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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Generators of Lorentz transformations
In Chapter 3 of Peskin and Schroeder's Introduction to Quantum Field Theory they write
For the rotation group, one can work out the commutation relations by writing the generators as differential ...
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Physical interpretation of generators of Lorentz boosts
Lorentz transformations can be thought of as pure rotations and pure boosts. The generators of the rotations are proportional to the angular momentum of a physical system. What is the physical ...
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Conservation of angular momentum using symmetry properties
Goldstein pg 59
It can be shown that if a cyclic coordinate $q_{j}$ is such that $d q_{j}$ corresponds to a rotation of the system of particles around some axis, then the conservation of its ...
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Question from Weinberg Lectures on quantum mechanics
In page number 37 of Weinberg's lectures of quantum mechanics book (2nd edition), After Eq.2.1.17, he states the following:
The Schrödinger equation (2.1.3) then takes the form
$E \psi(x) = -\frac{\...
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Atoms in a magnetic field
If we consider electrons circulating around the nucleus in circular orbits, they constitute current loops and hence have magnetic moments (due to their orbital motion). In a paramagnetic material, for ...
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Conservation of Momentum Paradox Thought-Experiment (Please Explain) [closed]
The Device:
Imagine a physical system involving two circular rings of distinct inner and outer radii, but equal in inertia. The smaller ring is superimposed inside the inner radius of the larger ring. ...
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Possible eigenstates and probabilities of a 2 spin particle system
I am trying to find the end states of a 2 spin particle system, in which the particle moving to the left has a spin-up z-component and the particle moving to the right a spin down z-component. The ...
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Direct experimental observation of magnetic orbital quantum number $m_l$
Is there an experimental way to observe magnetic quantum number $m_l$ values directly, the way electron spin was detected by Stern Gerlach experiment or proton's spin by nuclear magnetic resonance ...
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Is there a relationship between $p, d, f, d$... orbitals and $p, d$... block elements?
In the quantum mechanical model of an atom there are various orbitals around the nucleus as:
$p$ orbitals, $s$ orbitals, $f$ orbital, $d$ orbital etc.
And in the periodic table there are p block ...
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Eigenstates of $L_x$, $L^2$, and $L_z$ [duplicate]
$L_{x}$ and $L^{2}$ commute, while $L_{x}$ and $L_{z}$ do not. However, $L_{z}$ and $L^{2}$ also commute, and hence, they also have common eigenstates. So the problem is:
If $\lvert\psi\rangle$ is an ...
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Clebsch-Gordan coefficients and total angular momentum basis question
Suppose I have the total angular momentum basis for two particles, formed by $\{|J,M\rangle\}$, where $J\in |j_{1}-j_{2}|\leq J\leq j_{1}+j_{2}$ and $M=m_{1}+m_{2}$ with $ -J\leq M\leq J$. There is a ...
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Physical Significance of Irreducible representation of 3D rotation group
I read about the irreducible representation of groups in the book 'Lie Algebra in Particle Physics'. Since then, I have been wondering if there is a special physical meaning hidden in irreducible ...
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Is the total spin angular momentum $S=2$ symmetric or anti-symmetric?
I am aware that $S=0$ is anti-symmetric, whereas $S=1$ is symmetric. But what about $S=2$ and $S=\frac{3}{2}$? Is it symmetric or anti-symmetric? Is there any general formula for identifying whether $...
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Change in angular velocity of an initially non-rotating spherical object after a collision
In the theoretical scenario below:
Will the object rotate?
My first thoughts on this were:
The initial angular velocity is $0$. This means it does not have any angular momentum (right...?) and it ...
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Why do planets have a greater linear velocity closer to the Sun?
I get that the planets need to have a higher velocity to escape the gravitational well which is deeper when closer to the sun. What I don't understand is what causes this higher velocity. Am I missing ...
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Angular velocity across different reference frames
In classical mechanics:
Logically, it appears to me that if I draw a mark on a ball and let it roll, the amount of time that will pass before the mark reaches the same position (in terms of angles: ...
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A query regarding the definition of Conservation of Angular Momentum
Assuming two hollow cyliders $C_{\text{in}}$ and $C_{\text{out}}$ with radius $r_{\text{in}}$ and $r_{\text{out}}$ such that $r_{\text{in}} < r_{\text{out}}$. Each of the cylinder is uniform and of ...
