One atom. Four ways
to examine 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.

Series limit: 3.399572 eV; 364.7053 nm in vacuum (upper n → ∞).

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, m_e = 9.1093837139 × 10⁻³¹ kg, and m_p = 1.67262192595 × 10⁻²⁷ kg. Displayed digits are rounded model outputs; this tool does not propagate measurement uncertainties.

Explore atomic probability and radius

Compare volume density, radial probability, and the probability contained inside a sphere. The initial 1s example places the radius at one reduced-mass Bohr length, x = r/aμ = 1.

Probability inside selected radius

32.3324%

Probability outside selected radius

67.6676%

Selected radius in physical units

52.94654 pm

Radial probability at selected x

0.5413411

Angular-average density × aμ³

0.04307856

1s. The radius 1 aμ contains 32.3324% probability.

The 1s volume density is largest at the origin. Its radial probability is largest at x = 1 because spherical shells have volume proportional to r².

The chart ends at x = 8; 0.001632% of the normalized probability remains beyond that radius.

Several useful meanings of atomic radius

Mean radius ⟨r⟩

1.5 aμ

Median radius: 50% inside

1.33703 aμ

Radius containing 90%

2.66116 aμ

Root-mean-square radius √⟨r²⟩

1.73205 aμ

A radial probability is a density per unit x; its value at one point is not itself a probability for a finite shell. Integrate it over an interval to obtain that probability. The cumulative curve performs that integration from the origin. The most probable radius, the mean radius, and a probability-containing radius answer different questions.

The physical length uses aμ = ħ/(μcα), including reduced mass. These are stationary wavefunction distributions in the effective Coulomb model. The chart does not depict a classical electron orbit, an actual sequence of measurement outcomes, or a simulation of the SMM/DTT inner generative levels.

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, m_e = 9.1093837139 × 10⁻³¹ kg, and m_p = 1.67262192595 × 10⁻²⁷ kg. Displayed digits are rounded model outputs; this tool does not propagate measurement uncertainties.

Explore linked atomic scales

Change a declared coupling or nuclear mass and see several atomic quantities respond together. Electron mass, Planck’s constant, and the speed of light remain fixed. The initial state uses the reference constants.

Reference inputs

Current coupling α = 0.0072973525643.

Reduced-mass Bohr length aμ

52.94654 pm

Ground-state binding magnitude |E₁|

13.59829 eV

Relative ground-state RMS speed

2187691 m/s

Electron RMS speed in the centre-of-mass frame

2186500 m/s

Reference inputs. Radius 52.94654 pm; binding energy 13.59829 eV.

Read the response as a linked calculation

At fixed mass ratio, a 1% increase in α produces approximately a 1% decrease in radius, a 2% increase in binding, and a 1% increase in either RMS speed. These are local fractional sensitivities; the full nonlinear formulas determine the displayed finite changes.

The local mass sensitivity is q = 1/(1 + R) = 0.0005443206. A 1% increase in R changes the binding by approximately q%, the radius by −q%, and the electron COM RMS speed by q%. The relative RMS speed does not change when only R changes.

Meaning of the 137 control: the button enters α = 1/137 into an effective hydrogen calculation. It displays the consequences of that chosen input. Identifying a structural count with a physical coupling requires an independent matching derivation and any physical correction; moving this control does not supply that derivation. The reference mode uses measured constants, and all changed settings are marked illustrative.

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, m_e = 9.1093837139 × 10⁻³¹ kg, and m_p = 1.67262192595 × 10⁻²⁷ kg. Displayed digits are rounded model outputs; this tool does not propagate measurement uncertainties.

Explore shared level shifts and differential intervals

Synthetic illustration. Assign one correction to each state, then obtain every interval correction by subtracting the two state corrections. This shows why independent per-line adjustments can fail and why a shared response pattern may still be physically ambiguous.

The contact profile acts on S states (ℓ = 0). P and D states have zero contact weight in this simplified model. A and B are separately adjustable labels for exactly the same profile. C is a common offset. No measured spectral residuals are loaded.

Total contact coefficient A + B

+10 kHz

Signed correction: 2S → 8D

−1.25 kHz

Synthetic model. Total contact coefficient +10 kHz. The 2S to 8D correction is −1.25 kHz. The shared-level cycle closes.

1. Verify the cycle relation

For three levels a, b, and c, a shared energy model gives δν_a→c = δν_a→b + δν_b→c. The cancellation is algebraic, independent of the size of A, B, or C.

-8.75 + (-1.23046875) = -9.98046875 kHz
Cycle residual: 0 kHz (to numerical rounding).

2. Keep the lower S-state correction

For 2S → 8D, the upper D state has zero contact weight. The lower 2S state has weight 1/8, so the interval correction is −(A + B)/8. The contact contribution therefore remains present. Only an interval with equal contact weights cancels this profile.

3. Check whether the mechanism can be identified

Every interval depends on A and B through their sum. Moving 5 kHz from one coefficient to the other leaves all interval predictions unchanged. Even arbitrarily precise measurements of these same intervals could not separate A from B without additional physical information or observables with a different response pattern.

This is an illustration of a possible degeneracy with another contact contribution, such as a leading S-state finite-size term in an appropriate effective model. A real analysis must derive the coefficients, their dependence on mass and nuclear properties, their uncertainties, and the applicable perturbation operators. The equality of the two profiles here is deliberately stipulated to make the identifiability issue visible.

4. Specify the next theoretical calculation

A physical DTT correction needs a defined operator or state-energy map, a fixed normalization, and a coefficient derived from the source construction. A holonomy or structural quantity cannot be converted to kilohertz by selecting an arbitrary scale. Once that calculation is available, the same level-shift machinery can propagate it consistently across many intervals.

This tool contains synthetic coefficients and no measured residual dataset. Background and derivation requirements: differential comparisons, research status, and sources. The common formula is deliberately elementary so that every displayed value can be checked by hand.