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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.

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This is a convention. The operators given by $\hat S$ are written in terms of the Pauli matrices such that

$$\hat S = \frac{\hbar}{2} \hat \sigma$$

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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.

Notice that this will not always be true. There are other representations of the spin, and, in those representations, the spin will be given by different matrices.

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