Assumptions, results and research status
A gross interval is not a prediction of intensity, lifetime, fine structure, radiative shift, or every allowed multiphoton process.
What this explorer calculates
This explorer evaluates the nonrelativistic Coulomb spectrum with reduced mass. Its initial example is the Balmer 3 → 2 interval. The reference constants are external CODATA values, and all displayed wavelengths are vacuum wavelengths. A principal-number pair determines the gross interval, while angular quantum numbers and a nonzero dipole matrix element must be supplied separately to identify an allowed one-photon channel. [NIST CODATA]
Begin by predicting which way the interval moves when the upper level increases. The upper energy approaches zero, so the interval approaches the ionization energy of the chosen lower state. The arrows and numbers come from the same energy ladder. They are not observed intensities or a precision line profile.
Explore the model
Explore gross energy levels and spectral intervals
Select two principal levels. The tool calculates their energy difference in the reduced-mass Coulomb model and converts it into a frequency and a vacuum wavelength. The initial example is the n = 3 to n = 2 Balmer interval.
Balmer: n = 3 → 2
- Energy interval ΔE
- 1.888651 eV
- Frequency ν = ΔE/h
- 4.566738e+14 Hz
- Vacuum wavelength λ = c/ν
- 656.4696 nm
E₂ = -3.399572 eV; E₃ = -1.510921 eV.
Balmer interval 3 to 2: 1.888651 eV; 656.4696 nm in vacuum.
| Upper → lower n | ΔE / eV | ν / Hz | Vacuum λ / nm |
|---|---|---|---|
| 3 → 2 (selected) | 1.888651 | 4.566738e+14 | 656.4696 |
| 4 → 2 | 2.549679 | 6.165096e+14 | 486.2738 |
| 5 → 2 | 2.85564 | 6.904908e+14 | 434.173 |
| 6 → 2 | 3.021842 | 7.306781e+14 | 410.2935 |
| 7 → 2 | 3.122056 | 7.549097e+14 | 397.1236 |
| 8 → 2 | 3.187099 | 7.706370e+14 | 389.019 |
| 9 → 2 | 3.231692 | 7.814196e+14 | 383.6511 |
| 10 → 2 | 3.263589 | 7.891323e+14 | 379.9014 |
| 11 → 2 | 3.287189 | 7.948388e+14 | 377.1739 |
| 12 → 2 | 3.305139 | 7.991791e+14 | 375.1255 |
Series limit: 3.399572 eV; 364.7053 nm in vacuum (upper n → ∞).
All principal levels and degeneracies
The n² count is the number of orbital states in the spinless Coulomb model. Multiplying by two includes electron spin before spin-dependent interactions are added. These counts do not assert exact degeneracy after fine, hyperfine, or external-field effects.
| Level | Energy / eV | Orbital states | Including electron spin |
|---|---|---|---|
| n = 1 | -13.5982873 | 1 | 2 |
| n = 2 (lower) | -3.39957182 | 4 | 8 |
| n = 3 (upper) | -1.51092081 | 9 | 18 |
| n = 4 | -0.849892954 | 16 | 32 |
| n = 5 | -0.543931491 | 25 | 50 |
| n = 6 | -0.377730202 | 36 | 72 |
| n = 7 | -0.277516067 | 49 | 98 |
| n = 8 | -0.212473239 | 64 | 128 |
| n = 9 | -0.16788009 | 81 | 162 |
| n = 10 | -0.135982873 | 100 | 200 |
| n = 11 | -0.112382539 | 121 | 242 |
| n = 12 | -0.0944325504 | 144 | 288 |
What the interval does and does not specify
An energy difference supplies a photon frequency if a transition with that energy occurs. An electric-dipole transition also needs suitable angular quantum numbers and a nonzero dipole matrix element. For example, the familiar one-photon 2p → 1s line and the two-photon 2s → 1s decay require different transition physics even though their principal levels are the same in this gross model. This tool does not calculate transition probabilities, intensities, or lifetimes.
The displayed wavelengths are in vacuum. Comparisons with visible reference lines must account for any air-wavelength convention and for resolved fine-structure components. See the NIST hydrogen line tables.
Scope: point nucleus, nonrelativistic Coulomb interaction, reduced mass, and isolated atom. Fine structure, radiative corrections, hyperfine splitting, finite nuclear size, external fields, and recoil terms beyond reduced mass are outside this calculation. Reproducing these formulas with measured constants is a benchmark calculation; it does not derive the Coulomb Hamiltonian from SMM/DTT.
Constants: NIST 2022 CODATA. The calculation uses c = 299792458 m/s, h = 6.62607015 × 10−34 J s, e = 1.602176634 × 10−19 C, α = 7.2973525643 × 10−3, me = 9.1093837139 × 10−31 kg, and mp = 1.67262192595 × 10−27 kg. Displayed digits are rounded model outputs; this tool does not propagate measurement uncertainties.
A worked check you can reproduce
For 3 → 2 the dimensionless interval is 5B/36. The 1 → 2 interval is 3B/4, so their ratio is exactly 5/27. This check removes the common binding scale and tests the shape of the implemented spectrum. For equal-frequency two-photon excitation, each photon supplies half the full interval frequency.
The reference 3 → 2 vacuum wavelength is approximately 656.470 nm in this gross model. A quoted optical value in air, an individual fine component, or an unresolved multiplet needs a matching convention before comparison. [NIST lines]
Approximation and interpretation
A gross interval is not a prediction of intensity, lifetime, fine structure, radiative shift, or every allowed multiphoton process.
Use the reset control to restore the reference example. Exported tables should retain their input values and units. The reproducible-calculations guide explains how to distinguish numerical evaluation, uncertainty, model limitations, and independent prediction.
Continue reading
Sources for this guide
Source manuscripts, version notes, and standard references are listed in the reference library.