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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 since $\hat{\mathbf{J}} = \hat{\mathbf{L}} + \hat{\mathbf{S}}$. The equality holds for their matrix elements with respect to eigenstates of the $z$-component of the total angular momentum. So $$\langle j,m'_j| \hat{\boldsymbol{\mu}}|j,m_j\rangle = -g_J \frac{\mu_\text{B}}{\hbar} \langle j,m'_j|\hat{\mathbf{J}}|j,m_j\rangle$$ but this does not mean that $$\hat{\boldsymbol{\mu}} = -g_J \frac{\mu_\text{B}}{\hbar} \hat{\mathbf{J}}.$$

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 since $\hat{\mathbf{J}} = \hat{\mathbf{L}} + \hat{\mathbf{S}}$. The equality is between their matrix elements with respect to the eigenstates of the $z$-component of the total angular momentum for a given $j$. So $$\langle j,m'_j| \hat{\boldsymbol{\mu}}|j,m_j\rangle = -g_J \frac{\mu_\text{B}}{\hbar} \langle j,m'_j|\hat{\mathbf{J}}|j,m_j\rangle$$ but this does not mean that $$\hat{\boldsymbol{\mu}} = -g_J \frac{\mu_\text{B}}{\hbar} \hat{\mathbf{J}}.$$

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 matrix elements with respect to eigenstates of the total angular momentum. So $$\langle j,m'| \hat{\boldsymbol{\mu}}|j,m\rangle = -g_J \frac{\mu_\text{B}}{\hbar} \langle j,m'|\hat{\mathbf{J}}|j,m\rangle$$ but this does not mean that $$\hat{\boldsymbol{\mu}} = -g_J \frac{\mu_\text{B}}{\hbar} \hat{\mathbf{J}}.$$

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 since $\hat{\mathbf{J}} = \hat{\mathbf{L}} + \hat{\mathbf{S}}$. The equality holds for their matrix elements with respect to eigenstates of the $z$-component of the total angular momentum. So $$\langle j,m'_j| \hat{\boldsymbol{\mu}}|j,m_j\rangle = -g_J \frac{\mu_\text{B}}{\hbar} \langle j,m'_j|\hat{\mathbf{J}}|j,m_j\rangle$$ but this does not mean that $$\hat{\boldsymbol{\mu}} = -g_J \frac{\mu_\text{B}}{\hbar} \hat{\mathbf{J}}.$$

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.

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 matrix elements with respect to eigenstates of the total angular momentum. So $$\langle j,m'| \hat{\boldsymbol{\mu}}|j,m\rangle = -g_J \frac{\mu_\text{B}}{\hbar} \langle j,m'|\hat{\mathbf{J}}|j,m\rangle$$ but this does not mean that $$\hat{\boldsymbol{\mu}} = -g_J \frac{\mu_\text{B}}{\hbar} \hat{\mathbf{J}}.$$