Post quantum mechanics questions here

Hello, I was wondering if I am doing this correctly (Question 4 from HW2)? Iām not sure if my math works out for the eigenvector, I get a normalization constant of 1, would appreciate any feedback

(Also not sure if this is the right way to post questions?)

This looks like the usual result to me. There is some ambiguity about where the phase factor appears since usually I see this expression with e^{i\alpha} in the second vector entry (instead of e^{-i\alpha} in the first entry). Of course, this is accomplished by multiplying the eigenvector by an overall phase e^{i\alpha}, and in QM, overall phases do not matter. This is what you would need to do to turn your expression into the expression in the book.

Also, you are correct that the eigenvector is essentially already normalized.