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Do the operators $\hat{P}^i=-i\partial^i, ~\hat{J}^{ij}=i(x^i\partial^j-x^j\partial^i),$ ever arise in QFT as momentum and angular momentum operators?
The generators of the translation and the rotation group, acting on a classical field (say, a scalar field), are given by the differential operators $$\hat{P}^i=-i\partial^i, ~~\hat{J}^{ij}=i(x^i\...
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A confusion about the spin of a particle described by Dirac equation
The Hamiltonian of a particle described by the Dirac equation neither commutes with the orbital angular momentum nor with the spin angular momentum. However, it commutes with the sum of the orbital ...
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Is there such a thing like $Ψ(s)$ where $s$ is the state of the spin of the particle?
The wavefunction of a particle is most of the times a function of the particle's position. Is there a wavefunction as a function of the particle's spin (or both) and if yes how is it affected?
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How to demonstrate that light carries angular momentum by making an object rotate?
Electromagnetic fields, carry angular momentum. However, I want to demonstrate by an experiment and convince a bunch of high school students, that electromagnetic fields do carry angular momentum. To ...
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Problem similar to Feynman disk paradox. (violation of conservation of angular momentum)
I have a problem(paradox) similar to Feynman disk paradox. There is a example similar to Feynman disk paradox in Griffiths electrodynamics. Example 8.4 shows that the initial angular momentum of ...
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Spin angular momentum?
An electron spins around its axis and magnetic field is produced. It can spin either in clockwise $\left(\frac{1}{2}\right)$ or in counterclockwise $\left(\frac{-1}{2}\right)$ direction.
The spin ...
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Angular momentum and Conservation of energy [duplicate]
Suppose if a ballerina dancer was rotating at a constant angular velocity with her arms wide open. Then she retracted her hands which changed her moment of inertia and increased her angular velocity. ...
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Angular momentum conservation in collisions about point of impact
We use the idea that if a body collides with something, then its angular momentum is conserved with respect to the point of impact. Source
It is clear to me that all impulsive forces and the ...
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Impulse on a rod hinged at a point
I have a situation that is as follows:
A rod of mass $M$ is hinged at one of its end A on a smooth horizontal surface and can rotate about A without friction. A particle of mass $m$ moving on the ...
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An ant is sitting at edge of a rotating disc, if ant reaches the other end, after moving along diameter, then angular velocity will?
an ant is sitting at the edge of a rotating disc, if the ant reaches the other end, after moving along the diameter, then the angular velocity will increase, decrease, or remain constant? will it ...
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Intuition behind torque, rotational inertia and angular momentum
I'm reading about conservation of linear momentum and angular momentum. I understand the idea that angular momentum should be thought of as the "rotational analogue" of linear momentum, just ...
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Physical Pendulum: Sign of Torque
I was looking into some of the motion equations for a physical pendulum that oscillates in small angles (using the approximation $\sin(\theta) \approx \theta$). Specifically, I was interested in ways ...
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Forces on us on a rotating Earth
So the earth is constantly rotating but it doesn't need a force to rotate. It'll rotate indefinitely.(?) But we and other masses on earth need a force on us to continue rotating along with earth? And ...
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Newton's second law for rotating body with changing mass
Newton's second law for a body with changing mass given as $$F=ma + \frac{dm}{dt}v$$ I need the version for rotational motion. By inspection, it seems that it would be $$\tau = I\alpha + \frac{dI}{dt}\...
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Commutator of spin operators
Suppose we are given $\left[S_X, S_Y\right]$, $\left[S_Y, S_Z\right]$ and $\left[S_Z, S_X\right]$, that is the spin operator commutation relations, can we then determine the matrix representation of ...
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What happens when you replace an identity matrix with a matrix full of ones?
In physics, we often use resolutions of identity
$$\sum_n |n\rangle\langle n|=\mathbb{I}$$ to simplify expressions. Sometimes, the "full matrix" (for lack of a better term) $$\sum_{m,n}|m\...
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Composition of angular momentum (quantum): how do we know that finding common eigenspace of $J^2$ and $J_z$ is enough for degeneracy?
I have some basic question on composition of angular momentum (actually spin in my case), I forgot some basis.
The fundamental commutation relations between $J_x,J_y,J_z$ (the three components of the ...
