# How the unitary operator acts on this vector? [closed]

$$exp(-i S_{z} \phi (\hbar)^{-1}) | \alpha \rangle = e^{(-i \phi)/2} | + \rangle \langle + | \alpha \rangle + e^{(i \phi)/2} | - \rangle \langle - | \alpha \rangle.$$

$$Sz$$ is the spin in the $$z$$ component. For 1/2 spin case.

I can not understand how does this operator (of rotations) act on the alpha ket. I thought that we could obtain this equality expanding the exponent in Taylor series, but i was not able to do that, my terms was all messy.

• Is one of the $-\phi$'s meant to be a $+\phi$? Jan 26, 2021 at 17:07
• @jacob1729 ops, yes
– LSS
Jan 26, 2021 at 17:08
• Hint: in the basis $|+\rangle,|-\rangle$ $S_z$ is diagonal and should be easy to exponentiate. Jan 26, 2021 at 17:10
• You also haven't shown any of your "messy" work, it would make it easier to see where there was an issue. Jan 26, 2021 at 17:33