In my statistical physics book, it is said that, when a particle with total angular momentum $\vec J$ is placed in a weak external magnetic field, it has a dipole magnetic moment equal to $$\hat {\vec \mu}=g \mu_B \hat {\vec J}$$ where $g=1+\frac {j(j+1)+s(s+1)-l(l+1)}{2j(j+1)}$ is called Landé g-factor.
However, I don't understand what happens, when the operator $\hat {\vec \mu}$ is applied to a state for which the quantum numbers $j,l,s$ are not well defined, because there is not a unique g-factor.
For example, for the state $$| \alpha \rangle = a_1|j_1, l_1, s_1\rangle + a_2|j_2, l_2, s_2\rangle$$
My guess is that $$\hat {\vec \mu} | \alpha \rangle=(g_1 \mu_B \hat {\vec J})a_1|n_1, j_1, l_1, s_1, m_{j1} \rangle + (g_2 \mu_B \hat {\vec J})a_2|n_2, j_2, l_2, s_2, m_{j2} \rangle$$ where:
$$g_1= 1 + \frac {j_1(j_1+1)+s_1(s_1+1)-l_1(l_1+1)} {2j_1(j_1+1)} \\ g_2= 1 + \frac {j_2(j_2+1)+s_2(s_2+1)-l_2(l_2+1)} {2j_2(j_2+1)}$$
Is it correct correct?
