After, in the $S_z$-basis $|S_{z,\pm}\rangle$ denoted by $|\pm\rangle$ and in units $\frac{\hbar}{2}$, finding the spin operator in a general direction $\vec{n} = (\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$:
$$\hat{\vec{S}}\cdot\vec{n} = \sin\theta\cos\phi\begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix} + \sin\theta\sin\phi\begin{bmatrix} 0 & -i\\ i & 0 \end{bmatrix} + \cos\theta\begin{bmatrix} 1 & 0\\ 0 & -1 \end{bmatrix} = \begin{bmatrix} \cos\theta & e^{-i\phi}\sin\theta\\ e^{i\phi} \sin\theta & -\cos\theta \end{bmatrix}$$ I find the eigenvectors $$|S_{\vec{n},+}\rangle = \begin{bmatrix} \cos\theta/2\\ e^{i\phi}\sin\theta/2 \end{bmatrix}\text{ and }|S_{\vec{n},-}\rangle = \begin{bmatrix} \sin\theta/2\\ -e^{i\phi}\cos\theta/2 \end{bmatrix}$$ Now it is normally quite easy to see that these should be orthogonal but for some reason I just don't get $\langle S_{\vec{n},+}|S_{\vec{n},-}\rangle$ to equal zero: $$\langle S_{\vec{n},+}|S_{\vec{n},-}\rangle = \cos\frac\theta2 \sin\frac\theta2\left(1-e^{2i\phi}\right)$$ what is going wrong? thanks in advance.
