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I have three Pauli spin matrices corresponding to the spin states along each Cartesian axis. For some rotation about $\Phi_n$ about the $z$-axis we have:$$S_n=S_xsin(\Theta)cos(\Phi) +S_ysin(\Theta)sin(\Phi)+S_zcos(\Theta)$$ Where $\Theta$ is a rotation about the $z$-axis and $\Phi$ is a rotation about the $z$-axis in the $x$-$y$ plane. I don't understand how we go from this to the matrix representation: $$ {S_n=\frac{\hbar}{2} \begin{pmatrix} cos(\Theta) & sin(\Theta)e^{-i\Phi}\\ sin(\Theta)e^{i\Phi} & -cos(\Theta)\\ \end{pmatrix}}$$ Please can someone explain the origin of the complex exponential terms?

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It's just Euler's $\cos \Phi+i\sin \Phi= e^{i\Phi}$ compbined with the usual forms of Pauli's $\sigma_x$ and $\sigma_y$.

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