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53 views

Does total $\hat{S}^2$ always commute with total $\hat{S}_z$ even for interacting spins?

I was given the following operator $\hat{f}$ describing the interaction of two spin-$\frac12$ particles: $$\hat{f}=a+b{\hat{\bf S}_1}\cdot{\hat{\bf S}_2}.$$ I was told that I can prove that $\hat{f}$...
2
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2answers
83 views

Angular momentum coupling

I read about angular momentum coupling on wikipedia and there are a few things i dont understand. What does this mean "spin and orbital angular momentum of a single object belong to different Hilbert ...
0
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0answers
33 views

Conjugate of total spin operator

I got a lattice, and the total spin operator for x and for y, for that lattice. I know that the x component conmutes with an operator called staggered spin operator in y. I also know that the ...
2
votes
2answers
104 views

Commutation relations

Given that the Hamiltonian for Muonium spin in zero magnetic field is $$\hat{H} = a \vec I \cdot \vec J$$ where $\vec I$ is the spin of a muon, and $\vec J$ is the spin of the electron, what is the ...
0
votes
1answer
542 views

Uncertainty relation for $S_z$ eigenstates [closed]

I am content with the method of finding the uncertainty relation for $L_z$ eigenstates in a spin-1/2 system where $|\uparrow\rangle=|m=1/2\rangle$ and $|\downarrow\rangle=|m=-1/2\rangle$. I have used ...
0
votes
1answer
773 views

Show that the operators that commute with the spin-orbit Hamiltonian do, in fact, commute [closed]

I found the operators to be $J_z, J^2, L^2, S^2$, but how do I prove that they commute? My attempt: For $L^2$, we know that $[\vec{L},L^2]=0$, so $[\vec{L} \cdot \vec{S}, L^2]=0$. But I don't ...
0
votes
1answer
508 views

Commutator of spin and linear momentum

More specifically, what is $[S_z, p^2]$? This came up in a time-evolution problem for $\hat{S}_z(t)$, knowing that that it commutes with the non-kinetic part of some Hamiltonian $\leftrightarrow [S_z, ...
1
vote
1answer
381 views

Finding the commutator between components of spin operator in Schrodinger versus Heisenberg representations

I have a Hamiltonian $$H = a(t) \cdot (S_z) ^2$$ where $H$ and $S_z$ are operators, $S_z$ being the $z$-component of the spin operator. I am trying to show that the commutators for the spin ...
0
votes
0answers
57 views

Why can the y and z-components of spin be measured simultaneously? [duplicate]

I have a gut feeling that this is wrong. By the uncertainty principle where $x,y,z$ are the $x,y,z$ components of spin $$ \sigma_{y}\sigma_{z}\geq \frac{\hbar}{2}\langle x \rangle $$ and it can be ...
3
votes
1answer
460 views

Spin operator: tricky proof using gamma matrices

I have not dealt with the gamma matrices extensively so I am having a bit of trouble here. Basically I want to show that the spin operator defined by $$ \mathbf{\hat{S}} = \frac{1}{2}\gamma^5 \gamma^...
1
vote
0answers
80 views

Do the norms of the total and the orbital angular momentums commute? If yes, why is there a problem with 2p_{1/2}?

Question: For $\vec L$ the orbital angular momentum of an electron, $\bar S$ its spin, and $\vec J:=\vec L+\vec S$ the sum, do $\vec J^2$ and $\vec L^2$ commute? I assume it does: $[\vec J^2,\vec L^...
2
votes
1answer
152 views

Connection between half and whole integer eigenvalues for orbital angular momentum [duplicate]

I have been trying to follow this derivation from Sakurai and Shankar, pulling from both. I would like to see how the following derivation can be extended to orbital angular momentum, and thus find ...
0
votes
1answer
1k views

Spin operators commutation

Why do the spin operators $ S_{x1}$ and $S_{x2}$ of two particles along the $x$-axis commute i.e $S_{1x}S_{x2}-S_{2x}S_{1x}=0 $ ?
1
vote
2answers
218 views

Why angular momentum about three independent axes?

The generic commutation relations for the angular momentum operator are $[J_x, J_y] = i \hbar J_z$, where the $J_i$, $i = x,y,z$ are the components of the angular momentum vector operator, $\mathbf J$....
-1
votes
1answer
1k views

Commutator with Pauli spin matrices and the momentum operator

How is $\left[\vec\sigma \cdot \vec p, \vec \sigma \right]$ proportional to $\vec \sigma\times \vec p$, where $\sigma$ are the Pauli spin matrices and $p$ is the momentum operator?