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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How to understand zero elements in CG coefficient table?
I understand the standard theory of angular momentum and the reps. of SU(2). I found there are some zero elements in CG coefficient table. I can derive them by using the recurrence relations between ...
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How much should the disk/wheel spin for gyroscopic-precession to take place? Is there a threshold?
Suppose we have a spinning gyroscope whose disk is of mass $m$, spinning at angular velocity $\omega$, and attached to a rod of length $r$.
The precession of the gyroscope around the $z$-axis will be ...
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The angular momentum of zero mass limit of Kerr metric
The Kerr metric in Boyer coordinates is
$$
ds^2 = -\left(1 - \frac{2GMr}{r^2+a^2\cos^2(\theta)}\right) dt^2 +
\left(\frac{r^2+a^2\cos^2(\theta)}{r^2-2GMr+a^2}\right) dr^2 +
\left(r^2+a^2\...
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Dropping a spinning top into a black hole?
Is there a formal treatment for what happens when one drops a spinning top into a black hole? More precisely, if one has a spinning top, of mass $m$ and angular momentum $j$, and lets it drop into a ...
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Generalized electromagnetic angular momentum
Linear momentum has a generalization when electromagnetic vector potential is present:
$$m {\bf v} + q \bf{A} $$
Likewise for energy we have
$$\frac{mv^2} {2} + q \phi $$
I wonder if there is an ...
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Is there angular momentum conservation in models like the Ising model?
In Quantum Mechanics conservation laws are fundamental, I was thinking about spin altering models of interaction such as the Ising Model and realized that it isn't at all clear how angular momentum ...
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What forces are responsible for the circular motion of the center of mass of a gyroscope-like setup in precession?
Suppose we have a disk (of mass $M$) connected to a rod (of mass $m$) attached to a fixed pivot. (Note that initially I thought of the rod hanging off a rope, but I realized it would be better to ...
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How to write a hamiltonian in minimal basis as a spherical tensor
I am reading this paper where the authors write the atom-blocks of the hamiltonian in a minimal basis set and use some regression technique to fit the hamiltonian to data. This question is about how ...
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Dropped sign from Wigner-Eckart Theorem for matrix elements of $x$ in hydrogen atom $n = 2$ shell
Consider the $n=2$ states of the hydrogen atom, which we label by $|n\,l\,m_l\rangle$. I want to calculate whether or not there should be a sign difference in these specific matrix elements of $x$:
$$\...
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Does angular momentum of rotating body depend on its position?
This question might be silly. But angular momentum $\vec{L} = \vec{r} \times \vec{p}$. It always confuses me whether or not angular momentum depends on the origin because the position vector does ...
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How do we argue that there are only integer-spaced $m$ values for angular momentum?
I am following the usual development using ladder operators (in Ballentine Chapter 7) for the angular momentum spectrum (for the joint eigenbasis $|j,m \rangle$ of $J^2$ and $J_z$) and I am satisfied ...
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Does a rotating body resist acceleration in a direction that is perpendicular to the direction of the rotation of the body?
I would like to know if a rotating body resists acceleration in a direction that is perpendicular to the direction of the rotation of the body.
Say for example there is a bicycle wheel with a tire on ...
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Gyrating proton near magnetic pole of the Earth
Imagine a proton from space which approaches the magnetic pole in the Northern hemisphere of the Earth. The proton spirals around the magnetic field lines. The $\vec{B}$-field is stronger near the ...
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Lorentz force and conserved quantities
Imagine a permanent magnet at rest in empty space. A proton initially travels along a straight path with a velocity $\vec{v}$ and enters the field of the magnet with $\vec{v}$ perpendicular to $\vec{B}...
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Probability and the Magnetic Quantum Number
I am currently self-studying quantum mechanics, and I'm working problems on angular momentum. The problem I'm currently working on asks one to consider a particle subjected to a spherically symmetric ...
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Hyperfine structure
In Griffiths, the hyperfine structure is described as follows:
So the hyperfine structure is a result of a mechanism called spin-spin coupling, which is the interaction of the spin of the nucleus (...
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Why is the angular momentum needed in theories where the linear momentum is locally conserved?
