Infer a photon from an energy-level difference, not a level label

Begin with evidence

A single-electron energy diagram assigns an energy to each stationary state. A transition emits a photon when the final state has lower energy, and absorbs a photon when it has higher energy. Conservation gives photon energy |Ef−Ei|=hf=hc/λ. The magnitude of one level relative to zero is not generally the energy of a transition between two bound states.

Connect the model

The supplied hydrogen model uses En=−13.6/n² eV for n=1–4. It is intentionally separate from the slightly rounded-constant Bohr calculation. Use one consistent energy packet for a numerical inference rather than mixing its rounded levels with another model’s extra digits. The continuum begins at zero energy, so ionization from n requires at least 13.6/n² eV; excess energy can become electron kinetic energy.

Interpret the ledger

The workbench omits photon frequency and wavelength when initial and final levels are the same. That state change has zero energy difference and is not an emitted zero-frequency spectral line. A permitted energy gap also does not establish that the upper state is populated, how often a transition occurs, or the observed line intensity.

Test the boundary

Emission lines and absorption lines can correspond to the same energy gaps. Reverse the endpoints to preserve photon energy while changing whether energy enters or leaves the atom. Do not replace the diagram with a continuous thermal spectrum, and do not transfer this hydrogen formula unchanged to a many-electron atom.

Worked original case

For n=3→2, ΔE=−1.88889 eV, so the emitted photon has λ≈656.387 nm. Reversing 2→3 requires absorption of the same photon energy. Ionizing from n=3 needs 1.51111 eV, a different process.

Predict, compare and retain evidence

Will reversing a pair of levels change the photon wavelength? Capture 3→2, 2→3 and 2→2. Predict emission, absorption or no energy-changing event before checking. Compare the ionization threshold from each initial state with the bound-to-bound gap.

Changed-case practice

Find the photon energy for 4→2 and the minimum energy to ionize from n=4. Explain why they differ.

Check the reasoning after your attempt

E4=−0.85 eV and E2=−3.4 eV, so the emitted photon has 2.55 eV. Ionization to zero needs 0.85 eV. The endpoints differ.

Use more than one line to compare candidate sources

Different elements have different allowed level structures, producing characteristic sets of emission and absorption frequencies. Observing a matched energy gap can support a candidate identity, but one shared line, unresolved blends, absent excitation or unknown instrument response can leave alternatives. An absorption spectrum tests what a foreground material removes from a broadband source; emission tests what an excited source adds.

Predict and check with fictional libraries: candidate A has allowed lines at 2.0 and 3.0 eV, candidate B at 2.0 and 4.0 eV. A 2.0 eV line alone does not distinguish them. Observed 2.0 and 3.0 eV emission supports A over B under the declared complete-library and response assumptions. A 3.0 eV absorption feature supplies the reverse transition test if the lower state is populated. Absence of the 4.0 eV line alone does not prove B absent without an excitation and sensitivity premise. These are original hypothetical spectra, not empirical element identifications. (Current clause 15.3.A.4.)

Reflect and source-check

Why can an energetically allowed line still be absent from an observed spectrum?

Original prose and supplied numerical evidence; checked 30 October 2026 against the current AP Physics 2 framework, fall 2026 clarifications and OpenStax. Virtual-only: no physical source, exposure or apparatus is requested. Public examples and model captures are not independent mastery or supervised laboratory participation.

Clause-level source/action reconciliation checked 31 October 2026 against the linked current framework and Fall 2026 clarification. Added packets are original virtual comparisons; no new practical or learner-completion claim.