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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Spinning spheres colliding

In an ideal environment with no friction, in a vacuum, what happens to the velocity of the spin of two spheres spinning in perfect parity at two different velocities when they come into contact?
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327 views

Why does the electron spin with a particular tilt?

I found this image for the classical description of the electron spin at hyperphysics Can you explain why the axis of rotation makes an angle of 60° with the z-axis and how this particular ...
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321 views

An identity of Pauli matrices

I am studying spin recently, and textbook gives some identities of Pauli matrices, one said that for any two unit vectors $\bf m$ and $\bf n$, $[\bf m \cdot \bf{\sigma},\bf {n \cdot \sigma}]= ...
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134 views

How to associate a Hilbert space with a QM system?

I couldn't really find a fitting title for this question. I'm still relatively new to QM and am trying to get the basics down. I understand that a physical system is associated with a Hilbert Space, ...
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285 views

Commutator not transitive

I noticed the following: $$[L_{+},L^2]=0,\qquad [L_{+},L_3]\neq 0,\qquad [L^2,L_3]=0.$$ This would suggest, that $L^2,L_+$ have a common system of eigenfunctions, and so do $L^2,L_3$, but $L_+,L_3$ ...
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483 views

Quantum mechanical angular momentum and spin formalism/notation

I am currently stuck on the following notation: $\frac{1}{2}\otimes\frac{1}{2} = 0 \text{ (antisym) } \oplus 1 \text{ (sym) }$ No matter what I tried, I couldn't derive the identity. I am sure that ...
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1answer
406 views

What happens to a rotating rod that breaks in two?

I know that the approximation for the moment of inertia of an infinitely thin rod of mass $m$ and length $L$ spinning around an axis perpendicular to its own axis at its center is $\frac{mL^2}{3}$: ...
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124 views

Free rotation of a rigid body

So I am currently reading Fowles and Cassidy and there is something I'm confused about in the section about geometric description of free rotation of a rigid body. I will present the stuff first that ...
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90 views

Angular momentum conservation at quantum level

how angular momentum of system is conserved when electron jumps higher energy state to lower energy state and photon is emitted(circularly polarized)? i read somewhere that it is NOT conserved .Why?
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4k views

How does a spinning electron produce a magnetic field?

I learned in my undergraduate physics class that atoms have magnetic fields produced by the orbit of electrons and the spin of electrons. I understand how an orbit can induce a magnetic field because ...
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4answers
487 views

If angular momentum corresponds to linear momentum, what corresponds to energy?

Angular momentum is defined from linear momentum via $\vec L = \vec r\times\vec p$, and is conserved in a closed system. Since energy is the time part of the linear four-momentum, is there a quantity ...
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1k views

Can a linear momentum generate angular momentum at collision?

I'm trying to get the facts straight here. Suppose I'm throwing a ball with no angular momentum. It collides with the ground and Newton's third law tells us that a force opposite to the gravity will ...
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Quantization of orbital angular momentum

Probably a very simple question, but I can't find the answer on the Internet. I know nearly to nothing about quantum mechanics, but in statistical physics I'm confronted with the idea that the orbital ...
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3answers
213 views

Trilinear gauge couplings: Spin

In non-abelian gauge theories self interaction of gauge fields is permitted, allowing coupling such as $WWZ$ (i.e. $Z$-boson decaying to $W^+W^-$) or ggg (i.e. gluon splitting into two new gluons). ...
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880 views

Ice skater increase of energy

This may be a very basic question but I am not seeing how it works. Consider the standard example of an ice skate rotating about his/her center of mass and pulling in his/her arms. The torque is zero ...
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1answer
411 views

Angular Momentum Addition Theorem - Sanity Check

Looking back at my quantum mechanics notes, the angular momentum addition theorem is listed as: $j=j_1+j_2,j_1+j_2-1, ..., |j_1-j_2| $ (Using conventional notation) , but I'm a little unsure how to ...
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1answer
289 views

Conservation of total angular momentum in $\Phi$-meson decay

I am looking into the decay of a $\Phi$-meson decaying into $K^+$, $K^-$. My problem is, the $\Phi$-meson has a total angular momentum of 1 and the two Kaons have a total angular momentum of 0. On the ...
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347 views

How is angular momentum conserved if a bullet hits a wheel?