It is a well-known result that angular momentum conservation is related to the invariance of the Lagrangian respect spatial rotations, here a demonstration of how infinitesimal rotations do not alter ...
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How does Kepler's Second Law show that a planet further from the sun will move slower?
This is probably a very stupid question. We are told that due to Kepler's Second Law, which according to this very straightforward explanation:
"Kepler's second law of planetary motion describes ...
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Hydrogen spectrum for transitions of $\ell$, $m_{\ell}$
In this question, there is a spectrum exhibiting transitions corresponding to the azimuthal quantum number $\ell$ of a system corresponding to a spinning $\rm{Cs_2}$ molecule.
Regarding hydrogen-like ...
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Explanation about the rotation spectrum from cold $\rm Cs_2$ molecules
In the book Mécanique quantique (Jean Dalibard), there is an example to illustrate the quantification of the $L^2$ squared angular momentum.
This example is extracted from A. Fioretti et al, Eur. Phys....
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Bohr hydrogen atom model and quantum mechanics on quantisation of angular momentum
Bohr's model says that angular momentum is quantised to integral multiples of reduced Planck's constant, $$L = nh/2\pi$$
but in quantum mechanics, angular momentum operator has non-integer eigenvalues,...
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Does rigid body rotation contradict Newton's first law?
Per Newton’s law, an object will move along a straight line with a constant speed if no force is acting upon it.
No portion of a rotating ball is moving in a straight line (except those on the axis ...
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Find the angular momentum of a pulley system after the mass $M$ has descended through a height $h$ [closed]
Two masses $M$ and $m$ are connected by a light spring going over a pulley of radius $r$. The pulley is free to rotate about its axis which is kept horizontal. The moment of inertia of the pulley ...
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Why do the orbital quantum numbers need to be different for triplet states?
By definition, the spin wave functions for two electrons for the triplet are symmetric and the orbital wave functions for triplet are antisymmetric, so that the total wave function is going to be ...
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Why do states with $m_1 + m_2$ have to vanish in the Clebsch-Gordon expansion of combined states?
I'm currently studying QM and more specifically addition of spin angular momentum. I'm using the handbook from Griffiths.
A combined state $|s m \rangle$ can be written using Clebsch-Gordon ...
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Angular momentum about an axis?
We know that angular momentum of a body is defined about a point in space. Let us consider a solid cylinder whose radius is R and mass is M. It has a moment of inertia defined around the axis of ...
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Velocity of object under central force as radial and tangential components [closed]
Consider an object rotating about a point. A central force is always acting on the object. The force can be variable. Now suppose the object is not rotating in a perfectly circular motion but instead ...
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Doubt regarding derivation of general expression of angular momentum of circular orbits [closed]
The Newtonian angular momentum $J$ of a circular orbit for a gravitational potential $\Phi$ is given by the relation (eqn. 10 of this paper)
$$J=\left(r^3\dfrac{d\Phi}{dr}\right)^{1/2}.$$
For the ...
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How can we distinguish spin singlet and $m=0$ spin triplet states with spin measurements?
Say I have the singlet $|s\rangle = |0\rangle - |1\rangle$ and $m=0$ triplet $|t\rangle = |0\rangle + |1\rangle$ quantum spin states (say in the $S_x$ basis). If all I can do are spin measurements, ...
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Consistency of two statements on the angular momentum operator of a spin-1/2 particle
It says in the group theory that the angular momentum operator of a spin half particle is just half of the Pauli matrices. However, in the quantum field theory, the angular momentum can be expressed ...
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Is the dot product of spins the only way to create a scalar (Hamiltonian) invariant under spin rotation?
I wanted to generalize the result for the following question for four spins 1/2: Most general form of a spin rotation invariant Hamiltonian?.
Assume that we have a Hilbert space for four spins $(\vec{...
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Angular distribution of two-body decay with orbital angular momentum
I'm currently studying the two-body decay where a particle (Be) with spin-1 decays into a pion (spin-0) and spin-1 particle $X$. The system is prepared so that the Be is polarized with a spin ...
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Why dimension units of radius is not $\rm m/rad$ or $\rm cm/rad$? [duplicate]
Radius is not a just simple size or length between the two points. The radius shows the connection of linear and angular values. Something must indicate the information about a perpendicularity of the ...
