A particle with spin $\frac{1}{2}$ at $t=0$ is in a quantum state described by the wave function: $$\Psi=(|+\rangle +(1+\cos\theta) |-\rangle)f(r). $$ Temporal evolution is given by $$H=\frac{\omega} {\hbar} (L_x^2+L_y^2)$$
I have to calculate the expectation value of the operator $O=J_+J_-$
Because of the presence of the cosine I wrote the angular part of the quantum state with the spherical harmonics, I know that $$Y_1^0=\sqrt{\frac{3}{4\pi}}cos\theta$$ ($Y_\ell^m$) So $$cos\theta=\sqrt{\frac{4\pi}{3}}Y_1^0$$ I also know that $$Y_0^0=\sqrt{\frac{1}{4\pi}}$$ At the end I obtained (after a renormalization) $$\Psi=g(r) (|00\rangle|+\rangle+|00\rangle|-\rangle+\frac{1}{\sqrt{3}}|10\rangle|-\rangle)$$ Where $|L^2, L_z\rangle=|00\rangle=Y_0^0$ and $|10\rangle=Y_1^0$
With the temporal evolution obtained: $$\Psi_t=g(r) (|00\rangle|+\rangle+|00\rangle|-\rangle+\frac{1}{\sqrt{3}}e^{-\frac{i\omega t} {\sqrt{3}}}|10\rangle|-\rangle)$$ but now I should apply the composition of angular momenta, in order to calculate $\langle O\rangle$ and here is the problem, I have never applied the composition in a case of a wave function depending by two different values of $\ell$, I thought that I could treat separately the two parts with different $\ell$ but I don't know if it is possible! How can I solve this problem?