Scope of the light-matter and spectral-transitions examples page

What the three items establish. A two-level dipole-drive model: identification, resonant first-order coefficient and its comparison with the exact Rabi probability identifies declared two-level Pauli matter data with the linear two-level drive model of the driven-dynamics page, computes the resonant co-rotating first-order amplitude and shows that its co-rotating component coefficient is the leading term of the exact averaged Rabi probability. The full first-order coefficient includes both components and their interference, with a different small-time leading coefficient. Explicit dipole angular factors of the hydrogenic model: the allowed 1s-to-2p entry, the forbidden 1s-to-2s entry and the oscillator contrast computes explicit angular factors of the hydrogenic dipole matrix elements — values $\pm 1\mathrm{/}\sqrt{3}$, all with modulus $1\mathrm{/}\sqrt{3}$, for the allowed $1s$-to-$2p$ and $2p$-to-$1s$ angular entries, and zero for the forbidden $1s$-to-$1s$, $1s$-to-$2s$ and $m$-violating entries — together with the oscillator contrast $\mathrm{\mid }\mathrm{\Delta }n\mathrm{\mid }=1$. Two declared cutoff-dipole models with equal transition-energy density: the rate coefficient does not discriminate them, the formal photon-energy laws differ, with a detected-law reading at nonzero coupling proves that two declared cutoff-dipole models whose mode densities agree at the transition energy and are not proportional almost everywhere have the same rate coefficient but different formal single-vertex photon-energy laws at every positive finite time (normalized one-photon preparations require nonzero coupling), and that a declared sharp-line model is the idealisation approached in the limit rather than a property of the finite-time law.

What they rest on. The two-level example adopts the prescribed dipole-drive data and uses Stimulated dipole-drive transitions of the finite-level model: first-order amplitudes, equal emission and absorption coefficients, and the exact two-level identification together with the driven-dynamics suppliers cited in that example; its state, probability and laboratory-identification readings use P1, P2 and P7. The hydrogen angular-factor example uses the declared Coulomb dipole components, spherical harmonics, angular integration, the bound-state domain and oscillator suppliers, and Electric-dipole selection rules of the declared hydrogenic finite-level model: necessary angular conditions and the parity (Laporte) consequence. Its model reading uses P1 and the declared units, but its angular calculation uses neither the single-vertex photon law nor a detector or a P2/P7 inference. The spectral comparison uses the declared cutoff model The declared finite-level matter model and its cutoff dipole coupling to the transverse radiation sector, its spectral definition Spectral-transition readings of the coupled model: dipole coefficients, the single-vertex photon density and ideal energy-window detection, the kernel mathematics The first-order emitted-photon energy law: factorised kernel, normalisation, concentration at the transition energy and positivity on positive-measure windows, and The golden-rule limit of the single-vertex emission coefficient and its reading as a model rate coefficient, with the stated P1, P2 and P7 readings; its ideal detected-law reading requires nonzero coupling. The two-level example's parameters ${\omega }_0,d,E_0,\omega ,L,k$ and the comparison's densities $J_1,J_2$ are declared model data, not measured values.

What is not claimed. The page records no experiment, no apparatus, no calibration, no count, no frequency, no efficiency and no uncertainty budget, and none is invented. It does not claim that any laboratory atom, field, spectrometer or detector realizes the declared models; those are tested identifications with uncertainty under P7 (Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions). It asserts no measured rate, lifetime, linewidth or exponential decay from any of its comparisons, and no statistical inference. Comparing the declared model laws with data would test the combined model and apparatus assumptions under a declared sampling and measurement model, would carry finite-sample fluctuations, selection effects and systematic uncertainties, and would not prove a model; spectra, counts and lifetimes remain research candidates for a separate reported-experiment item, and energy differences or successful comparisons alone do not establish that the framework is a proof of nature.