To construct the rotation operator for spin, we use the concept of spin angular momentum in quantum mechanics. The rotation operator for a spin-$ \frac{1}{2} $ particle (like an electron) is particularly important. Here's the step-by-step construction:
For a spin-$ \frac{1}{2} $ particle, the spin operators are given by:
$$ S_x = \frac{\hbar}{2} \sigma_x, \quad S_y = \frac{\hbar}{2} \sigma_y, \quad S_z = \frac{\hbar}{2} \sigma_z $$
where $ \sigma_x, \sigma_y, \sigma_z $ are the Pauli matrices, and $ \hbar $ is the reduced Planck constant.
The general rotation operator $ R(\hat{n}, \theta) $ around an axis $ \hat{n} = (n_x, n_y, n_z) $ by an angle $ \theta $ is given by:
$$ R(\hat{n}, \theta) = e^{-i \frac{\theta}{\hbar} (\hat{n} \cdot \mathbf{S})} $$
For spin-$ \frac{1}{2} $, this becomes:
$$ R(\hat{n}, \theta) = e^{-i \frac{\theta}{2} (\hat{n} \cdot \mathbf{\sigma})} $$
where $ \hat{n} \cdot \mathbf{\sigma} = n_x \sigma_x + n_y \sigma_y + n_z \sigma_z $.
To find the matrix form of $ R(\hat{n}, \theta) $, we can use the properties of the Pauli matrices and the exponential of operators:
$$ R(\hat{n}, \theta) = \cos \left( \frac{\theta}{2} \right) I - i \sin \left( \frac{\theta}{2} \right) (\hat{n} \cdot \mathbf{\sigma}) $$
This can be expanded as:
$$ R(\hat{n}, \theta) = \cos \left( \frac{\theta}{2} \right) I - i \sin \left( \frac{\theta}{2} \right) (n_x \sigma_x + n_y \sigma_y + n_z \sigma_z) $$
Rotation around the $ x $-axis:
$$ R(\hat{x}, \theta) = \cos \left( \frac{\theta}{2} \right) I - i \sin \left( \frac{\theta}{2} \right) \sigma_x $$
Rotation around the $ y $-axis:
$$ R(\hat{y}, \theta) = \cos \left( \frac{\theta}{2} \right) I - i \sin \left( \frac{\theta}{2} \right) \sigma_y $$
Rotation around the $ z $-axis:
$$ R(\hat{z}, \theta) = \cos \left( \frac{\theta}{2} \right) I - i \sin \left( \frac{\theta}{2} \right) \sigma_z $$
The rotation operator for spin-$ \frac{1}{2} $ particles allows us to describe rotations in quantum mechanics. Using the Pauli matrices and the exponential form, we can construct rotation matrices for any axis and angle of rotation.