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Torque on a rotating axle [closed]
The problem is from the book Classical Mechanics by David Morin. Here's my attempt
One of the principal axes will be along the rod ($\hat x_1$) and other two will be passing through the center ...
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$J=1/2$ and $J=3/2$ Baryons with $uds$ quarks
Why are there two different $J = \frac{1}{2}$ baryons with quark content $uds$ (the $\Lambda_{0}$ and $\Sigma_{0}$) but only one $J = \frac{3}{2}$ baryon (the $Σ_{*,0}$) with the same quark content?
I ...
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Angular Momentum Operators and Spherical Harmonics in Higher Dimensions
Suppose we have a $d$-dimensional quantum system with a rotationally symmetric Hamiltonian $\hat{H}$. Extrapolating from the two and three dimensional cases, one might expect that the eigenstates of $\...
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Angular momentum of a spinning solid sphere about an axis other than the spinning axis [closed]
A solid sphere is spinning about z axis. I know that its angular momentum about the z axis will be L=Iw. Where I is the moment of inertia about its central axis and w is its angular speed about z axis....
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What Wigner rotation matrix elements to use when calculating the angular dependence?
In order to work out the angular dependence of the differential cross section of an interaction, we can use the Wigner rotation matrix elements, which quantify the probability of rotating through some ...
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Why don't ladder operators mix angular momentum eigenstates?
The crucial step in the typical (see e.g. Sakurai Chapter 3.5.1) proof of the eigenvalues of the angular momentum operators $J^2$ and $J_z$ is (after defining the ladder operators $J_\pm :+ J_x \pm ...
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Ladder Operators for Quantum Angular Momentum
Consider the operators
\begin{equation*}
\mathbf{J}^{2} = J_{x}J_{x}+J_{y}J_{y}+J_{z}J_{z}
\end{equation*}
where the $J_{i}$ are the generators of infinitesimal rotations. Choosing $\mathbf{J}^{2}$ to ...
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Angular momentum of $N$ particles
I am reading Goldstein's Classical Mechanics book; I have difficulty understanding these lines. Why do the last two terms vanish? I am reading this and thinking $r'$ is a null vector, but the second ...
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Shape of atomic orbitals
I am a physics bachelor student and currently learning quantum mechanics. In my course we derived the wave function for the hydrogen atom.
I know that the quantum number L is connected with the shape ...
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Intrinsic Parity of the $𝐾^+$ Meson
Why it is not possible to determine intrinsic parity of the $𝐾^+$ mesons from the $𝐾^+ → 𝜋^+ 𝜋^0$ decay?
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Confused about angular momentum conservation from deflection angle derivation
I have started going through A. Zee's "Einstien Gravity in Nutshell" after a good number of years away from studying physics (perhaps too many) and I have been stuck on a derivation that I ...
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Under what conditions can we assume action-reaction force pairs are along the line joining two objects (strong form of Newton's third law)?
In many circumstances we take it that when object $A$ applies a force on object $B$ and vice-versa, the two forces are central forces or along the line joining $A$ and $B$. This is used to derive ...
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Can a body rotate indefinitely about its axis in space?
Wondering if a rigid body can keep rotating about its axis with constant angular velocity without the application of any force or torque, to the object. Assuming that the object is not in influence of ...
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Angular Momentum Operator
Can a system be in an eigenstate of the total angular momentum operator $L^2$ without being in an eigenstate of any of $L_x$, $L_y$, and $L_z$? What would such a state look like?
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From where does the perpendicular component of force arise in this lever? Does it conflict with the strong form of Newton's third law?
Newton's third law states that for every force applied by object $A$ on object $B$, there is an equal an opposite force by object $B$ on object $A$. The strong form of Newton's third law states that ...
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How does a spaceship respond to a centrifuge?
If you are in a spaceship and you spin a balanced centrifuge with a rock in it, does the spaceship rotate in the opposite direction as the centrifuge with the same angular, kinetic energy as the ...
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How do I prove that a fast spinning disk requires more force to stop it than a slowly spinning disk?
I am trying to prove that with all things the same, a rapidly spinning disk requires more force to stop it than a slowly spinning one.
For example, when you first hit the brakes in a fast moving car, ...
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