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for compute:

$$e^{\frac{iS_z\phi}{\hbar}}S_x e^{\frac{-iS_z\phi}{\hbar}}$$

so, if we use $$S_x=(\frac{\hbar}{2})[(|+><-|)+(|-><+|)]$$

$$e^{\frac{iS_z\phi}{\hbar}}(\frac{\hbar}{2})[(|+><-|)+(|-><+|)] e^{\frac{-iS_z\phi}{\hbar}}$$

so, why that is equal to $$(\frac{\hbar}{2})[\frac{i\phi}{2}|+><-|\frac{i\phi}{2}+\frac{-i\phi}{2}|-><+|\frac{-i\phi}{2}]$$

??

$$e^{\frac{iS_z\phi}{\hbar}}S_x e^{\frac{-iS_z\phi}{\hbar}}$$

so, if we use $$S_x=(\frac{\hbar}{2})[(|+><-|)+(|-><+|)]$$

$$e^{\frac{iS_z\phi}{\hbar}}(\frac{\hbar}{2})[(|+><-|)+(|-><+|)] e^{\frac{-iS_z\phi}{\hbar}}$$

so, why that is equal to $$(\frac{\hbar}{2})[\frac{i\phi}{2}|+><-|\frac{i\phi}{2}+\frac{-i\phi}{2}|-><+|\frac{-i\phi}{2}]$$

??

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