The Rydberg constant from Schrödinger's equation itself

Why does hot hydrogen emit a barcode of sharp fixed colours instead of a smooth rainbow — and does the empirical Rydberg constant that indexes that barcode really fall out of Schrödinger's equation with nothing but the Coulomb potential put in?

Units: m⁻¹ (Rydberg constant R_∞, CODATA 2018)

▶ Run this simulationRead how it works

The finding

The Rydberg constant from Schrödinger's equation itself: a Numerov shooting solver coded with ONLY the Coulomb potential returns R_∞ = 10973731.568 m⁻¹ (rel 2.6e-11) and R_H = 10967758.340 vs the spectroscopic 109677.5834 cm⁻¹ (rel 3.9e-13) — the 1/n² ladder (p = 2.000000001), the l-degeneracy (8e-10) and the four visible Balmer lines all emerge from a normalizability boundary condition, with no Bohr formula, no 1/n² and no quantization postulate coded

Method

Solve the radial Schrödinger equation u'' = [l(l+1)/x² − 2Z/x − 2ε]u in units ℓ₀ = 4πε₀ħ²/(me²), E₀ = ħ²/(mℓ₀²) built from CODATA constants alone — the equation has no free parameter and no closed-form eigenvalue anywhere in the code. Integrate outward by Numerov from an 8-term power-series seed at the origin and bisect the energy on the interior NODE COUNT: quantization enters exactly where it does in the physics, as a boundary condition (normalizability), never as a postulate. Read the emergent ladder the way Rydberg read the lines in 1890: least-squares slope of 1/λ against (1/n₁² − 1/n₂²) across twelve lines of three series (Lyman n₂=2..5, Balmer 3..6, Paschen 4..7), with the electron mass giving R_∞ and the electron–proton reduced mass giving R_H. Gates: grid invariance over 24 jittered discretizations; a Z = 1..4 perturbation sweep (Z only ever enters the coded potential); and two rivals — the same solver machinery on a harmonic well, and the classical Rutherford atom integrated under Larmor radiation loss.

Measurements, controls & cross-checks

Fit intercept over R

2.9100e-11

R H

Recovered

1.0968e+7

Known spectroscopic

1.0968e+7

Rel error

3.8900e-13

Note

spectroscopic R_H = 109677.5834(24) cm⁻¹; grid-invariant to worst-seed 1.3e-10 over 24 jittered meshes (spread 1.7e-10) — physics, not mesh

Ladder exponent

Fitted p

2

Law

|E_n| ∝ n^−p over n = 1..7

Note

the 1/n² crowding toward a finite ionization limit emerges; never coded

L degeneracy

Worst rel spread

8.0000e-10

Note

E(n,l) independent of l for all l < n ≤ 5 — the Coulomb 'accidental' degeneracy, exact for 1/r and recovered at the shooting floor

Balmer lines nm

Recovered

• 656.47
• 486.27
• 434.17
• 410.29

Known vacuum

• 656.46
• 486.27
• 434.17
• 410.29

Worst rel

1.4600e-5

Note

gross structure only; the 1.5e-5 residual IS the neglected fine structure (α² ≈ 5.3e-5 scale)

Ionization eV

Recovered

13.598287

Known

13.598435

Rel error

1.0800e-5

Note

the offset is the deliberately omitted fine-structure + QED shift (~1.1e-5), disclosed, not absorbed into tolerance

Perturbation Z sweep

Fitted exponent

2

Abs error

9.1000e-13

Note

E₁(Z) ∝ Z² over Z = 1..4 with Z entering only the coded potential — the hydrogenic law behind He⁺'s Pickering lines at λ_H/4

Control harmonic

Spacing ratio

1

Coulomb spacing ratio

5.4

Note

the SAME Numerov/node-count machinery on V = x² gives an equally spaced, unbounded ladder — no 1/n² crowding, no ionization limit, no Rydberg series. The series is Coulomb physics, not a solver artifact.

Control classical

Collapse time s

1.5560e-11

Textbook

~1.6e-11 s

Numeric vs analytic rel

1.0000e-9

Note

the classical Rutherford atom under Larmor radiation spirals from a₀ into the nucleus in 16 ps with a continuous spectrum — no stable ground state, no sharp lines. Hydrogen's permanent barcode falsifies classical electrodynamics at the atomic scale.

Gates

16/16 in ~5 s (11 recovery + 5 honest-module cert)

Module certificate

Summary

The derisk EXECUTES the shipped HydrogenSpectrumModule.ts (sha256-pinned, 34 exact strip pairs + 2 counted bulk passes, run via new Function under Babylon/DOM recorder stubs) and certifies a fully STATIC world in the blackbody frozen-proof pattern.

Exec bit exact

init measurement === op-order statics replica BIT-FOR-BIT: a₀ 5.2917721090608536e-11 m, E₁ −13.598287264178731 eV, R fit slope 10967758.340192972 m⁻¹, intercept 3.104e-9, all 12 line records {series,n1,n2,λ,x,colour}, all 4 Balmer values; executed En/lambda/wavelengthColor/lsq match the replica at fresh probes; all 18 thin-instance buffers (6 rungs, ionization limit, 12 arrows, 4-line visible strip) bit-exact — the ladder is drawn at exactly the measured values.

Frozen proof

600 engine render() calls: no fixedUpdate declared, ZERO DOM writes after init, all 18 buffers frozen (same reference, setBuffer once), HUD === exact template replica, chart SVG === replica === fresh _buildChart(); write counts + HUD/chart sha256 pinned.

