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 \text { where }|\uparrow\rangle \text { and }|\downarrow\rangle \text { are the electron's spin up and down states in the } z \text {-direction, respectively. } \text { (a) Construct }\langle\psi| \text { and calculate }\langle\psi \mid \psi\rangle (b) Calculate the probabilities of finding spin up and spin down in an experiment. (c) What is the expectation value of the electron spin in the z-direction?

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