5
votes
Why does $J^2$ only commute with one of $Jx, Jy, Jz$ and not all?
It's certainly true that $J^2$ commutes with each of $J_x$, $J_y$ and $J_z$. Additionally, it is true that for each of $J_x$, $J_y$ and $J_z$, a set of simultaneous eigenstates between that operator ...
- 416
4
votes
Accepted
Extra energy by applying torque?
In both cases the collisions are inelastic (as the bullet stops inside the block), so some of the initial kinetic energy is converted to heat due to friction between the block and the bullet. The ...
- 2,007
3
votes
Beta decay of 6-Helium
Hans Bethe and Philip Morrison give the following "hand-waving" answer to this question on pages 226-227 of the 2nd edition of their influential non-textbook on nuclear physics titled ...
3
votes
Possible errata Landau and Lifshitz in $\S29$ Matrix elements of vectors in Quantum Mechanics Third Edition
I don't know what is wrong with your reasoning, since it seems to return the right result in the special case where $A_{yy}=A_{zz}$. That said, I never saw this approach, and I searched in Landau'book ...
- 3,162
3
votes
Accepted
Possible errata Landau and Lifshitz in $\S29$ Matrix elements of vectors in Quantum Mechanics Third Edition
The identity to use is:
$$[p_i,r_j]=-i\hbar\delta_{ij}$$
You are doing something wrong:
$$[L_z,r_x]=[r_xp_y-r_yp_x,r_x]=r_x[p_y,r_x]-r_y[p_x,r_x]$$
$$=0-r_y (-i\hbar \delta_{x,x})=i\hbar r_y$$
It ...
- 5,831
2
votes
What is predicted to happen for electron beams in the Stern-Gerlach experiment?
In 2015, someone did this experiment. The electron beam does split!
Observing the spin of free electrons in action(Stern-Gerlach experiment by free electrons)
I'll quote a paragraph in the result of ...
- 99
2
votes
Total orbital angular momentum of closed orbital shell
Can anyone please help me in understanding why total orbital angular momentum of a closed orbital is zero?
The total orbital angular momentum is obtained by constructing "multiplets."
The ...
- 23.3k
2
votes
Collision of rotating sticks
The angular velocity vectors produce a resultant vector just like any other vectors do. The combined spin axis is shown in the diagram below as the dashed red diagonal line.
The red point labeled B ...
- 10.1k
2
votes
Why is the spin degree of freedom associated with angular momentum?
I will give a more mathematical take on the answer, mainly taken from the following lecture notes: https://scholar.harvard.edu/files/noahmiller/files/representation-theory-quantum.pdf
It boils down to ...
- 304
2
votes
Why is the spin degree of freedom associated with angular momentum?
Spin is linked to angular momentum because spin is linked to symmetry under rotations.
When a Lagrangian has symmetry under rotations, there is a conserved quantity associated with that symmetry. We ...
- 61.8k
2
votes
Why is the spin degree of freedom associated with angular momentum?
Because it is angular momentum. More correctly, it's an angular momentum operator. Spin operators follow the conmutation relation
$$[S_i,S_j]=i\hbar\varepsilon_{ijk}S_k.$$
Any three operators $S_i$ ...
- 3,430
2
votes
How to test whether a 2-particle Hamiltonian is rotationally invariant?
Rotational invariance means the hamiltonian is invariant under all rotations, so it commutes with all generators of such. If you only had one particle, so if you extinguished the second one, $\hat{H} =...
- 66.3k
2
votes
Why does $J^2$ only commute with one of $Jx, Jy, Jz$ and not all?
$\vec{J}^2$ actually commutes with all three $J_x$, $J_y$ and $J_z$. So $\vec{J}^2$ have common eigenstates with $J_x$ and have common eigenstates with $J_z$. How is it possible? Those are different ...
- 8,739
2
votes
In a 2-electron system (such as Helium atom), is the Total Angular Momentum quantum number $j = \ell$?
No. The Pauli principle is a statement for the entire state, not just the spin part. It is perfectly possible to have $s=0$ - the antisymmetric spin singlet - if the spatial part is symmetric - say $\...
- 47.8k
2
votes
Is linear momentum always conserved in absence of external forces
Both linear and angular momentum must be conserved, regardless of whether or not mechanical energy is conserved.
If impacting the rod at the end produces less linear momentum in the rod than impacting ...
- 77.9k
1
vote
Hamiltonian is time independent in rotating frame
While it's true that the dot product form of $H$ suggests rotational invariance, we must remember that the spin operator $\vec{S}$ itself transforms under rotations.
Let's denote the components of the ...
- 764
1
vote
Accepted
Deriving the normalization factors of $SU(2)$
From the commutation relations of the SU(2) algebra we get
$$J_3 J^\pm | m,\alpha \rangle = J^\pm J_3 | m,\alpha \rangle \pm J^\pm | m,\alpha \rangle = (m \pm 1) J^\pm | m,\alpha \rangle.$$
This ...
- 26
1
vote
Accepted
Topological magnon bands, chiral edge states and broken time-reversal symmetry
If you are looking at say Fig. 4(a) in J. Phys.: Condens. Matter 28 386001 or Fig. 1(b) in PRL 117, 227201, the way they get the band structure is by diagonalizing the Hamiltonian on a slab geometry, ...
- 1,058
1
vote
How to show that $\int d^3 x\; e^{ikx}(x^1 \partial^2-x^2\partial^1)\varphi (x)=0$?
The integral doesn't always identically vanish. Strictly speaking it only vanishes if you approach spatial infinity using a cylindrical geometry.
I will do the problem in 2D because the third ...
- 438
1
vote
Is linear momentum always conserved in absence of external forces
Conservation of linear momentum always applies. In the case of the ball hitting the center of the rod, you know by symmetry that the rod has no rotational momentum or energy. That means that you can ...
- 593
1
vote
Accepted
Graphical representations of the vector model of quantum angular momentum
Your text should indicate this is a graphical shared metaphor in the community to summarize the quantum picture geometrically, of help in rough estimates. Skipping the real McCoy, the cartoon metaphor ...
- 66.3k
1
vote
How to test whether a 2-particle Hamiltonian is rotationally invariant?
To determine whether a Hamiltonian possesses a given symmetry, you determine how the symmetries act on the Hilbert space (i.e. the wave functions) and then check if the actions commute with the ...
- 3,749
1
vote
Is there an Ehrenfest-like result for the expectation value of orbital angular momentum?
Yes, indeed, it is possible to derive such a result... although it might look somewhat trivial, if compared to position and momentum in an arbitrary potential.
Hamiltonian of angular momentum in ...
- 65k
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