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In your calculations you are getting the state $$|\psi \rangle = \frac{1}{\sqrt{2}}\begin{pmatrix} i \\ 1 \end{pmatrix}$$ instead of what you are expecting $$|\phi \rangle = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ -i \end{pmatrix}.$$ Well it turns out that in quantum theory these two states are considered the same! This is because they only differ up to a ...