Reproducible hydrogen calculations: inputs, units, uncertainty, and independence

1. Record the question that the calculation answers

Begin by naming the atom, states, Hamiltonian, and observable. “Hydrogen frequency” is not sufficient: the calculation may concern a gross principal-level interval, a selected fine-structure component, a hyperfine centroid, or the photon frequency in a two-photon excitation.

Also state the intended claim. A reference calculation reproduces the consequences of known constants in a supplied model. A source prediction uses independently computed theoretical inputs and a derived effective operator. A synthetic example explores a declared hypothetical perturbation. Each can be useful, but their evidential roles differ.

Paper 12 makes this distinction central to the hydrogen programme: constants can be externally calibrated or independently supplied, and the origin of the Coulomb envelope remains visible in the dependency chain. [Paper 12, input and conditional-reconstruction sections]

A compact calculation record should contain the source version, input values and units, approximation order, numerical method, and target observable. The record makes later changes interpretable: a revised constant, a corrected sign, and a newly derived operator are different kinds of progress.

2. Choose an independent set of inputs

For a simple electronic hydrogen calculation, one convenient set is α, m_e, x = m_e/m_p, h, c, and the elementary charge e for conversion between joules and electron volts. In the present SI, h, c, and e have exact defining values; α and the mass quantities carry empirical uncertainty in the reference mode. [NIST 2022 CODATA table]

Do not treat a mass, a mass ratio, and a second mass computed from that ratio as three independent measurements. Likewise, a Rydberg constant calculated from α and m_e is not a new independent validation datum for the same calculation.

3. Reproduce the reduced-mass Coulomb example

First compute ℏ = h/(2π), μ = m_e/(1 + x), the length a_μ = ℏ/(μcα), and the positive binding energy B = μc²α²/2. The reference inputs give

For the gross 1S–2S energy interval, calculate ΔE = 3B/4 and then ν = ΔE/h. This gives about 2.46603842 × 10¹⁵ Hz. Its equivalent vacuum wavelength is c/ν ≈ 121.56845 nm. In equal-frequency two-photon excitation, each photon supplies half this frequency and twice this wavelength.

A dimensionless implementation can calculate E_n/B = −1/n² first and restore physical units only at the end. This separates the spectral shape from the external scale and makes exact checks easier.

These values reproduce the stated gross model. They are not a complete precision prediction of the physical 1S–2S interval, because fine structure, radiative, recoil-beyond-reduced-mass, and nuclear terms have not been added. Reporting the approximation alongside the result prevents numerical precision from being mistaken for physical completeness.

4. Audit the units at every interface

Energy divided by h gives ordinary frequency in cycles per second. Energy divided by ℏ gives angular frequency in radians per second. Their ratio is 2π. A tick rate, phase angle, angular velocity, and photon frequency must therefore be labelled separately before they are related.

Length also needs a physical map. A dimensionless source count does not become metres by itself, and a dimensionless holonomy does not become hertz without an energy or clock-response law. The canonical 137 construction distinguishes a structural selection from the subsequent coupling and physical matching tasks. [Canonical construction, §§14–16]

For numerical interfaces, keep energy internally in one unit and convert explicitly. Use nuclear masses in reduced-mass formulas. Distinguish vacuum wavelengths from wavelengths in air. State whether a frequency is a full atomic interval or one photon’s share in a multiphoton process.

Dimensional analysis also catches less obvious errors. The leading finite-size energy contains μ³c⁴r_p²/ℏ²; with c² it would have units of mass. Such checks are inexpensive and should precede any interpretation of a surprising numerical agreement.

5. Propagate covariance, not only separate error bars

Let θ be an independent parameter vector with covariance Σ, and let f(θ) be a vector of calculated observables. Linear uncertainty propagation gives C_f = JΣJ^T, where J is the Jacobian of f. This includes cross terms from correlated inputs.

For a gross interval with fixed h and c, ν is proportional to μ α². Thus d lnν = d lnμ + 2d lnα. Using independent parameters m_e and x = m_e/m_p,

The coefficients in these expressions determine the corresponding uncertainty terms. Treating every tabulated quantity as independent can either inflate or understate the result, depending on the correlations.

