an electron is prepared in the spin state |\psi\rangle=\frac{4}{5}|\uparrow\rangle+\frac{3}{5} \mathrm{e}^{2 \pi i / 3}|\downarrow\rangle where |T) and [J) are the electron's spin up and down states in the z-direction, respectively.

\text { (a) Calculate the expectation values for the spin operators } S_{x}, S_{y} \text {, and } S_{z} \text {. } \text { (b) Calculate the variances }\left(\Delta S_{x}\right)^{2},\left(\Delta S_{y}\right)^{2}, \text { and }\left(\Delta S_{z}\right)^{2} (c) Verify that the se variances obey the uncertainty relation \Delta A \Delta B \geq \frac{1}{2}|\langle[A, B]\rangle| \text { where }[A, B] \equiv A B-B A \text { is the commutator. }