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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Linear speed from conservation of angular momentum and energy
A massless spring with constant $k$ is mounted on a turntable of rotational inertia $I$, as shown in the figure. The turntable is on a frictionless vertical axle, though initially it's not rotating. ...
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Transformation of coordinate in Lagrangian
Lagrangian for a Central force problem is:
$$\mathcal{L} = \frac{1}{2}\mu(\dot{r} + r^{2}(\dot{\theta}^{2} + sin^{2}\theta\cdot \dot{\varphi}^{2})) - U(r)$$
We know that angular momentum is defined as:...
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Question Regarding the conservation of momentum in an inelastic collision of two rods
I am tasked with solving this question but am facing some intuition difficulty.
consider this system:
The empty circle signifies a nail that is stuck in the wall.
I am unsure if there is conversion ...
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Equation to define the change in orbital radius given a situation where angular momentum is conserved but energy is lost
I am considering the motion of two satellites around a Protostar. I need an equation to define the change in orbital radius given a situation where angular momentum is conserved but energy is lost.
<...
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Spin vector representation for 1/2 spin
I know that, for 1/2 spin systems, the projection of the spin vector along one of the base's axis can be represented using Pauli's matrices as $\hat{S}_i = \frac{\hbar}{2}\sigma_i$.
While studying ...
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Determining $L_z$ eigenvalues via superposition of eigenstates
This is exercise 12.3.2. from Shankar's 'Principles of Quantum Mechanics'
Part 1: "By considering the superposition of two allowed $l_{z}$ eigenstates $$\psi(\rho,\phi)=A(\rho)e^{i\phi l_{z}/\...
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Mass difference between two baryons
I have been trying to make sense out of this (unsuccessfully for days). It's an exercise on Particle Physics. Exercise asks to calculate the mass difference between baryons ($cuu$) with
\begin{...
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Calculate total angular momentum of object rotating about 2 axis (e.g. Earth)
Consider the Earth. It rotates about its own axis (going through the poles) with some angular velocity $\vec\omega$, and around the sun, with some angular velocity $\vec\Omega$.
In every textbook/web ...
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How to solve Problem 2 chapter 12 Wald's book? [closed]
In here you find the problem from Wald's book.
Could you plz help in solving it?
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How do you show that a state is a simultaneous eigenstate of $\hat L_z$ and $\hat L^2$?
what is the general process for showing that a given state is a simultaneous eigenstate of the angular momentum operators $\hat L_z$ and $\hat L^2$? I've searched for a while but I'm not really ...
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Which force exerted a torque?
Suppose a horizontal disc fixed in the center with a vertical shaft passing perpendicular to the plane, is rotating at some angular speed and there is an insect sitting initially at center. The insect ...
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How is spin defined in Quantum Mechanics, exactly?
I know that spin gets a proper and complete definition in Quantum Field Theory, when we account for relativity in our quantum theory. This question is not about this.
I am instead interested in ...
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How angular momentum composition works?
I am having some real difficulties in understanding how to work with the total angular momentum $J$. Especially regarding change of basis between the base $|l_1,m_1,l_2,m_2\rangle$ (or $|l,m,s,s_z\...
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Angular momentum of a purely rotating body about any axis
It's proved in my K&K mechanics textbook that in pure rotation about an axis passing through the body it's angular momentum is $I\omega$.
What about if I want to find the angular momentum about ...
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Non-conservation of angular momentum example but is an external torque really required?
Say a man is twisting his body using internal body muscle contractions and torque interactions between his feet and ground (no slipping).
From the answers of a previous question I raised, these ground ...
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Where is my blunder in this angular momentum “identity”?
I was searching for ways to reduce the following expression
$$
A = \sum_{m_3=-l_3}^{l_3}C_{l_1m_1,l_3m_3}^{l_2m_2} Y_{l_3m_3}(\hat u)
$$
where $\hat u$ is a unit vector, $Y_{lm}(\hat u)$ is a ...
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Integral of the product of 4 spherical harmonics
Recently, I saw a closed formula for the integral of the product of three spherical harmonics in two dimensions here
Integral of the product of three spherical harmonics
and I was wondering if someone ...
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What experiment confirms $\mathbf{J}^2 = \hbar^2 j(j+1)$?
