= Weak-field Zeeman splitting
{title2=$\Delta E=\mu_B g_Jm_JB$}
Use the leading nonrelativistic magnetic coupling for an electronic level with total <angular momentum operator> $\mathbf J$, and choose the $z$ axis along a uniform external <magnetic field> $\mathbf B=B\hat{\mathbf z}$, $B\ge0$. Write its <magnetic quantum number> as $m_J$, so $J_z|J,m_J\rangle=\hbar m_J|J,m_J\rangle$. There are $2J+1$ possibilities from $-J$ to $J$. With <Landé g-factor> $g_J$, the first-order shift is
$$
\Delta E=\mu_Bg_Jm_JB,\qquad
\mu_B=\frac{e\hbar}{2m_e},\qquad
\Delta\nu_{ul}=\frac{\mu_BB}{h}(g_um_u-g_lm_l).
$$
Here $e>0$ is the magnitude of the <Electron> <electric charge>, $m_e$ its <mass>, $h$ the <Planck constant>, and $\mu_B$ the <Bohr magneton>. Both $g_J$ and $m_J$ are dimensionless; with $B$ measured in <tesla (unit)> and $\mu_B$ in <joules> per <tesla (unit)>, $\Delta E$ is in <joules>. The <photon> <frequency> shift $\Delta\nu_{ul}$ is in <hertz>, not <angular frequency>; indices $u,l$ denote upper and lower levels. NIST states this weak-field convention and its orbital/spin assumptions explicitly.<Atomic Spectroscopy - A Compendium of Basic Ideas, Notation, Data, and Formulas by William C. Martin and Wolfgang Wiese/12. Zeeman Effect>[Atomic Spectroscopy - A Compendium of Basic Ideas, Notation, Data, and Formulas by William C. Martin and Wolfgang Wiese; Atomic Spectroscopy, §12, Eqs. (6)–(8): magnetic-sublevel shift, tesla convention and magnetic splitting factor]{ref}
The sign follows directly from <magnetic dipole potential energy>. Within this electronic $J$ multiplet, the projected <magnetic dipole moment> is $\boldsymbol\mu=-g_J\mu_B\mathbf J/\hbar$. Therefore $H_Z=-\boldsymbol\mu\cdot\mathbf B$ has the displayed positive coefficient. Subtracting the shifts of the two levels and using <photon> <energy> $h\nu_{ul}=E_u-E_l$ gives the transition formula. For the orbital case, $\mu_B/h=e/(4\pi m_e)$ gives the same leading <frequency> spacing as the classical charged oscillator. This explains the historical sensitivity to charge divided by <mass>; the <quantum state> derivation here is modern.
“Weak” compares interaction <energies>: mixing with neighbouring <fine structure> levels must be small, so $J$ remains a useful <quantum number>. The formula keeps the term linear in $B$ and neglects quadratic shifts. It also omits <hyperfine structure>; if that is resolved and controls the coupling, the appropriate total electronic-plus-nuclear <angular momentum> and splitting factor must replace $J,g_J$. Weak compared with <fine structure> does not automatically mean weak compared with <hyperfine structure>.
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