The Pauli matrices for three spin-$\frac{1}{2}$ particles are $\vec{\sigma}_1,\ \vec{\sigma}_2$, and $\vec{\sigma}_3$, respectively. The dimension of the Hilbert space required to define an operator $\hat{O} = \vec{\sigma}_1 \cdot \vec{ \sigma}_2 \times \vec{ \sigma}_3$ is ________
Spin Particle Hilbert Space Dimension
To determine the dimension of the Hilbert space required for the operator $\hat{O} = \vec{\sigma}_1 \cdot \vec{ \sigma}_2 \times \vec{ \sigma}_3$, we need to consider the Hilbert space of the system composed of three spin-$\frac{1}{2}$ particles.
Single Particle Hilbert Space
A single spin-$\frac{1}{2}$ particle is described by a 2-dimensional Hilbert space. This is because the spin state can be represented by a 2-component spinor (e.g., spin up $|\uparrow\rangle$ and spin down $|\downarrow\rangle$).
Combined System Hilbert Space
For a system composed of multiple independent particles, the total Hilbert space is the tensor product of the individual Hilbert spaces. If particle $i$ has a Hilbert space dimension $D_i$, then a system of $n$ particles has a total Hilbert space dimension $D_{total} = D_1 \times D_2 \times \dots \times D_n$.
- In this case, we have three particles ($n=3$).
- Each particle is spin-$\frac{1}{2}$, so its Hilbert space dimension is $D_1 = D_2 = D_3 = 2$.
Calculating Operator Dimension
The total dimension of the Hilbert space required to define operators acting on this three-particle system is calculated as:
$D_{total} = D_1 \times D_2 \times D_3 = 2 \times 2 \times 2 = 2^3 = 8$
The specific form of the operator $\hat{O} = \vec{\sigma}_1 \cdot \vec{ \sigma}_2 \times \vec{ \sigma}_3$, involving the Pauli matrices $\vec{\sigma}_1, \vec{\sigma}_2, \vec{\sigma}_3$ for each particle, requires this 8-dimensional space to be defined. Each $\vec{\sigma}_i$ acts on the $i$-th particle's Hilbert space.
Therefore, the dimension of the Hilbert space required is 8.
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
