# Quantum mechanics and spin $1/2$

I was reading a book but, I am getting confused in a part which claims that:

$$\exp\left(\frac{-i \vec S \cdot \hat n \, \phi}{\hbar}\right)=\exp\left(\frac{-i \vec \sigma \cdot \hat n \, \phi}{2}\right)$$

where $$\vec S=(S_x,S_y,S_z)$$ and $$\vec\sigma = (\sigma_1, \sigma_2, \sigma_3)$$ (Pauli).

As the book doesn't show a proof, I think this should be immediate to me. Yet, I can not see why.

This is a convention. The operators given by $$\hat S$$ are written in terms of the Pauli matrices such that
$$\hat S = \frac{\hbar}{2} \hat \sigma$$
For spin 1/2, one possible representation uses the Pauli matrices with $$S_i = \dfrac{\hbar}{2}\sigma_i.$$ There is a subsection discussing this here.