An atom offers a network of related tests. A model can predict energies, transition frequencies, spatial distributions and responses to controlled fields. Agreement across that network is more informative than agreement with a single value used to set the model's scale.
Begin with a benchmark
Consider the leading nonrelativistic Coulomb model for hydrogen with a fixed, infinitely heavy nucleus. Its bound-state energies have the form Eₙ = −A/n², where n is a positive integer and A is approximately 13.6 electronvolts. This is a useful reference calculation, not the full theory required for precision hydrogen spectroscopy.
For a transition from an upper level to a lower one, the emitted photon's energy equals the positive difference between the levels. The corresponding frequency is that difference divided by Planck's constant h. Thus one scale A determines many frequencies at this level of approximation.
| Transition | Energy difference | With A = 13.6 eV |
|---|---|---|
| 2 → 1 | 3A/4 | 10.2 eV |
| 3 → 1 | 8A/9 | 12.0889 eV |
| 3 → 2 | 5A/36 | 1.8889 eV |
These values are calculations within the stated idealization. Reporting extra decimal places would not make that idealization a more accurate description of the physical atom.
An energy interval alone does not determine whether a particular single-photon transition is allowed, nor its intensity or lifetime. Those questions require angular states and interaction matrix elements. The table isolates the level-spacing pattern.
A ratio without the scale
Divide the 3 → 1 energy difference by the 2 → 1 difference. The unknown scale cancels, leaving (8/9)/(3/4) = 32/27. A measurement of one transition can calibrate A; a separate measurement of the ratio then tests the pattern of levels.
The relation Δ₃₁ = Δ₃₂ + Δ₂₁ also follows, but for a different reason. It is simply the subtraction identity among three well-defined level energies. Many alternative spectra satisfy it. Tests should therefore be chosen with an understanding of which alternatives they can distinguish.
This distinction is useful across research: a consistency relation can detect an error without uniquely selecting the proposed mechanism. A discriminating relation changes between the competing accounts under consideration.
One fit, two predictions
To make that distinction concrete, introduce an explicitly invented alternative spectrum Eₙ = −A/n² + B/n³. The B term is a teaching deformation, not a proposed substitute for the known corrections in atomic physics. It gives
Both models can fit the same Δ₂₁ by adjusting A. If the target is 10.2 eV and B = 0.08 eV is fixed in advance, the deformed model needs A ≈ 13.693333 eV. It then predicts Δ₃₁ ≈ 12.094815 eV, rather than the undeformed value of approximately 12.088889 eV.
Eliminating A makes the difference especially transparent:
A second transition tests the deformation after the common calibration. If B is also fitted using that transition, a further independent observable is needed to test the enlarged model. Adding a parameter changes the evidential role of the data.
Return to the physical atom
Real comparisons require finite nuclear mass, relativistic and radiative effects, nuclear properties, experimental conditions and uncertainty. A residual from the leading formula can be caused by omitted established physics. Calling every such residual evidence for a new ontology would make an approximation error do work it cannot support.
A useful comparison specifies the baseline theory, the proposed additional term, the independently fixed inputs and the observables reserved for testing. Correlated uncertainties matter when several frequencies share the same calibration or fitted constants.
The Hydrogen & Spectroscopy application supplies a practical route through atomic scales and linked spectral quantities. Its calculations should be read with the assumptions stated in the application, then compared with a precision model appropriate to the experimental question.
A generative research target
For SMM/DTT, the substantial target is to derive an effective atomic dynamics from the proposed inner generation and outer completion, identify its measurable states and calculate several connected consequences. Recovering a representation or a phase structure is one ingredient; selecting the physical interaction and its scale is another.
The published real-commutant phase-selection paper is a technical destination for the structural part of that programme. Its presence does not make the illustrative B term above a derived SMM/DTT prediction. The examples here instead explain what a future atomic prediction would have to face.
Sources and further reading
- CODATA 2022 tables and adjustment for the reference constants and atomic scales.
- Constants and material scales; circular derivations.
- DoT book contents; the hydrogen research section.