Electric-dipole transition rates

If an optical wavelength is much larger than the atom, the electric field changes little across its electronic wavefunction. The leading interaction is $H_{\mathrm{int}}=-\mathbf d\cdot\mathbf E$. Terms involving magnetic dipole or electric quadrupole operators appear at higher multipole order and can dominate when the electric-dipole element vanishes. Spontaneous emission requires coupling the atom to quantized radiation; an isolated stationary state under an electronic Hamiltonian alone only accumulates phase and does not decay.

For specified initial and final states, summing over free-space photon directions and polarizations gives a rate proportional to angular frequency cubed and the squared vector dipole matrix element. The frequency factor combines the field normalization and the photon mode density. Near a cavity, surface, or structured electromagnetic environment, the mode density changes and the free-space expression need not hold. The displayed expression therefore assumes weak coupling to a free-space vacuum and the usual exponential-decay regime.

The electric dipole operator is odd under parity, so exact parity eigenstates must have opposite parities for a nonzero E1 element. It is also a rank-one tensor under rotations. Consequently the total-J triangle rule permits changes zero or plus or minus one, excluding zero to zero. Its spherical components change the magnetic projection by zero or plus or minus one. These constraints are necessary conditions, not promises of a large radial integral or a visible experimental signal.

In a one-electron central-field description, the orbital rule becomes $\Delta l=\pm1$ while the principal quantum number is unrestricted by symmetry. In pure LS coupling the operator does not act on spin, so $\Delta S=0$; that rule becomes approximate when spin-orbit mixing combines terms. A weak intercombination line can therefore appear without violating the rigorous total-J and parity conditions. When hyperfine structure is resolved, coupling to F introduces analogous triangle constraints and relative strengths through angular-momentum coefficients.

An upper state can decay through several final channels. Its total population-decay rate is the sum of the partial rates, and its lifetime is the reciprocal of that sum when other losses are absent. A branching fraction is the probability of using a specified radiative channel, obtained by dividing its partial rate by the total. A table entry for one spectral line is not automatically the inverse lifetime. Nonradiative quenching introduces additional rates and changes the observed radiative yield.

For an optically thin source, a line’s emitted power is proportional to upper-state population times its partial rate times photon energy. Absorption and reemission can invalidate this simple relation in an opaque sample. A transition called forbidden means forbidden in a specified multipole or approximation, rather than impossible under all interactions. Magnetic-dipole, electric-quadrupole, two-photon, or mixing-assisted channels can provide long but finite lifetimes. The two-state lab displays coherent pure-state probabilities and phase, so it cannot itself generate stochastic radiative decay or emitted photons.