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
- 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?