I learned that if we measure the spin angular momentum of an electron in
one direction $J_z$, we get $\pm \frac{1}{2} \hbar$. But if we measure
the magnitude of the angular momentum $\mathbf{J}^2$, ...
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Where does this formula come from?
I am doing revision for my module stellar & galactic astrophysics and have come upon this formula which I cannot seem to derive. Could someone please explain where it comes from?
"For an ...
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Probability of finding certain value of the $z$-component of the angular momentum when the wave function contains multiple $l$-values [closed]
Suppose that $$ \psi = \frac12 Y_{00}+\frac1{\sqrt 3}Y_{11}+\frac 12 Y_{1,-1}+\frac1{\sqrt6}Y_{22}.$$ This wave function is not an eigenstate of $\hat{L}_z$. If a measurement of the $z$-component of ...
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Angular Momentum of a rigid body w.r.t to an axis about which it is not rotating
suppose we need to calculate $\overrightarrow{L}$ about an axis, but the rigid body is not rotating about this axis. Can we define the $\overrightarrow{L}_{axis}$ still? I think we should be able to ...
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How can the 'spin' angular momentum vector NOT always be parallel to the spin angular velocity?
From Wikipedia:
The orbital angular momentum vector of a point particle is always parallel and directly proportional to the orbital angular velocity vector $\omega$ of the particle, where the ...
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Is it easier to apply angular momentum to a rocket with the engine off?
Let's say I've got a rocket that I need to rotate for my next maneuver. Assuming it's flying through full vacuum, I can turn off my engine first, and then use reaction control thrusters to rotate, or ...
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Uniform / rigid rotation in fluids, velocity gradients and resulting friction
In the derivation of the Navier-Stokes equation e.g. in Landau & Lifshtiz Volume 6 on fluid mechanics it is stated that the viscous stress tensor
$\sigma_{ij}^{\prime}$ must also vanish when the ...
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Exponential of spin operator
I was working with time evolution of a spin state under constant magnetic field and I stumbled upon this operator:
$$\exp\left[{\frac{i \omega _0 t}{2} S_x}\right]$$
where $S_x$ (also called $\sigma ...
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The relation between of three angular momentum coupling basis
This is the note about three angular momentum coupling. But I don't know whether I can use Eq.(9) directly.
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How to determine the summing range of the $j_{12}$ for $ 6j$ symbols?
$$
|j_{1},j_{2}j_{3}(j_{23});j\rangle = \sum_{j_{12}}(-1)^{j_{1}+j_{2}+j_{3}+j}\hat{j}_{12}\hat{j}_{23}
\begin{Bmatrix}
j_{1}&j_{2}&j_{12}\\
j_{3}&j&...
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Commutation relations of generators of $SO(N)$
That is, $J_{i j}$ is a tensor. We can take this a step further, and let $R^{\prime}$ itself be an infinitesimal rotation, of the form $R^{\prime} \rightarrow 1+\omega^{\prime},$ with $\omega_{i j}^{\...
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How to find angular momentum about other axes?
Two particles each of mass m are attached by a light rod (massless and cannot bend or stretch) of length l. A particle "A" of same mass strikes B. The collision is perfectly elastic. Find ...
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Do good quantum numbers matter for the Wigner-Eckart theorem?
I have a question related to the following passage in the quantum mechanical scattering textbook by Taylor,
Here Taylor makes the choice to use a basis of total angular momentum eigenvectors instead ...
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Decomposing spin $n$ into $n$ spin $1$'s
I would like to write a spin $n$ field $\psi^n_m$ as a tensor field $\psi^{\mu_1\cdots\mu_n}$ (where $\mu_i=1,2,3$), and I have boiled down the problem to decomposing the state $|n,m\rangle$ into a ...
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Gravitoelectromagnetic effect of a rotating disc
Assume a thin disc in a vacuum with 2 objects placed along the axis of the disc - one above the disc and one below, each an equal distance from the disc. Each object will feel an identical force of ...
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Indistinguishability in Spin-1/2-system
In terms of statistical physics I thought the microcanonical partition function can be interpreted as summing over all possible quantum numbers. Neglecting indistinguishability in the case of two ...
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Non-homogeneous cylinder collides with small mass
I seek some guidance with regard to the following.
Question: The distribution of mass on a cylinder C of radius R and height H is given by $\rho(r)=\rho_0(1-r/R)$, where r is the radial distance from ...
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Do pseudovectors also transform differently to vectors under spatial dilation, not just reflections and parity?
