Questions tagged [angular-momentum]
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 happens when you shine circularly polarized light at a pole of a rotating black hole?
Circularly polarized light carries spin angular momentum (SpAM 😉), so shining it into a pole of a spinning black hole from a point on the rotation axis of the BH should raise the angular momentum of ...
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Prove $d/dt\langle L\rangle=\langle N\rangle$ without computing over a specific basis
In Griffith 3rd Edition, Problem 4.23 asks for the proof of
$$\dfrac{d}{dt}\langle\mathbf{L}\rangle=\langle\mathbf{r}\times-\nabla V\rangle=\langle\mathbf{N}\rangle$$
One evaluates $[\mathbf{L},H]$ ...
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No net torque about principal axis
I’m trying to follow the proof for why no net torque is required to sustain constant rotation about a principal axis.
I am confused by the proof by contradiction above. How does “rotating the axis and ...
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Connection between infinitesimal rotations and angular momentum (classical)
What is the connection between infinitesimal rotations and angular momentum, classically? I am not interest in any QM theory. How does one regard infinitesimal rotations as a generator of angular ...
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Confused about selection rules in optical transitions
This question is motivated by this paper in particular (let me know if it is not open access). They measure transitions between electronic levels inside an ion, placed inside a crystal. In Fig. 2, ...
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Electron spin in Pauling's General Chemistry book
I have a question regarding the interpretation in General Chemistry by Pauling (p. 77 - 78) about the spin of the electron/ it's angular momentum vector direction in a magnetic field:
It was ...
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$\vec L$, $\vec S$ and $\vec J$ of the atom or electron when considering the Zeeman effect
I am confused regarding the different angular momenta considered when dealing with the Zeeman effect.
I don't know whether the different angular momenta are attributed to the entire atom, exposed to ...
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Limit of angular momentum and "alone" black hole singularity
I just read in a book that a black hole has an angular limit at which the event horizon would disappear, leaving the singularity alone. This question is not about this said statement — even though I ...
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Equations for rotational motion for fluids [duplicate]
For a solid particle, we write $\mathbf{F} = m\mathbf{a}$ and $\mathbf{T} = I \mathbf{\alpha}$ to account for linear and angular momentum balance. As far as I know, these two momentum conservation ...
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Projective representations and reduction of half-integer spin representations under $C_{\infty v}$
Suppose we have an orthonormal basis of states $|j,m,p\rangle$ where
$j=\frac{1}{2},\frac{3}{2},\ldots$ is the angular momentum quantum number associated with some angular momentum operator $\mathbf{...
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Forces on rotating mass from losing counter mass
If you have a massive rotating body, a flywheel with a less massive body attached to it at one point along the edge via some cable or rod etc. what happens to each body if the smaller body is released?...
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Proving quantum operator relationship used in derivation of Radial Schrodinger Equation
In the derivation of the Radial Schrodinger Equation for central potentials, I have seen the following relationship used:
$$
r^2p^2 = L^2+(\bf{r}\cdot\bf{p})^2-i\hbar(\bf{r}\cdot\bf{p})
$$
and I have ...
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How to show the ratio (spin angular momentum density)/(energy density) for circularly-polarized gravitational plane wave = $\pm 2/\omega$?
It is well understood that an infinite monochromatic, circularly-polarized electromagnetic plane wave has no angular momentum density. However, a finite monochromatic, circularly-polarized ...
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Angular mometum of particles in FQHE hierarchy states
I'm reading through David Tong's notes on the quantum hall effect, and I'm stuck understanding his argument for the filling fraction of hierarchy states. The relevant discussion is under A Quantum ...
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Sketching Angular Momentum Vector Field
The following is a question from the book Gravitation by Misner, Thorne, and Wheeler. They define the angular momentum vector in Euclidean Space by the formula $L_j=\epsilon_{jkl}{x^k}\frac{{\partial}}...
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Is a black hole spherical?
Black holes are usually created when massive stars use up all their fuel and collapse due to gravitational collapse.
All stars rotate.
However, since angular momentum must be conserved even when they ...
