= Solution
Interpret the source's $\mathbf J$ as the dimensionless angular-momentum operator, with $J_z|J,m_J\rangle=m_J|J,m_J\rangle$. If <angular momentum> is written with physical units, the magnetic-moment formula instead contains $\mathbf J_{\rm phys}/\hbar$; this is a convention, not an extra factor in the requested <energy>.
The field fixes the quantization axis, so
$$
\mathcal H=-g\mu_BBJ_z,\qquad
\boxed{E_{m_J}=-g\mu_BBm_J,\quad m_J=-J,-J+1,\ldots,J.}
$$
There are $2J+1$ Zeeman levels equally spaced in <energy> by $g\mu_BB$. For positive $gB$, the state with largest positive moment has lowest <energy>. The given sign defines $\mathbf J$ through the specified moment relation; inserting an additional electron-charge minus sign into it would change the source convention.
Back to article page
