Given a spin 1/2 particle in state $|\alpha\rangle=\begin{bmatrix}a \\b\end{bmatrix}$, what is the probability of it being measured in the $S_{y+}$ state. Is this equivalent to, if $S_y$ is measured on this particle, what is the probability the result being $\hbar/2$? I think it's supposed to be like $|\langle S_{y+}|\alpha\rangle|^2$ right? And $|S_{y\pm}\rangle=\frac{1}{\sqrt{2}} \begin{bmatrix}1 \\\pm i\end{bmatrix}$ But then I get $a^2/2 +b^2/2$ for both. That doesn't make sense. What am I doing wrong?
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Figured out what I was doing wrong. I had to write it out explicitly considering, a,b complex.
The answer is $\frac{1}{2}(|a|^2 + |b|^2 + i(a^*b - b^*a))$ Does anyone know a more succinct way of writing that last term? It's definitely real so no worries about that.