I am a beginner on quantum information and right now I study it from the lectures by Peter Shor. The lectures can be found on edX and can be viewed on YouTube as well.
The lecture that I'm struggling to understand is one in which Shor's explain about continuous rotation on the Bloch sphere. Near the end of the lecture, the Prof. gives an example that the rotational matrix $e^{-i\theta \sigma_z/2}$ can be used to rotates by 90 degree from the state $\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$ to $\frac{1}{\sqrt{2}}(|0\rangle+i|1\rangle)$. A few seconds later, he substitutes $\theta=\frac{\pi}{4}$ into the formula of rotational matrix. It's here that confused me. Why $\frac{\pi}{4}$? It looks wrong because, when multiplying the matrix with $\theta=\frac{\pi}{4}$ with $\begin{bmatrix}\frac{1}{\sqrt{2}}\\\frac{1}{\sqrt{2}}\end{bmatrix}$, the result isn't $\begin{bmatrix}\frac{1}{\sqrt{2}}\\\frac{i}{\sqrt{2}}\end{bmatrix}$.
