Pauli Spin Matrix Properties
Pauli spin matrices, denoted by $\sigma_\alpha$ (where $\alpha$ can be x, y, or z), are fundamental in quantum mechanics, particularly in describing spin-1/2 particles. They satisfy specific commutation and anticommutation relations.
Commutation Relation
The commutation relation defines how the order of multiplication affects the result for these matrices. It is given by:
$ \sigma_\alpha \sigma_\beta – \sigma_\beta \sigma_\alpha = 2i \sum_\gamma \epsilon_{\alpha\beta\gamma} \sigma_\gamma $
Here, $\epsilon_{\alpha\beta\gamma}$ is the Levi-Civita symbol, which is antisymmetric in any pair of indices and equals 1 for $\epsilon_{xyz}$. The factor '$i$' is the imaginary unit. This relation matches Option 2.
Anticommutation Relation
The anticommutation relation defines the sum of products in different orders. It is given by:
$ \sigma_\alpha \sigma_\beta + \sigma_\beta \sigma_\alpha = 2 \sum_\beta \delta_{\alpha\beta} I $
Where $\delta_{\alpha\beta}$ is the Kronecker delta, which is 1 if $\alpha = \beta$ and 0 otherwise. $I$ represents the identity matrix. This simplifies to $2I$ when $\alpha = \beta$, and 0 when $\alpha \neq \beta$. The relation is correctly represented as $\sigma_\alpha \sigma_\beta + \sigma_\beta \sigma_\alpha = 2\delta_{\alpha\beta}$ when considering the identity matrix is implicit or contextually understood (often written as $2\delta_{\alpha\beta}I$). This matches Option 4.
Conclusion
The Pauli spin matrices satisfy both the commutation relation (Option 2) and the anticommutation relation (Option 4).
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?
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
Consider a spin $S = \hbar/2$ particle in the state $| \phi \rangle = \frac{1}{3} \begin{bmatrix}2 + i \\2 \end{bmatrix} $. The probability that a measurement finds the state with $S_x = + \hbar/2$ is
