How do I combine the unitary rotation operators about the $x$, $y$ and $z$ axes to get the unitary rotation operator about a generic axis $u$?

Question

I have the following in my lecture notes

In a past evaluation I was asked to combine the rotation operators about the $x$, $y$ and $z$ axes to get the rotation operator about a generic axis $u$ with $u$ the unit vector in the direction of the axis Therefore I must write (4.30) in terms of $R_z(\theta)= exp(-i\theta L_z)$, $R_x(\theta)= exp(-i\theta L_x)$, $R_y(\theta)= exp(-i\theta L_y)$. I have already tried it myself but I don't come up with anything, actually I arrive at eq (4.31) with an "=" which is wrong. I also googled it but I was unable to find it. Can someone shed some light on how it's done?

Answer 1

From (4.29), you work out $L_z$ near the identity, $R_z(0)={\mathbb I}$, $$ L_z=\lim_{\theta \to 0}~ i~\partial_\theta R_z(\theta), $$ and likewise for the x and y axes rotations. You can check they are all Hermitean.

You then immediately have $$ R_{\mathbf u}(\theta)= \exp \bigl( -i\theta (u_xL_x + u_yL_y+u_zL_z)\bigr )\\ = \exp \Biggl ( \theta \left (u_x \lim_{ a \to 0} \partial_a R_ x( a)+ u_y \lim_{ b \to 0} \partial_b R_ y( b)+ u_z \lim_{ c \to 0} \partial_c R_ z( c) \right) \Bigg) . \tag{4.30} $$