It is frequently expressed online that the only difference between vectors and pseudovectors is a change in sign with reflection/parity transformations etc.
For instance the pseudovector angular ...
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Can all pseudotensors be expressed as components of higher rank proper tensors?
Axial vectors like angular momentum $L^k$ are not proper vectors and thus do not follow the vector transformation law: $\bar v^i = \Lambda^i_j v^j$. Instead they are 'pseudovectors' which follow the ...
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Gravitational collapse of a gas/dust cloud into a disk and subsequent evolution (minimizing energy?)
Assuming a relatively isolated cloud of dust and gas with a well-defined total mass and well-defined net angular momentum. Perhaps also assume that the size (or density) of the initial cloud is ...
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Is it possible that the energy of the energy eigenstates $|LSJM_J\rangle$ depend on $M_L$ and/or $M_S$?
If a many-electron Hamiltonian $H$ commutes with ${\vec L}^2, {\vec S}^2, {\vec J}^2$, and $J_z$ but not $L_z$ and $S_z$, the energy eigenstates are designated by $|LSJM_J\rangle$. Since there is no ...
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How is a tensor operator defined in terms of commutators?
If $J_i$ represent the angular momentum operators, then a scalar operator $S$ (rank-0 tensor) is defined as an operator which satisfies $$[S,J_i]=0$$ for $i=1,2,3$.
$A_i$ is a vector (rank-1 tensor) ...
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An application of Conservation of Angular Momentum
One of the applications of the law of conservation of angular momentum involves a helicopter with a single propeller. A/c the book, a helicopter with one propeller would rotate itself in the opposite ...
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An ice skater with only one arm extended
Imagine an ice skater spinning around its axis. It is well-known that if she extends her arms, she is spinning slower and by moving her arms towards the body, she is spinning faster.
But what happens ...
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Spherical pendulum torque and angular momentum
I want to know if the angular momentum equation $\frac{dL}{dt}=\vec\tau$ holds for the spherical momentum. Suppose that the angle $\theta$ is constant and the mass goes on uniform circular motion ...
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How to build an intuition regarding QM angular momentum?
I am trying to build an intuition on how angular momentum algebra works.
From what I currently understand there is a set of rules we must know to deal with angular momentum:
The commutator of angular ...
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Physics of pushing the top of a box a little bit to make it rock back and forth
Context:
I'm trying to make a simulation of a box that you can apply force to the top and it will oscillate back and forth until the energy in the system reaches equilibrium again.
Visual ...
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About the Lie algebra of the angular momentum Poisson bracket structure [duplicate]
The components of the classical angular momentum $L_i$, satisfy the Poisson bracket relation $$\{L_i,L_j\}=\epsilon_{ijk}L_k,\tag{1}$$ and forms a Lie algebra (i.e, anti-symmetric, obeys the Jacobi ...
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What does it “physically” mean, in terms of uncertainty and measurements, for a commutator to be different than zero in quantum mechanics?
Let's consider the commutator $[L_i,L_j] = i \hbar L_k$ of the angular momentum. The consequence of this equation is that two components of the angular momentum cannot be simultaneously measured. I ...
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Wigner-Eckart theorem: what is $q $? How does $T^{(k)}_{q}$ relate to an operator?
We saw in class that the Wigner-Eckart theorem is,
$$
\langle \alpha', j',m'|T^{(k)}_{q}|\alpha, j, m \rangle = \langle j,k;m,q|j',m'\rangle \frac{\langle \alpha', j'||T^{(k)}||\alpha, j \rangle}{\...
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Does a non-rotating satellite always show the same side towards the main body?
If an initially non-rotating (zero angular momentum) satellite starts moving around a main body, such as the earth,
will then the satellite continue to have zero rotation or
will it show the same ...
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About the angular momentum of the particle system relative to the center of mass reference
I know its correct expression:$L=\sum_{i}^{n}\left(\vec{r_i}\times m_i\vec{v_i}\right)$
But the textbook thinks:$\vec{F_i}=\frac{d}{dt}\left(m_i\vec{v_i}\right)$,But the speed here is in the center of ...
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Kepler's second law in General Relativity
Kepler's second law, that planets in orbits sweep equal area in equal time, is a consequence of orbital angular momentum conservation. In the case of Schwarzschild spacetime, the angular orbital ...
