Explore coupling, atomic size, and binding energy

Mohamed Haj Yousef

Assumptions, results and research status

Parameter variation is illustrative. The explorer does not infer a new constant from atomic data or calculate the source-level correction to structural 137.

What this explorer calculates

The Coulomb model relates several observables through the same coefficients. This explorer holds the electron mass, Planck constant, and speed of light fixed while allowing a declared coupling or nuclear mass to change. Its initial setting uses external reference constants. Other settings are hypothetical models that help explain the mathematical dependencies. [NIST CODATA]

Predict the direction of a change before moving a control. A stronger attraction reduces the characteristic radius and increases the magnitude of binding. A heavier nucleus increases the reduced mass slightly. The result is a linked response, not a separate fit of each displayed quantity.

Explore the model

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.

Choose the effective model inputs

Inverse coupling α⁻¹ 137.035999178

Increasing α⁻¹ decreases α. The number field retains the full input precision; the slider moves in steps of 0.001.

Nucleus/electron mass ratio R 1836.1526734

The logarithmic slider covers R = 100 to 10000. Reference hydrogen uses the proton/electron mass ratio.

Use an infinitely heavy nucleus (μ → m_e)

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

Every bar is divided by its own reference hydrogen value. The dashed line marks a ratio of one. The horizontal scale adapts so that all current values remain visible.

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

Current values compared with reference hydrogenScroll sideways to see every column.
Quantity	Current	Reference	Change
Radius aμ / pm	52.94654095	52.94654095	0%
Binding \|E₁\| / eV	13.59828726	13.59828726	0%
Relative RMS speed / c	0.007297352564	0.007297352564	0%
Electron COM RMS speed / c	0.007293380465	0.007293380465	0%
Reduced-mass fraction μ / mₑ	0.9994556794	0.9994556794	0%
2 → 1 vacuum interval / nm	121.5684456	121.5684456	0%

Read the response as a linked calculation

μ/m_e = R/(1 + R); aμ = ħ/(μcα); |E₁| = μc²α²/2;
v_rel,rms/c = α; v_e,COM,rms/c = (μ/m_e)α.

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.

Why there are two RMS speeds

The Coulomb wavefunction describes relative separation. Its momentum gives a relative RMS speed αc in the ground state. In the centre-of-mass frame, the electron’s momentum is the relative momentum, but its velocity is obtained by dividing by m_e, so v_e,COM,rms = (μ/m_e)v_rel,rms. This factor approaches one for an infinitely heavy nucleus.

RMS speed measures momentum variance. It does not imply a classical circular trajectory. In particular, a real stationary 1s wavefunction can have zero probability current and nonzero momentum variance.

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.

A worked check you can reproduce

μ = m_eM/(m_e + M); a_μ = ℏ/(μcα); B = μc²α²/2.

At fixed masses, increasing α by ten percent multiplies the radius by 1/1.1 and the binding energy by 1.1² = 1.21. The ground-state relative RMS speed scales as αc; the electron RMS speed in the centre-of-mass frame includes the factor μ/m_e. These are quantum momentum statements, not a planetary orbit model.

A structural selection such as 137 has a separate theoretical role. Choosing α = 1/137 in an explorer evaluates that hypothetical input; it does not establish the physical matching between a count and an electromagnetic coupling. [Canonical construction]

Approximation and interpretation

Parameter variation is illustrative. The explorer does not infer a new constant from atomic data or calculate the source-level correction to structural 137.

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.