Can anyone explain to me why the following operators are rotation operators:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

\begin{align*}R_x(\theta) &= e^{-i\theta X/2}=\cos(\frac{\theta}{2})I-i\sin(\frac{\theta}{2})X=

\left(\!\begin{array}{cc}\cos(\frac{\theta}{2}) & -i\sin(\frac{\theta}{2}) \\ -i\sin(\frac{\theta}{2})& \cos(\frac{\theta}{2}) \end{array}\!\right)\\

R_y(\theta) &= e^{-i\theta Y/2}=\cos(\frac{\theta}{2})I-i\sin(\frac{\theta}{2})Y=\left(\!\begin{array}{cc}\cos(\frac{\theta}{2}) & -\sin(\frac{\theta}{2}) \\ \sin(\frac{\theta}{2})& \:\:\cos(\frac{\theta}{2}) \end{array}\!\right)\\

R_z(\theta) &= e^{-i\theta Z/2}=\cos(\frac{\theta}{2})I-i\sin(\frac{\theta}{2})Z=\left(\!\begin{array}{cc}e^{-i\theta/2} & 0\\ 0 & e^{-i\theta/2} \end{array}\!\right)\end{align*}.[/tex]

I understand that when considering the 2-d case, any complex number z can be rotated anti-clockwise by an angle [itex]\theta[/itex] with the transformation [itex]z\mapsto ze^{i\theta}[/itex]. This has no factor of 1/2 so where does it come from in the rotation operators?

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# Rotation operators on Bloch sphere

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