= Solution
An <electric dipole transition> has amplitude proportional to $\langle f|\mathbf d\cdot\boldsymbol\epsilon|i\rangle$. The <electric dipole operator> is odd under <parity>, so the initial and final states must have \b[opposite <parity>]. As a rank-one spherical <tensor> it also requires \b[$\Delta J=0,\pm1$, excluding $J=0\leftrightarrow0$, and $\Delta m_J=0,\pm1$]. These follow from angular-momentum addition and the <Wigner-Eckart theorem>; vanishing <Clebsch-Gordan coefficients> can impose further restrictions. Relative to the quantization axis, $\pi$ <polarization> drives $\Delta m=0$, while $\sigma^\pm$ drive $\Delta m=\pm1$.
For a single active <electron>, <parity> and the orbital triangle rule combine to give $\Delta l=\pm1$. In <LS coupling>, the spin-independent electric dipole operator gives $\Delta S=0$ and $\Delta L=0,\pm1$, excluding $L=0\leftrightarrow0$; total <parity> must still change. Spin-orbit mixing can weakly relax the approximate <spin> rule. For resolved hyperfine states the corresponding triangle rule is $\Delta F=0,\pm1$, excluding $F=0\leftrightarrow0$, with $\Delta m_F=0,\pm1$. These labels require the appropriate coupling regime. There is no universal E1 rule demanding a particular change in <principal quantum number>. An E1-forbidden line may occur through another multipole or a multiphoton process.
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