Use signed electronic energy changes without literal orbital trajectories

Choose the zero of energy

The supplied approximation E_n = −13.6/n² eV describes a hydrogen bound electron with the ionization continuum at zero. Negative energy means bound relative to a separated electron and proton in this model. A larger principal number has energy closer to zero, not a larger negative magnitude. This reference choice makes ionization from level n require positive energy −E_n.

Keep two signs distinct

The electron change is final minus initial energy. A downward change has negative ΔE and can release energy; an upward change requires energy input. The matching photon energy is the positive magnitude of that gap. Frequency and wavelength are positive, so putting a signed negative electron change directly into hc/λ would be physically misinterpreted.

Calculate a reversible pair of labels

For n = 3, energy is approximately −1.5111 eV; for n = 2 it is −3.4000 eV. The 3→2 electron change is approximately −1.8889 eV, while 2→3 gives +1.8889 eV. Both correspond to the same photon-energy magnitude and approximate wavelength 656.4 nm using the rounded 13.6 eV model. The sign distinguishes emission from absorption.

Test spacing and the continuum

Compare the 4→3 gap with the 2→1 gap. Adjacent high-n levels are closer in energy even though their integer labels differ by one in both cases. At n = 5 the ionization requirement is only 13.6/25 = 0.544 eV under this approximation. Equal initial and final labels give zero level change; the model does not invent a zero-energy spectral line with an infinite wavelength.

Understand what the historical model does not establish

Discrete hydrogen energies explained observations that a simple classical orbit picture could not. The modern description uses quantum states and probability distributions, not a tiny planet tracing a known path. This simplified energy ladder also omits fine structure, transition probabilities and selection rules. A computed gap is an energy compatibility condition, not proof that every transition appears with equal strength.

Transfer and species limits

For an ideal one-electron ion with nuclear charge Z, the simple Coulomb energy scale grows as Z². A Z = 2 version would have four times the corresponding hydrogen gap. That relation is not a valid replacement for solving a many-electron atom with repulsion and shielding. State the one-electron assumption before applying a familiar formula to a different species.

Sources and model boundary

Conceptual references checked 30 October 2026: OpenStax hydrogen energy model, NIST 2022 CODATA constants. All lesson prose, numerical packets and diagrams are original; no textbook figures or exercises are copied. Synthetic parameter choices are not measured atomic spectra. The activity does not request radiation sources, vacuum equipment, particle beams or chemical manipulation. Saved calculations remain public practice, not independent mastery or physical laboratory evidence.