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Reaction torque of angled nutrunner
I have a question regarding the behavior of an right-angled, air/electrical powered nutrunner. More specifically, the reaction torque that the operator is subject to during operation. When the ...
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Rotations of spherical harmonics and Wigner $D$-matrices
I seem to be having trouble understanding how Wigner D-matrices rotate spherical harmonics. I asked this question on the Maths Stack Exchange but decided to cast my net a bit wider and ask the ...
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Basics about angular momentum in quantum mechanics
As usual let $|l,m\rangle$ denote an eigenstate of $\vec{L}^2$ and $L_z$. I know that
\begin{align}
\vec{L}^2 |l,m\rangle &= \hbar^2 l(l+1) |l,m\rangle, \\
L_z |l,m\rangle &= \hbar m |l,m\...
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What is the conversion ratio of linear to angular momentum when a ball hits a rod in space?
If the ball hits the rod at 90 degrees then the rod will start spinning, while also following the original trajectory of the ball. On what factors does the ratio between the two types of momentum ...
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Spin without quantum mechanics?
In Emergence of spin from special relativity some answers discuss how spin can arise in non-relativistic quantum mechanics (let's not enter into those details here). However it is also argued that you ...
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Emergence of spin from special relativity
I have pulled up and read as many answered questions as I can find here on why spin emerges as a consequence of making quantum mechanics compatible with special relativity- and still have problems ...
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How to prove Nitrogen atom with total angular momentum $L=2$ and $L=1$ are not anti-symmetric?
I'm working on Problem 5.13(d) in "Griffiths 《Introduction of Quantum Mechanics》 2nd Edition".
It asked to determine the nitrogen electron configuration by Hund's rule. And here is the ...
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Why Goldstein's book is claiming that radius and angle doesn't contain time variable even there is $\dot{r}$ and $\dot{\theta}$?
$$L=\frac{1}{2}m(\dot{r}^2+r^2\dot{\theta}^2)-V(r)$$
$$p_\theta=\frac{\partial L}{\partial \theta}=mr^2\dot{\theta}$$
$$\dot{p}_\theta=\frac{d}{dt}(mr^2\dot{\theta})$$
Goldstein wrote that $\dot{P}_\...
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Pulling a mass moving in a circle inwards [closed]
In this question, I couldn't derive the right answer if I equate
$$\text{tension} = \text{radial acceleration} = \frac{v^2}{r}$$
Why shouldn't I equate that?
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Why is the rotation about COM? [duplicate]
Suppose a ring is given to us with no hinge as such. Now a bullet comes and strikes the ring and gets embedded in the ring.
The ring will now have linear momentum and some rotation going on. Okay, the ...
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General solution of a ball elastically colliding with a spinning rod
I am working on finding the general solution to a disc colliding with a thin spinning rod in two dimensions floating in free space. The collision is perfectly elastic. The width of the rod is ...
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Inequalities of orbital angular momentum eigenvalues (show that is bounded)
If we apply the rising/lowering operators for angular momentum to a state $|{l,m}\rangle$, we get:
\begin{align}
L_+|{l,m}\rangle = C_+(l,m)|{l,m+1}\rangle \\
L_-|{l,m}\rangle = C_-(l,m)|{l,m-1}\...
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Does Initial Angular Velocity Affect Loss of Angular Velocity in Collisions?
I derived the following formula for change in angular velocity after a collision using conservation of angular momentum.
$$I_1 ω_i+I_2 ω_2i=I_1 ω_f+I_2 ω_2f$$
$$I_1 ω_i-I_1 ω_f=I_2 ω_2f-I_2 ω_2i$$
$$...
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Why have we not found an interior Kerr solution?
The Schwarzschild interior solution was found not so long after the exterior solution was found. I understand that Kerr solution is significantly more complicated and there are more conditions at the ...
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Changing the RPM of a frictionless spinning wheel in a box
Imagine a spinning wheel built into a hand size vacuum box. There is no friction between the axe bearings of the wheel and the box. Let's say that the wheel rotates with 60 RPM. Am I right if I assume:...
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What is the physical significance of $\langle L_x\rangle=\langle L_y\rangle$ and $\langle L_x^2\rangle = \langle L_y^2\rangle$?
If we find the expectation value $$\langle L_x\rangle = \langle L_y\rangle = 0$$ and $$\langle L_x^2\rangle = \langle L_y^2\rangle,$$ what is the physical significance that their values are equal?