Earned

Shown a₀ ('5.2918e-11') AND shown R_H ('1.09678e+7') both ROUND ONTO the module's own cited theory labels at display precision while the doubles differ (a₀ Δ 2.11e-17, R Δ 0.34 m⁻¹) — the coincidence resolves only at full precision; the fit is REAL (slope vs closed-form R = −E₁/(hc) rel 3.33e-16, intercept/R 2.83e-16, both at the double-precision floor, tolerances 5e-15 ≈ 15× observed); the fitted ladder predicts a fresh Brackett-α (4→6) line never in the fitted set to 2.22e-16.

Priced vs oracle

The DISCLOSED-CIRCULAR display is PRICED against the emergent Numerov recovery with ceilings derived in-run, no free constants: R module-vs-oracle rel 1.80e-11 ≤ 1.65e-10 (the oracle estimator's own 24-mesh full spread, gate H); Balmer worst rel 2.01e-10 ≤ 3.46e-9 and |E₁| rel 7.38e-12 ≤ 2.40e-10 (the documented h-independent shooting floor 4e-11 abs in ε propagated per line as 3·2·floor/|Δε|). Provenance scan: the Bohr closed form is PRESENT in the generator (that IS the disclosure), RH_THEORY/A0_THEORY appear zero times in the measurement path (display labels only), no R/a₀/λ/E₁ answer digits in the 2331-ch scanned code, planted-violation self-test catches; the stored air-reference values (656.3/486.1/434.0/410.2) are write-only ('.ref' reads: 0).

Correction

The previous finding claimed the on-screen value 'agrees with the emergent recovery to ~4e-13' — measured by execution this run, the module-vs-oracle gap is 1.80e-11 (the ~4e-13 was the ORACLE-vs-spectroscopic figure, misattributed to the display); the display's closed-form R_H sits 1.76e-11 rel from the spectroscopic value, all far inside the 1.65e-10 mesh-spread ceiling.

Tamper

Triad surgical: known_value → only RINF fails (recovery unchanged, exit 1); cert module_sha256 → only CERT-EXEC fails; post-strip 1-ulp tamper of the executed Bohr denominator (8 → 8.000000000000002) → caught by all four bit-exactness gates (CERT-EXEC/BITEXACT/FROZEN/EARNED) while CERT-PRICED stays green — the pricing gate measures physics distance, the bit-exact gates carry the ulp sensitivity, and the recovery gates are untouched.

What it reduces to

Bohr's 1913 energy ladder E_n = −R hc/n² and Rydberg's 1890 empirical constant, derived as Schrödinger derived them in 1926 (Ann. Phys. 79, 361): as the eigenvalues of the Coulomb Hamiltonian. Non-circular in the same sense as the emwave world's c = 1/√(μ₀ε₀): the recovery path contains the potential −Z/x and a normalizability boundary condition, NEVER the eigenvalue formula — the −1/(2n²) coefficient, the l-degeneracy, the n⁻² exponent and the origin intercept are all measured outputs, and R_∞ = mₑe⁴/(8ε₀²h³c) is loaded only to score them (in SI the CODATA constants are mutually adjusted, so the test's content is precisely that the Schrödinger spectrum produces the −1/2 coefficient and the 1/n² structure, which is Bohr's formula). MODULE SYSTEMATIC, DISCLOSED: the on-screen HydrogenSpectrumModule display is CIRCULAR — it synthesizes its twelve lines from Bohr's closed form E_n = −μe⁴/(8ε₀²h²n²) (module line ~83) and fits R back out of its own synthesis, so its 'measured' R_H = 1.09678e7 is guaranteed by construction. This oracle closes that gap non-circularly and the on-screen value AGREES with the emergent recovery to 1.80e-11 (measured by execution; an earlier ~4e-13 claim misattributed the oracle-vs-spectroscopic figure to the display), so the display is honest in value while circular in provenance; the module doc-comment's 'derives, not assumes' claim properly belongs to this derisk, not to the display path. CERTIFIED HONEST-MODULE this run (zero-edit): the derisk now sha-pins and EXECUTES the shipped module, proves the init measurement and all 18 buffers bit-exact vs an op-order replica, proves the display frozen over 600 engine calls, proves the shown strings earned (both round onto their theory labels, resolved only at full precision; the fit is real at the float floor; a fresh Brackett line is predicted), and prices the disclosed circularity against the emergent Numerov recovery with in-run-derived ceilings — see result.module_certificate.

Confidence & reproduction

Confidence

high

Validation

derisk-pass

Re-run the check

npm run derisk -- hydrogen (scripts/hydrogen-derisk.mjs)

Oracle

scripts/oracles/hydrogen.reference.json

Sources

E. Schrödinger, Ann. Phys. 79, 361 (1926). N. Bohr, Phil. Mag. 26, 1 (1913). J. J. Balmer (1885); J. R. Rydberg (1890) — spectroscopic R_H = 109677.5834(24) cm⁻¹. CODATA 2018: R_∞ = 10973731.568160(21) m⁻¹. NIST ASD: Balmer vacuum wavelengths 656.46/486.27/434.17/410.29 nm; H ionization 13.598434599702 eV. Classical collapse t = a₀³/(4r_e²c) ≈ 1.6×10⁻¹¹ s (Griffiths, Introduction to Electrodynamics, prob. 11.14).