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The following is from 'Modern Quantum Mechanics' by J.J. Sakurai, page 159.

[tex] \left( \frac{ \hbar }{ 2 } \right) \exp \left( \frac{i S_{z} \phi}{\hbar} \right) \left{ left( \mid + \rangle \langle - \mid \right) + \left( \mid - \rangle \langle + \mid \right) right} \exp \left( \frac{- i S_{z} \phi}{\hbar} [/tex]

[tex] = \left( \frac{\hbar}{2} \right) \left( e^{i \phi / 2} \mid + \rangle \langle - \mid e^{i \phi / 2} + e^{- i \phi / 2} \mid - \rangle \langle + \mid e^{- i \phi / 2} \right) [/tex]

How do I get from the first to the second?

[tex] \left( \frac{ \hbar }{ 2 } \right) \exp \left( \frac{i S_{z} \phi}{\hbar} \right) \left{ left( \mid + \rangle \langle - \mid \right) + \left( \mid - \rangle \langle + \mid \right) right} \exp \left( \frac{- i S_{z} \phi}{\hbar} [/tex]

[tex] = \left( \frac{\hbar}{2} \right) \left( e^{i \phi / 2} \mid + \rangle \langle - \mid e^{i \phi / 2} + e^{- i \phi / 2} \mid - \rangle \langle + \mid e^{- i \phi / 2} \right) [/tex]

How do I get from the first to the second?

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