-1
votes
1answer
151 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?
1
vote
1answer
178 views

The Holstein-Primakoff Representation (approximation)

I have a question regarding the Holstein-Primakoff representation. In the HP-representation we define the spin operators in terms of bosonic creation and annihilation operators. $$ S_j^+ = \sqrt{2S ...
2
votes
1answer
120 views

Intrinsic parity of particle and antiparticle with spin zero

I need to prove that the intrinsic parities of a particle and antiparticle with spin zero are the same. Can I prove that by an argument that operator of $P$-inversion commutes with charge conjugation ...
0
votes
0answers
87 views

Diagonalize a dot product with Pauli matrices

How can I diagonalize the following operator? $$\lambda \hat{\vec{\sigma}}\cdot\vec{r}$$ where $\lambda$ is a real constant, $\hat{\vec{\sigma}}=(\hat{\sigma_{x}},\hat{\sigma_{y}},\hat{\sigma_{z}}) ...
1
vote
1answer
123 views

Ensemble average of product of spin operators?

How do you evaluate the canonical ensemble average of a product of spins, e.g.: $$[S_zS_x]$$ Where: $$S_x = \frac{\hbar}{2} \begin{pmatrix} 0 & 1\\ 1 & 0\\ \end{pmatrix}$$ $$S_y = ...
3
votes
1answer
84 views

The square of Pauli-Lubanski operator

Let's have Pauli-Lubanski operator: $$ \hat {W}^{\alpha} = \frac{1}{2}\varepsilon^{\alpha \beta \gamma \delta}\hat {J}_{\beta \gamma}\hat {P}_{\delta} = \frac{1}{2}\varepsilon^{\alpha \beta \gamma ...
0
votes
0answers
168 views

Operator of full angular momentum and spin operator

In relativistic mechanics classical angular momentum of one particle with center of energy vector creates angular momentum tensor. It's algebra is the same as Poincare group algebra. So in operator's ...
0
votes
0answers
48 views

What physical value is described by following operator?

Let's have the system of point-like non-interacting particles and it's own angular momentum $$ \mathbf L_{1} = \mathbf L - [\mathbf R_{E} \times \mathbf P], $$ where $\mathbf R_{E}$ - center of energy ...
11
votes
3answers
1k views

How to tackle 'dot' product for spin matrices

I read a textbook today on quantum mechanics regarding the Pauli spin matrices for two particles, it gives the Hamiltonian as $$ H = \alpha[\sigma_z^1 + \sigma_z^2] + ...
4
votes
2answers
375 views

When you apply the spin operator, what exactly is does it tell you?

The example I'm trying to understand is: $ \hat{S}_{x} \begin{pmatrix} \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}} \end{pmatrix} = 1/2 \begin{pmatrix} \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}} ...
2
votes
1answer
228 views

How could $\textbf{S}^2$ not be a multiple of the identity?

I'm self-studying quantum mechanics with Sakurai's book (Modern Quantum Mechanics, 2nd edition) and came across the following in reference to the operator $\textbf{S}^2$: As will be shown in ...
1
vote
1answer
637 views

Probability of getting a particular spin

I'm a beginner in quantum mechanics, and I'm a bit confused about states and the probability to measure certain values. I would like to understand at least the following simplified situation: ...