Simplifying Pauli Matrix Expression
The problem requires simplifying the expression $2\sigma_x \sigma_y + \sigma_y \sigma_x$ using the properties of Pauli matrices $\sigma_x, \sigma_y, \sigma_z$. We need to find the value of this expression.
Pauli Matrix Properties
Recall the fundamental commutation and anti-commutation relations for Pauli matrices:
- $\sigma_x \sigma_y = i\sigma_z$
- $\sigma_y \sigma_x = -i\sigma_z$
These relations are key to solving the problem.
Expression Evaluation
Substitute the known products of Pauli matrices into the given expression:
- Start with the expression: $2\sigma_x \sigma_y + \sigma_y \sigma_x$
- Substitute $\sigma_x \sigma_y = i\sigma_z$: $2(i\sigma_z) + \sigma_y \sigma_x$
- Substitute $\sigma_y \sigma_x = -i\sigma_z$: $2(i\sigma_z) + (-i\sigma_z)$
- Simplify the terms: $2i\sigma_z - i\sigma_z$
- Combine the terms: $(2 - 1)i\sigma_z$
- Final result: $i\sigma_z$
Therefore, the expression $2\sigma_x \sigma_y + \sigma_y \sigma_x$ simplifies to $i\sigma_z$.
Important Questions from Spin Electron Spin Pauli Matrices
- Atomic numbers of V, Cr, Fe and Zn are 23, 24, 26 and 30, respectively. Which one of the following materials does NOT show an electron spin resonance (ESR) spectra?
Consider two non-identical spin $\frac{1}{2}$ particles labelled $1$ and $2$ in the spin product state $|\frac{1}{2}, \frac{1}{2}\rangle_1 |\frac{1}{2}, -\frac{1}{2}\rangle$. The Hamiltonian of the system is
$H = \frac{4\lambda}{\hbar^2} \vec{S}_1 \cdot \vec{S}_2$,
where $\vec{S}_1$ and $\vec{S}_2$ are the spin operators of particles $1$ and $2$, respectively, and $\lambda$ is a constant with appropriate dimensions. What is the expectation value of $H$ in the above state?- A spin $\frac{1}{2}$ particle is in a spin up state along the $x$-axis (with unit vector $\hat{x}$) and is denoted as $|\frac{1}{2}, \frac{1}{2}\rangle_x$. What is the probability of finding the particle to be in a spin up state along the direction $\hat{x}'$, which lies in the $xy$-plane and makes an angle $\theta$ with respect to the positive $x$-axis, if such a measurement is made?
- Pauli spin matrices satisfy
An electron with mass $m$ and charge $q$ is in the spin up state $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ at time $t = 0$. A constant magnetic field is applied along the y-axis, $\vec{B} = B_0 \hat{j}$, where $B_0$ is a constant. The Hamiltonian of the system is $H = -\hbar \omega \sigma_y$, where $\omega = \frac{q B_0}{2m} > 0$ and $\sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$. The minimum time after which the electron will be in the spin down state along the x-axis, i.e., $\frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ -1 \end{pmatrix}$, is
