+r^2 ...
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2answers
67 views

How can $J_1^2, J_2^2, J_{1z}, J_{2z}$ commute mutually?

I'm reading through J. J. Sakurai's Modern Quantum Mechanics book and currently looking at the "Angular-momentum addition" part. Here, it says you have two options and that one option is to ...
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1answer
63 views

Does mass equal angular momentum?

At the wikipedia pages for angular momentum ($L$) and moment of inertia ($I$) we find the equations: $$L=I \omega$$ $$I=m r^2$$ where $m$ is mass and $r$ is the distance between said mass and ...
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70 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 ...
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45 views

Calculating the components of angular momentum of a rigid body

You have a rigid body with 1 fixed point in space (the origin). It's self-explanatory how to get the following equation for the angular momentum: $\vec L = \sum_n m_n\vec r_n\times\vec v_n$ ...
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92 views

Shell model of an odd-odd nucleus: $^6$Li

Lithium-6 isotope has an approximate magnetic momentum of $0.88\ \mu_N$ in its fundamental nuclear state. I'm trying to find its angular momentum and parity. I found in a standard table: $I=1^+$ and ...
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2answers
172 views

Angular momentum - maximum and minimum values for $m_{\ell}$

I want to work out the maximum and minimum values for $m_{\ell}$. I know that $\lambda \geq m_{\ell}$, therefore $m_{\ell}$ is bounded. In the lectures notes there is the following assumption: $$ ...
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2answers
78 views

Lagrangian point or dark matter?

We know that spiral galaxies spin in a way such that we have to assume that dark matter is responsible for the extra mass required to do so. My question is, can Lagrangian points (L1 and L2) be used ...
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29 views

Wikipedia's derivation of torque related to angular acceleration [duplicate]

Wikipedia derivation of the relationship between a torque and an angular acceleration is given here. Could someone help me to see how the following: $$\vec{\tau} = \left(-\sum^n_{i=1}m_i [\Delta ...
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1answer
68 views

Angular momentum of anyons

Why is it true that anyons can have angular momentum taking any real value? Why aren't they restricted to the $j(j+1)$ integer values most are familar with?
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1answer
61 views

State with non-zero angular momentum - cannot be described by spherical harmonic?

For a state with non-zero angular momentum, why is it that it cannot be described by the spherically symmetric spherical harmonic?
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1answer
44 views

Pulsars with accreting disk in binary system

Following this line, I am wondering about the following question. Accreting pulsars in binary systems are usually thought to accrete from a prograde disk, so increasing their spin in the process. ...
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1answer
246 views

ket vector with two “entries”

This is a very simple question. I am learning about angular momentum. In my lecture notes, the symbol $|\lambda,m_l \rangle$ was defined as a eigenfunction of a central potential. Two assumptions are ...
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1answer
243 views

Apparent violation of Newton's 3rd law and the conservation of angular momentum for a pair of charged particles interacting magnetically

Consider a system of the two identical point positive charges situated in the space (isolated from influence of any other external fields) as shown in the figure.Particle1 is at (a,a,0) and Particle2 ...
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60 views

QM: Commutation relations between irreducible vectors and angular momentum $[J^2,T_q^k]$

reading about the irreducible tensors and its commutation relations with the angular momentum one can find relations for $J_{z}$, $J_{+}$, $J_{-}$, but I was wondering, what about $J^2$ ? from ...
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1answer
57 views

What are phase conventions in angular momentum and rotation calculations?

I work with complicated angular momentum calculations related to atomic physics; nevertheless, I never need to use anything related to a phase convention (apparently because it's taken care of in a ...
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2answers
67 views

What would cause a spinning fluid to stop spinning?

I once saw a demonstration where an electric current caused a drop of mercury to spin. The drop contained bits of iron, which could be seen flowing around in a circular pattern. As soon as the ...
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0answers
37 views

Spherical harmonics for two particle?

For one particle angular momentum eigenket, we can represent it in position space, such as $\langle\theta,\phi|l,m\rangle=Y_{lm}$ where $Y_{lm}$ means spherical harmonics. But if we have we have ...
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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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What is the eigenvalue of $J_z$?

In the calculation of the Zeeman Effect, the most important calculation is $$\langle J_z + S_z\rangle.$$ Suppose we want to find the Zeeman Effect for $(2p)^2$, meaning $l = 1$. In Sakurai's book, ...
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3answers
501 views

Why is $ \vec{S}^{(A)} \otimes \vec{S}^{(B)} = \frac{\hbar^2}{4}(\sigma_x \otimes \sigma_x + \sigma_y \otimes\sigma_y + \sigma_z \otimes \sigma_z)$?

I haven't been taught tensor product in class but they have taught us addition of spin. I looked up online in this link->http://homepage.univie.ac.at/reinhold.bertlmann/pdfs/T2_Skript_Ch_7.pdf (pg ...
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1answer
94 views

Clebsch-Gordan Coefficients for two spin-1 particles - Why is there a ∣0⟩∣0⟩ ket?

I have used the rules for addition of angular momenta to work out the Clebsch-Gordan coefficients, which all seem right except for state $\lvert0,0\rangle$: For n = 1 \begin{align} \lvert1,1\rangle ...