= Solution
<Potassium> has a closed 18-electron core and one <valence electron>. In $3d\,{}^2D_{5/2}$ the orbital label specifies <principal quantum number> $n=3$ and single-electron <orbital angular momentum> $l=2$. The term letter D specifies total <orbital angular momentum> $L=2$, the superscript $2S+1=2$ specifies $S=1/2$, and the subscript is total electronic <angular momentum> $J=5/2$. The $5p\,{}^2P_{3/2}$ state instead has $n=5$, $l=L=1$, the same <spin>, and $J=3/2$. The <closed electron shell> that forms the core contributes no net $L$ or $S$ in this alkali-like approximation. The dash identifies a pair of levels rather than by itself specifying the direction of emission.
The leading <electric dipole transition> rules for one active <electron> are
$$
\boxed{\Delta l=\pm1,\quad\Delta S=0,\quad
\Delta J=0,\pm1\ \text{except }J=0\leftrightarrow0,\quad
\Delta m_J=0,\pm1.}
$$
The electronic <parity> must change, since the <electric dipole operator> is odd under <spatial reflection>. The <electric-dipole selection rule> imposes no general restriction on changing $n$ by more than one. The rule for <magnetic sublevels> depends on the radiation's <polarization> relative to the <quantization axis>: the $q=0$ component couples equal $m_J$, whereas $q=\pm1$ supplies the other changes. In a many-electron description using <LS coupling> the total orbital rule is $\Delta L=0,\pm1$, excluding $L=0\leftrightarrow0$; for a single active <electron> the stronger $\Delta l=\pm1$ applies. Thus the displayed D–P pair satisfies the leading rules. Relativistic mixing and higher-multipole transitions can permit weaker lines outside the approximation of <electric dipole transitions>.
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