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Circular motion about a central point or axis
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Rotating a wavefunction to get an exponential of $L^2$ and $L_z$
Note that the question is not about finding an expression for the general rotation but about finding any rotation that can be expressed as an exponential that includes $L^2$ and possibly $L_z$ but no other … Also, if it exists, getting the rotation in terms of spherical coordinates $\theta$ and $\phi$ is great if that is easier. …