# All Questions

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### 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 $\hat{f}$...
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 ...
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 ...
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 ...
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 ...
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 ...
1answer
508 views

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 ...
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 $S_{1x}S_{x2}-S_{2x}S_{1x}=0$ ?
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, $\mathbf J$....
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?