For a two-nucleon system in spin singlet state, the spin is represented through the Pauli matrices $ \sigma_1, \sigma_2$ for particles 1 and 2, respectively. The value of $( \sigma_1 \cdot \sigma_2)$ (in integer) is ________
For a two-nucleon system in a spin singlet state, the total spin quantum number is $ S=0$. The spin operators for individual particles are related to the Pauli matrices by $ \mathbf{S}_i = \frac{\hbar}{2} \boldsymbol{\sigma}_i$, where $i=1, 2$. The spin singlet state is an eigenstate of the total spin operator $ \mathbf{S} = \mathbf{S}_1 + \mathbf{S}_2$ with eigenvalue $ S=0$.
Spin Operator Value Calculation
We consider the square of the total spin operator:
$ \mathbf{S}^2 = (\mathbf{S}_1 + \mathbf{S}_2)^2 = \mathbf{S}_1^2 + \mathbf{S}_2^2 + 2 \mathbf{S}_1 \cdot \mathbf{S}_2 $Rearranging to find the term $ \mathbf{S}_1 \cdot \mathbf{S}_2$:
$ 2 \mathbf{S}_1 \cdot \mathbf{S}_2 = \mathbf{S}^2 - \mathbf{S}_1^2 - \mathbf{S}_2^2 $The eigenvalues for the square of spin operators are:
- $ \mathbf{S}^2 $ eigenvalue for $ S=0 $ is $ S(S+1)\hbar^2 = 0(0+1)\hbar^2 = 0 $.
- $ \mathbf{S}_i^2 $ eigenvalue for spin $ s=1/2 $ is $ s(s+1)\hbar^2 = \frac{1}{2}(\frac{1}{2}+1)\hbar^2 = \frac{3}{4}\hbar^2 $.
Substitute these eigenvalues back into the equation:
$ 2 \mathbf{S}_1 \cdot \mathbf{S}_2 = 0 - \frac{3}{4}\hbar^2 - \frac{3}{4}\hbar^2 $ $ 2 \mathbf{S}_1 \cdot \mathbf{S}_2 = - \frac{3}{2}\hbar^2 $Pauli Matrices Scalar Product
Now, substitute $ \mathbf{S}_i = \frac{\hbar}{2} \boldsymbol{\sigma}_i$:
$ 2 \left( \frac{\hbar}{2} \boldsymbol{\sigma}_1 \right) \cdot \left( \frac{\hbar}{2} \boldsymbol{\sigma}_2 \right) = - \frac{3}{2}\hbar^2 $ $ 2 \frac{\hbar^2}{4} (\boldsymbol{\sigma}_1 \cdot \boldsymbol{\sigma}_2) = - \frac{3}{2}\hbar^2 $ $ \frac{\hbar^2}{2} (\boldsymbol{\sigma}_1 \cdot \boldsymbol{\sigma}_2) = - \frac{3}{2}\hbar^2 $Solving for $ (\boldsymbol{\sigma}_1 \cdot \boldsymbol{\sigma}_2)$:
$ \boldsymbol{\sigma}_1 \cdot \boldsymbol{\sigma}_2 = \frac{- \frac{3}{2}\hbar^2}{\frac{\hbar^2}{2}} $ $ \boldsymbol{\sigma}_1 \cdot \boldsymbol{\sigma}_2 = -3 $The value of $ (\sigma_1 \cdot \sigma_2)$ for a two-nucleon system in the spin singlet state is -3.
Important Questions from Spin Electron Spin Pauli Matrices
- Consider the Pauli matrices $\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$, $\sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$, $\sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$.
The value of $\text{Tr}(\sigma_z [\sigma_x, \sigma_y])$ is - The Hamiltonian of two interacting spin-1/2 particles is $H = \frac{A}{\hbar^2} \vec{S}_1 \cdot \vec{S}_2$, where $\vec{S}_1$ and $\vec{S}_2$ are the spin angular momenta of particles 1 and 2, respectively. Here, $A = 10.56 \text{ eV}$. The energy in eV required to induce an excitation from the ground state to the excited state (rounded off to two decimal places) is _____
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 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?