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Do all the planets orbiting a star lie in one plane? [duplicate]
If you look at our solar system and our galaxy, the stars and planets are generally all in one plane. So, are all galaxies in one plane? And why are they in one plane?
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Translational motion of a rigid body and summing angular momenta
I'm trying to understand the most accurate way to determine the angular momentum of a rigid body that is undergoing translational motion around a point. Perhaps I missed something in my classical ...
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Why does it make sense to say that "full shells" have 0 angular momentum even when not using the configuration model?
When talking about the periodic table, one is often confronted with the notion of a 'configuration of electrons', i.e. we have the configuration $1s^22s^22p^2$ for carbon for example. This notion ...
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What are physical laws behind the vibrational energy transfer?
My orbital sander causes objects sitting on the surface where the force is applied to levitate slightly and rotate in the opposite direction to the spin of the sander. The harder I push down on the ...
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Can a hypothetically massless object/particle of finite size spin around its own center in a fixed position in space? [closed]
Imagine a hypothetically spinning massless object or particle of finite size. I am not referring to orbital spin but spinning around its own center like a flywheel spinning with a fixed center. This ...
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Electric and magnetic field lines in a plane wave of finite extent
In an infinite plane wave propagating in the $z$ direction, the momentum density is $\mathbf{p}=(4π)^{-1}(\mathbf{E} × \mathbf{B})$ which points in the $z$ direction; therefore, the angular momentum ...
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Exact meaning of the mass $M$ in the Kerr metric event horizon?
Posting this as I have so far not been able to find a straightforward answer to the following question. The formula for the outer event horizon of a kerr black hole is given by the following equation:
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Frame dragging caused by a stream of particles with aligned spins
In theory, does an elementary particle, with intrinsic spin, induce an immeasurably tiny amount of rotational frame-dragging, i.e. can quantum spin cause a Lense-Thirring effect?
There is the Einstein-...
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Expectation value of angular momentum operator
In a question from the book by Cohen Tannoudji, it was asked to find the expectation value of the angular momentum operator $J$. I want to know if this is equivalent to finding $\langle J_x\rangle$, $\...
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How to calculate linear momentum from angular momentum? (Newtonian) [closed]
None of the mechanics textbooks I own directly provide a method to calculate linear momentum from angular momentum.
If there is a way, it would be worth knowing, and have consequences worth discussing....
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Solution of the classical 2D quartic oscillator
Consider a classical oscillator in two spatial dimensions with Hamiltonian
$$H=\frac{1}{2}\vec{p}^2 + \frac{1}{2}\omega r^2 + ar^4.$$
This system has two conserved quantities, the energy and angular ...
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What is the commutator of the lowering operator $J_-$ and the exponential of $J_z$, arranged so that the lowering operator is always to the right?
I'm trying to perform a computation closely related to the problem below. It's a tricky little problem which I imagine has been tackled in the literature before, but I've had no luck finding it. ...
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Why is there no "internal" part of linear momentum? [closed]
One of the ways to derive the expression for how angular momentum operators act on fields from the corresponding action on coordinates is to define spin as $$M_{\mu \nu} \Phi(0) = S_{\mu \nu} \Phi(0) \...
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Understanding conservation of angular momentum with examples
I want to be sure I understand fully the conservation of angular momentum or/and the angular momentum itself with 3 examples, the first one I understand but the other 2 not really.
The famous ...
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Conservation of Angular Momentum in rotating arm
Consider an arm that can rotate along a central axis which is fixed at its left end. On the right end is a (homogeneous) disk that can rotate along its own central axis (disk axis) as shown in my ...
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Proving the Plane Harmonic Oscillator move in an ellipse
My problems states that we have $r(t)$ satisfying $m\ddot{r}(t)=-kr(t)$
And in the first section we were asked to evaluate the derivative of $r(t)\times \dot{r}(t)$
And by cross product derivative law ...
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Why do the forces on a Feynman Sprinkler balance out?
Apparently, in a Feynman sprinkler, the momentum imparted to the tube from sucking the fluid in is exactly balanced by the momentum imparted in the other direction when the fluid turns the bend. I get ...
