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The nuclear beta decay is fast compared with the electronic orbital timescale. In the sudden approximation, the electronic wavefunction does not have time to change during the increase from $Z_i=1$ to $Z_f=2$. The subsequent probability of a particular daughter energy eigenstate is therefore its squared overlap with the initial state.

For the final $1s$ state, write ${\beta }_i=1/a_0$ and ${\beta }_f=2/a_0$. The sudden hydrogenic charge change gives the amplitudeThusThe final $2p$ states have spherical harmonics $Y_{1m}$, whereas the initial state is proportional to $Y_{00}$. Their angular overlap $\int Y_{1m}^{\ast }{Y}_{00}d\Omega$ vanishes by orthogonality. Accordingly $\boxed{P_{2p}=0}$ in the ideal spherically symmetric sudden-change model. Nuclear recoil or interactions with the emitted particles can produce small corrections. This is an overlap selection rule for a sudden change, not a photon-induced electric dipole transition.

Inverting the definitions of the creation operator and annihilation operator gives $x=\sqrt{\hslash /(2m\omega )}(a+a^{†})$. Their actions and the orthogonality of the oscillator energy eigenstates giveThus the electric dipole transition selection rule iswith the lower alternative absent for $n_1=0$. The field exchanges one oscillator quantum in first-order dipole coupling. For a long weak monochromatic drive, its frequency must also be resonant with the level separation, here $\omega$, to give a substantial transition probability. Higher-order multiphoton processes are beyond the stated single-position-matrix element approximation.

Potassium has a closed 18-electron core and one valence electron. In $3d{}^2{D}_{5/2}$ 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 $5p{}^2{P}_{3/2}$ 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 areThe 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 $q=\pm 1$ supplies the other changes. In a many-electron description using LS coupling the total orbital rule is $\Delta L=0,\pm 1$, excluding $L=0\leftrightarrow 0$; for a single active electron the stronger $\Delta l=\pm 1$ 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.

The limit near 121.8 nm lies about $hc/(121.8\mathrm{nm})\simeq 10.18$ eV above its lower term. Subtracting from the approximately $11.88$ eV ground-state ionization energy puts that lower term near $1.70$ eV: the $3d{}^2{D}_{3/2,5/2}$ pair. The first upper levels are the already identified $4p{}^2{P}_{1/2,3/2}$ pair. Their allowed decays areThe fourth possible pairing, $P_{1/2}\to D_{5/2}$, has $|\Delta J|=2$ and is forbidden as an electric dipole transition. The three lines therefore arise from doublet terms, not a spin-triplet term: “triplet series” counts three spectral lines.

For example, the first two wavelengths together with the $4p_{3/2}$ energy give $E(3d_{3/2})\simeq 1.693$ eV and $E(3d_{5/2})\simeq 1.700$ eV. The longer 854.2 nm wavelength from the same upper level places its lower D level higher. The 866.2 nm transition independently returns the lower D energy within the rounding precision. Increasing the upper principal quantum number gives $np{}^{2}P\to 3d{}^{2}D$ triplets with two nearby lower-term ionization limits near 121.8 nm. The arrows are included in b; their emission direction is P to D, not the reverse. The independent tabulation confirms the D-term position and ordering.^[ref]

A helium atom electronic state with total spin zero. Its antisymmetric two-spin function requires a symmetric spatial wavefunction. Its approximate spectroscopic family is distinct from the triplet family under spin-conserving electric dipole transitions.