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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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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2answers
515 views

Peskin and Schroeder Equation 3.23

I've been trying (for a while) to prove that $S^{\mu\nu}:=\frac{i}{4}\left[\gamma^\mu,\,\gamma^\nu\right]$ is a representation of the Lorentz Lie algebra, that is, to prove that it satisfies the ...
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3answers
564 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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2answers
11k views

Torque And Moment Of Inertia

I am reading the two concepts mentioned in the title. According to the definition of torque and moment of inertia, it would appear that if I pushed on a door, with the axis of rotation centered about ...
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1answer
484 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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2answers
249 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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1answer
114 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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1answer
474 views

Find angular momentum about any point

How do I find the angular momentum of a body about any point? We know that $L=I\omega$ for a body rotating in space, where $L$ denotes the angular momentum, $I$ denotes the moment of inertia and ...
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4answers
549 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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1answer
2k 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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2answers
1k views

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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2answers
1k 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
478 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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2answers
62 views

Why do tops move opposite to each other when colliding, not tangentially?

When two well-balanced tops collide, they tend to bounce directly away from each, in other words along the line connecting their centers. Intuitively I would expect the tops to move tangentially, not ...
3
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2answers
651 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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1answer
116 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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1answer
445 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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3answers
570 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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4answers
2k views

Angular momentum conservation in pion decay?

I have seen the charged pion decay $$\pi^{-}~\to~ \bar{\nu}_{\ell} +\ell^{-}$$ represented with diagrams containing a $W^-$ in the $s$-channel. The $\pi^-$ and $W^-$ have angular momentum $0$ and $1$ ...
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1answer
185 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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1answer
106 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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4answers
355 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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2answers
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Mech stability through gyroscope

I recently read up about gyroscopes, angular momentum and mechs (the big Cockpit controlled robots) and was wondering if it would be possible to get a stable walking mech (only as example, not meant ...
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2answers
108 views

Conservation of angular momentum while sitting on a spinning chair

Today my friend was sitting on a spinning char. By moving his top part of the body left to right and his bottom part of the body the opposite he managed to spin. As I understand Conservation of ...
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1answer
108 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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1answer
151 views

Why is orbital angular momentum quantized according to $I= \hbar \sqrt{\ell(\ell+1)}$?

I simply have no idea how this result is found $$I=\hbar \sqrt{\ell(\ell+1)}.$$ The result seems to just be dumped in textbooks rather than explained. I can get the result that $I_z=\hbar m_j$. ...
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1answer
2k views

Definition respective derivation of angular momentum formula

I am reading An Introduction to Mechanics by Kleppner and Kolenkow (2014). On page 241 is the definition of the angular momentum: Here is the formal definition of the angular momentum $\vec{L}$ ...
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2answers
86 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
929 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 $m_I$ ...
3
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1answer
191 views

Noether Charge For Scalar Fields Under Lorentz Transformations

The conserved charge associated with the Lorentz transfomation of a scalar field is given by $Q^{\alpha\beta}=\int d^3x\frac{1}{2}(x^\alpha T^{0\beta}-x^\beta T^{0\alpha})$. The quantities $Q^{ij}$ is ...
3
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1answer
322 views

How does $SU(2)$ group enters quantum mechanics?

What is the reason that $SU(2)$ group enters quantum mechanics in the context of rotation but not $SO(3)$? What really rotates and which space it rotates? It cannot be the physical electron that ...
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3answers
680 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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2answers
251 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 ...
3
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1answer
106 views

Does this commutation relation hold?

I was wondering whether it is true that $[L_x^2,x^2+y^2+z^2]=0$. I could not find it in the internet and therefore I wanted to ask here whether anybody here knows that this is true or false.
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1answer
289 views

Calculating the path of a ball with spin moving across a table

A ping pong ball is rolling over a smooth (but not frictionless) table. During its travel, a clockwise spin is placed on the ball. The ball's path is changed to move to the right (in perspective from ...
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3answers
437 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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2answers
937 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
171 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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2answers
6k views

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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2answers
557 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 ...
3
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1answer
46 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
24 views

Which force transfers angular momentum in tidal locking?

The moon is in tidal lock with the earth, but a long time ago it was not. As the moon became tidally locked with the earth, its angular momentum changed and the delta went into it's orbit and possibly ...
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2answers
63 views

Intuitive explanation of rotational inertia with respect to angular momentum

I understand that there are proofs (e.g. proof, another proof) of why the angular momentum about two points for an object is the same. However, could someone give an intuitive explanation of why this ...
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1answer
42 views

Conservation of Angular Momentum in Einstein - de Haas effect

I am not really sure why the law of conservation of angular momentum should hold true in the Einstein - de Haas effect. Consider the following excerpt about the phenomenon (taken from Magnetism in ...
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1answer
70 views

How skydivers control their angular momentum?

As the title says, how do skydivers stop spinning from an out of control spin while in freefall? Doesn't conservation of angular momentum doom them to keep spinning, even if they make a small mistake ...
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1answer
141 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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1answer
185 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
117 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
245 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
108 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?