Shared levels generate another covariance. If independent level uncertainties contribute to ν₁₂ = (E₂ − E₁)/h and ν₁₃ = (E₃ − E₁)/h, then their covariance includes Var(E₁)/h². Subtracting the two intervals cancels that common level contribution.

Parameter uncertainty and omitted physics are separate. A very small propagated constants uncertainty does not cover a fine-structure or radiative term absent from the chosen model.

6. Keep source selection independent of the reserved comparison

A noncircular prediction requires more than declaring a calculation “alpha-free.” Its state construction, branches, regulator choices, and normalization must remain fixed when the target spectroscopy information is removed. Paper 11 states this independence requirement as part of its source-construction programme. [Paper 11, input-removal and provenance requirements]

Document which inputs come from source theorems, independent measurements, calibration data, or illustrative choices. If a branch or coefficient was chosen because it improved agreement with a line, that line helped calibrate the model. It cannot simultaneously be counted as an independent prediction.

There is also indirect dependence. The CODATA adjustment uses hydrogen and deuterium spectroscopy together with other information; its Rydberg and nuclear-radius results therefore have a documented relationship to those observations. [CODATA review, §§I.B.3 and III]

A file hash records which inputs were used, but it does not prove that they were selected independently. The substantive evidence is the dependency graph and the prior definition of the construction. Reserved comparisons should be identified only after that graph has been inspected.

7. Specify the reference before calling a difference a residual

A precision reference should identify its Dirac, recoil, self-energy, vacuum-polarization, nuclear, and hyperfine conventions. Form all intervals from one consistent table or model of level energies. State whether the comparison uses a resolved component, a centroid, or an unresolved blend.

If a source calculation derives a term already present in the reference, count it once. If it predicts an additional contribution, document the matching that separates the new operator from the conventional one. The same requirement prevents a nuclear-radius correction from being silently relabelled as a temporal holonomy effect.

Residual uncertainty should reflect the measurement, the reference theory, the input parameters, and relevant shared dependencies. Where the measurement also influenced the input adjustment, their cross covariance matters. Simply adding two nominal error bars in quadrature assumes independence that may not hold.

A short calculation note should consequently distinguish four outputs: the numerical value of the chosen model, its parameter uncertainty, its approximation limit, and any actual comparison residual. Keeping them separate makes both a successful match and a disagreement easier to interpret.

8. Use exact identities and limiting cases as checks

Several checks follow directly from the mathematics and do not require a precision experimental data set.

• Infinite nuclear mass: as m_p → ∞, μ → m_e and the reduced-mass scales approach their fixed-nucleus values.
• Weak coupling: as α → 0 with masses fixed, B → 0 and a_μ → ∞. A calculation should not leave a finite Coulomb binding scale behind.
• Dimensionless levels: E₂/E₁ = 1/4 and E₃/E₁ = 1/9.
• Interval ratio: ν_2→3/ν_1→2 = 5/27.
• Shared states: every signed transition cycle closes when its intervals come from the same energy vector.
• Contact scaling: one first-order contact coefficient gives δE_2S/δE_1S = 1/8 and a zero leading contact matrix element for D states.

A numerical source model needs further convergence checks. Refine its regulator without retuning physical parameters, compare stable observables, and estimate the remaining error. Agreement at a single discretization does not establish a continuum or physical finite-regulator limit.

9. Make the next researcher’s calculation possible

A useful research handoff includes the equations or executable method, the exact input set, the source version, the output units, and a small set of reference results with their expected approximation. It should identify every unresolved coefficient directly rather than burying it in a normalization convention.

For SMM/DTT, the most valuable replacement for an empirical input is a source output accompanied by a derivation, a domain of validity, and an independent physical-unit map. The hydrogen formulas then propagate that output into several linked observables. Paper 21’s synthesis is designed around this direction of dependence. [Paper 21, dependency and calculation protocols]

The interactive pages provide reference and synthetic calculations for learning. The research objective is stronger: a reproducible source construction whose operator, scale, and correction pattern are fixed before their atomic consequences are assessed. That is the standard by which the hydrogen programme can become both clearer to readers and more useful to collaborators.

Continue reading

• Reproduce and vary the atomic scales
• Check gross energy intervals
• Analyse independent spectral response directions
• See completed results and remaining source calculations