Suppose my system involves: 1) A mounted wheel with some outward flap 2) A bullet already in motion Initially the net angular momentum is 0 and the net kinetic energy is just that of the speeding ...
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1answer
157 views

Paradox of angular velocity

For a torque-free symmetric top, the Inertia tensor has an inverse $I^{-1}$, and $L=I\omega$. Which implies that $\omega=I^{-1}L$. But since $I, L$ are constants, $\vec\omega$ is a constant. However, ...
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3k views

Example where angular momentum and angular velocity are not parallel

I am unable to visualize any case where angular momentum and angular velocity of an object are not parallel.
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101 views

Why doesn't this equation for orbital motion change with position in the orbit?

The question and answer are on pg.8-10 of this PDF: At first, I went through it, thinking nothing of it. But then, I wondered: "What if we picked a final state in which the space junk was NOT at ...
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753 views

Why, for a spin-½ particle, are the possible outcomes of measuring spin projection along any direction the same?

If one measures the projection of spin of a spin half particle along the $x$ axis one will always get $\pm\tfrac12\hbar$. Measuring it along the $y$ axis one will always get $\pm\tfrac12\hbar$. ...
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341 views

N particles, will there be any rotation after a period of time or everything will collapse

This is in context of classical Newtonian physics. Consider a system of n different point mass particles. Initially all are spread around on one plane. No particle possess any velocity to begin with. ...
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126 views

How does the kinetic energy of a ballerina increase? [duplicate]

When a ballerina pulls her arms in, her rotational kinetic energy increases because angular momentum is conserved. That means that work must have been done on her. I saw somewhere that there is work ...
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100 views

What's the significance of the difference between the quantum numbers, $\ell$ and $m_{\ell}$?

I know that $m_{\ell}$ is associated with the projection of the angular momentum vector onto the $z$ axis and $\ell$ is associated with the length of the angular momentum vector. To me this implies ...
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78 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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1answer
546 views

A tensor product of two spin-1 particles

I'm rather confused, and I was hoping if someone could help me figure out this (probably rather elementary) issue. I have two particles with spin 1, whose state I describe by $m_S$ and $m_I$ ...
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401 views

Rotational invariance and operator-squares

My mind is drawing a blank right now. In systems with spin and orbital angular momentum, I know that rotational invariance implies that $[H, \mathbf{J}]=0$ where $\mathbf J=\mathbf L+\mathbf S$. But ...
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400 views

Why does $\ell=0$ correspond to spherically symmetric solutions for the spherical harmonics?

In quantum mechanics why do states with $\ell=0$ in the Hydrogen atom correspond to spherically symmetric spherical harmonics?
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397 views

Where do the conservation laws come from?

I know the conservation of energy comes from Noether's theorem via the time-translational symmetry, and if I remember correctly, the conservation of momentum comes from space-translational symmetry. ...
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725 views

Space Quantization of Quantum Angular Momentum

I am trying to understand what my book is trying to convey. Quantum angular momentum is $L_z = m_l \hbar$ "Choosing arbitrarily a z axis and using an appropriate experimental technique, we measure ...
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1answer
166 views

Mathematically, how do we deduce that angular momentum is bounded?

So, how do we know $J_{+}|j,(m=j)\rangle =|0\rangle$? I.e. that m is bounded by j. We know that $J_{+}|j,(m=j)\rangle =C|j, j+1\rangle$, but how do I know that gives zero? Is it by looking at its ...
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What causes precession or nutation in a spinning object?