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Tilting due to Earth's rotation at a distance
I just stumbled upon a video on spacetime and a weird fact caught my attention.
In the video, it explained that a free falling body dropped onto the Earth would change its orientation and would be ...
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Angular momentum conservation and isotropy of space
If we consider a spin-less particle at some state $|\psi\rangle$ belonging to the state space $\mathcal{H}_r$ then the following holds true:
Invariance of the quantum mechanical system under rotation ...
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What is Kerr doing here?
On pages 19-20 of Kerr's conference about how he discovered his metric, he basically performs several coordinate transformations until reaching:
$$ds^2 = dx^2 + dy^2 + dz^2 - dt^2 + \dfrac{2mr^3}{r^4 +...
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Spinning top moving in curved spacetime
If I have a spinning top in empty space, it would take work to change the orientation of the angular momentum vector of the top. Suppose I throw a spinning top in flat space such that the direction of ...
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The Fields of a Moving Point Charge and special relativity
David J. Griffiths Introduction to Electrodynamics page 456 shows the following figure:
... then page 461:
... and page 462:
The following thought experiment has three electrons, negative charges, ...
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Sign of components of spinor spherical harmonics
In Bjorken and Drell's text on relativistic QFT, the spinor spherical harmonics for a spin $\frac{1}{2}$ particle with orbital angular momentum $\ell$ are given as $$\chi_{j,m}=\begin{bmatrix} \sqrt{\...
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Angular momentum of matter halos?
How much momentum (or more specifically, angular momentum) could a matter halo (like a dark matter halo, a neutrino halo...) have? Is it possible that they may have high amounts of momentum?
For ...
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In equation (20) from lecture 10 in Leonard Susskind’s ‘Classical Mechanics’, why is there a summation involved?
Here is the equation
$$\{x_i,L_j\}=\sum\limits_{k}ϵ_{ijk} x_k.$$
Is this equation generalised for any number of dimensions? In which case, would the following example be correct assuming 4 dimensions?
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Does A Pivot Exert A Force
On a frictionless horizontal table, a uniform stick is pivoted at its middle, and a ball collides elastically with one end, as shown in Fig. 8.10. During the collision, what are all the quantities ...
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Confusions in conservation of angular momentum and torque
Say someone is on a spinning object with heavy weights in his hands. He then pulls those heavy weights to himself and then the total angular speed is faster. We can explain this with conservation of ...
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Hund's rules in Helium level scheme
If I am not wrong, Hund's rules state that:
I) The highest $2S+1$ has the lowest energy.
II) For a given $S$, the lowest $L$ has the highest energy.
However, I have found in various textbooks the ...
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A puzzle about relativistic spin
I'm suffering from a confusion about relativistic spin. I don't believe my question has been asked before, and I'm sure I've made some silly mistake somewhere, but I can't spot it. So I'm appealing to ...
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How to understand gyroscopes by using vectors?
I’ve seen plenty of videos explaining how gyroscopes work, but almost non of them why they are constrained to work in such way. From seeing only angular momentum vectors, how could one predict ...
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Is the kinetic energy of a rotating system just the sum of the instantaneous kinetic energies of the individual particles?
Take a look at this construction which shows linearly moving particles giving the illusion of circular motion. (Follow the instructions on the screen).
Now the question is when I calculate the total ...
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Calculating the expectation value of the angular momentum operator
I'm not looking for the exact answer to the question, but rather why a certain way of solving it is chosen. We agree on the answer, but why is the approach different. I'm afraid it's a sign of me not ...
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Energy eigenvalue of hydrogen-like atoms using Laplace-Runge-Lenz vector
I have a basic question about a few calculations involving the quantum mechanical Laplace-Runge-Lenz vector.
In classical mechanics there is the Laplace-Runge-Lenz vector, which for a hydrogen-like ...
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Doubt on conservation of angular momentum for Kepler's laws
Just before proving Kepler's laws, my Professor claimed that if $\vec{F}$ is a central force with center $O$ and it is the only force acting on a point $P$, then the trajectory of $P$ is a curve plane....
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