The Expectation Value of $L_x$

What is the expectation value of $L_x$ in the state $\psi(\vec{r})=R(r)(\sqrt{2\over 3}Y_{11}(\theta,\phi)-\sqrt{1\over 3}Y_{10}(\theta,\phi))$?

We need $L_x=(L_++L_-)/2$.

$$\begin{eqnarray*} \langle\psi\vert L_x\vert\psi\rangle &=&{1\over 2}\left<\sq... ...t> \\ &=&{\hbar\over 2}{2\over 3}(-1-1)=-{2\over 3}\hbar \\ \end{eqnarray*}$$