In general, $\hat{\boldsymbol{\mu}} \neq (-g_J \mu_\text{B}/\hbar) \hat{\mathbf{J}}$ as operators. Indeed, unless the g-factors of the individual angular momenta are the same, the two operators cannot be proportional. The equality only holds for their expectation values with respect to eigenstates of the total angular momentum.
Vincent Thacker
- 17.4k
- 17
- 46
- 72