What causes precession in a spinning object? What causes nutation in a spinning object? What causes a top, gyroscope, and the earth to wobble? Just because it's a simple question, I'm not ...
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482 views

Why is there a phase factor when the two composite angular momentum is exchanged in Clebsch–Gordan coefficients

An identity exists for CG coefficients: $$\langle j_1 m_1 j_2 m_2 |J M \rangle = (-1)^{j_1+j_2-J} \langle j_2 m_2 j_1 m_1|J M\rangle,$$ But why is there a phase factor $(-1)^{j_1+j_2-J}$? It seems ...
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1answer
42 views

What determines a progenitor's fate as a spiral or elliptical?

I was thinking about my answer to Are the inner planets on planar orbits because there was more dust in the inner solar system (early on in planetary accretion)? - when it occurred to me that maybe I ...
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1answer
72 views

What makes us twist in a somersault?

In a backwards straight somersault you can decide whether you twist early or late. Twisting early means, that you induce the twisting movement before you rotatation hits 180° and twisting late means, ...
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63 views

Correct vector space of eigenkets of angular momentum

When we say an particle is in the state: \begin{equation} |l,m\rangle, \end{equation} what is the underlying state space, as a vector space? Is it a tensor product vector space, of dimension: ...
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1answer
77 views

Angular momentum-torque relationship in a rotating frame?

I have read that $$\vec\tau=\frac{\mathrm{d}\vec L}{\mathrm{d}t}$$ holds true whenever the origin is not accelerating. But I cannot see why this holds true for a rotating frame of reference (such as ...
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3answers
77 views

Why is acceleration significant to generating power in racket sports?

I'm reading a paper on badminton stroke power (see Figure 4 on page 8). It says: For most of the trials the racket head reached peak speed just at the time of impact. The racket head showed ...
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1answer
166 views

Paradox: electric current in a coil on a disc - will this disc spin if the circuit is opened?

I encountered this problem in a book, but there were no solution written there. The setup: there is a plastic (insulator) circular disc, that is suspended in a way, that can very easily rotate (so ...
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2answers
101 views

Introducing angular momentum for the first time to a class

What is the best way to introduce the notion of angular momentum to a class without making it appear an unnecessary and artificial construction?
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1answer
76 views

Why does matter gather as discs around massive objects? [duplicate]

Why do discs, like rings of Saturn and the spiral shape of our galaxy form around massive objects, instead of just a (spherical?) cloud of matter?
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1answer
172 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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1answer
78 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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2answers
180 views

How do you measure proton's spin? [duplicate]

I've probably read it somewhere in Sakurai but I cannot recall it at the moment. So how does one really measure the proton's spin? I mean the proton's spin and not its constituents. Do you measure ...
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1answer
1k views

Hamiltonian matrix off diagonal elements?

I'm trying to understand how Hamiltonian matrices are built for optical applications. In the excerpts below, from the book "Optically polarized atoms: understanding light-atom interaction", what I ...
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1answer
164 views

Angular momentum of a rotating black hole

Is there an upper limit to the angular momentum of a rotating (Kerr) black hole?
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2k views

Conservation of angular momentum in helicopter

I have a small RC-controlled toy helicopter with removable tail rotor. Suppose I remove the tail rotor, hold the tail with my hand, start the rotor until it moves with constant angular velocity and ...
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1answer
636 views

Equation that tells me the rpm and mass of a spinning disk needed to keep a second large mass stable using gyroscopic effects

I am trying to figure out how large of a mass and how quickly I need to spin said mass to keep a two-wheeled robot stable. Ideally, I am looking for a formula that relates m1=mass of robot, m2=mass of ...
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1answer
155 views

generalizing spin rotations

I have a question about the relation: $\exp(-i \vec{\sigma} \cdot \hat{n}\phi/2) = \cos(\phi/2) - i \vec{\sigma} \cdot \hat{n} \sin(\phi/2)$. In my texts, I see $\phi\hat{n}$ always as c-numbers. My ...